An operational approach to spacetime symmetries: Lorentz transformations from quantum communication

In most approaches to fundamental physics, spacetime symmetries are postulated a priori and then explicitly implemented in the theory. This includes Lorentz covariance in quantum field theory and diffeomorphism invariance in quantum gravity, which are seen as fundamental principles to which the final theory has to be adjusted. In this paper, we suggest, within a much simpler setting, that this kind of reasoning can actually be reversed, by taking an operational approach inspired by quantum information theory. We consider observers in distinct laboratories, with local physics described by the laws of abstract quantum theory, and without presupposing a particular spacetime structure. We ask what information-theoretic effort the observers have to spend to synchronize their descriptions of local physics. If there are"enough"observables that can be measured universally on several different quantum systems, we show that the observers' descriptions are related by an element of the orthochronous Lorentz group O^+(3,1), together with a global scaling factor. Not only does this operational approach predict the Lorentz transformations, but it also accurately describes the behavior of relativistic Stern-Gerlach devices in the WKB approximation, and it correctly predicts that quantum systems carry Lorentz group representations of different spin. This result thus hints at a novel information-theoretic perspective on spacetime.


Introduction
Spacetime symmetries are powerful principles guiding the construction of physical theories. Their predictive power comes from the constraints that arise from demanding that the fundamental physical laws are invariant under those symmetries. A paradigmatic example is given by quantum field theory [9], where Lorentz covariance severely constrains the physically possible quantum fields and successfully predicts the different types of particles that we find in nature. Similarly, diffeomorphism invariance guides our attempts to construct a quantum theory of gravity [10,11,12,13].
However, diffeomorphism invariance in particular reminds us of an important insight that played a fundamental role in the construction of general relativity: that spacetime has a fundamentally operational basis. The description of spacetime in terms of a metric field can be derived from the equivalence principle, which is obtained as a statement about what an observer in a free-falling elevator can or cannot get to know by performing measurements. Spacetime symmetries, like the Lorentz transformations in special relativity, can be interpreted as "dictionaries" that translate between distinct descriptions of the same physics given by different observers, and many of their properties can be understood by analyzing operational tasks like clock synchronization.
All these operational protocols must be performed within the laws and limits of quantum physics. Thus, it is natural to take an "inside point of view", and to ask the fundamental question how different descriptions of the same physics by different observers are related, if we only assume the validity of quantum theory. One might expect -or at least hope -that the spacetime symmetry group does not have to be postulated externally, but instead emerges from the structure of physical objects themselves, that is, from the structure of quantum theory.
This general idea is not new, but has been pursued in several ways during the last decades. In his "ur theory", von Weizsäcker [27] argued that the spacetime symmetries are inherited from the SU(2) symmetry of the quantum two-level system. Wootters [34,35] pointed out the relation between distinguishability of quantum states and spatial geometry, and several authors [36,37,25,26] have analyzed aspects of the relation between properties of space and the structure of quantum theory. This research has substantial overlap with questions in quantum information theory regarding the use of quantum states as resources for reference frame agreement [38,1].
In this paper, we approach this old question from a new angle by using the ideas and rigorous vocabulary of (quantum) information theory. We consider an information-theoretic scenario involving two agents, Alice and Bob, who cooperate to solve a simple communication task. The task is that Alice sends a request to Bob in terms of classical information, and Bob is supposed to answer by sending the concrete physical object that was described by Alice's message. Then we ask for the smallest possible group of transformations that Alice has to apply to her request to make sure that the task succeeds. We argue that this gives an operational definition of the notion of a reference frame. Furthermore, this derives the reference frame transformations from the informational relation between Alice and Bob, established by communication with quantum systems. It does not rely on any externally given spatial or spacetime symmetries.
After proving some general properties of this scenario in Section 2, we consider in Section 3 the special case that the objects to transmit are stand-alone finite-dimensional quantum states. Under the hypothesis that many different sorts of quantum systems can be measured in the same "universal" devices, we show that the resulting transformation group, relating Alice's and Bob's reference frames, will be the orthogonal group O(3). This confirms von Weizsäcker's intuition [27], but puts it on firm operational grounds, by specifying detailed physical background assumptions that are sufficient (and probably necessary) to arrive at this conclusion, and by pointing out how these assumptions are realized in actual physics. Furthermore, we also derive the fact that quantum systems carry projective representations of SO(3) of different spin.
In Section 4, we argue that the previous derivation contained one crucial oversimplification: namely that Alice and Bob agree beforehand how the finite-dimensional quantum systems are embedded in the (possibly infinite-dimensional) total Hilbert space or operator algebra. Dropping this assumption leads us to consider different types and behaviors of quantum systems. Analyzing the communication task in this more general setting, it turns out that Alice's and Bob's descriptions will be related by a Lorentz transformation, an element of O + (3, 1). These transformations act as isometries between different finite-dimensional Hilbert spaces.
Thus, we arrive at the Lorentz group, without having assumed any aspects of special relativity beforehand. Furthermore, the resulting formalism turns out to correctly describe the behavior of relativistic Stern-Gerlach measurement devices, under a WKB approximation where the spin degree of freedom does not mix with the momentum of the particle. This provides further evidence for a spacetime interpretation of this abstractly derived Lorentz group.
A B physical answer request (description) T Figure 1: The communication task. Alice requests a physical object by sending a classical description, and Bob answers by sending the corresponding physical system in the desired state. If Alice and Bob use different maps ϕ A and ϕ B to describe physical objects, a correcting transformation T must be applied to Alice's message before it arrives at Bob's laboratory. This T is seen as the "information gap" between Alice and Bob; agreeing on T corresponds to agreeing on a common frame of reference with regards to the desired physical object.
step of the game, Alice sends Bob a "classical request", telling him to prepare a certain physical system. In the second step, Bob answers this request by sending an actual instance of the requested physical system with the desired property.
Clearly, if Alice and Bob have never encountered before, and thus have never agreed on a common reference frame, Bob will not know what map ϕ A Alice is using to describe the desired physical quantity. The best he can do is to guess ϕ A -or, equivalently, hope for the unlikely fact that ϕ A = ϕ B -and send a physical system x such that ϕ B (x) equals the description that Alice sent. If Alice checks whether she received the desired physical object, she will in general detect failure, except for the unlikely case that ϕ A = ϕ B .
To correct for this problem, Alice can apply a correcting transformation T before the classical information is actually sent to Bob. It is easy to see that the protocol succeeds if Alice applies T = ϕ B • ϕ −1 A to the classical description she sends out -the problem is, of course, that Alice may not know ϕ B and thus does not know T . However, if she knows ϕ B , she can make the protocol succeed even if Bob does not know ϕ A . Regardless whether Alice knows ϕ B or not, we may regard T as the relation between Alice's and Bob's local descriptions. Not knowing T may be regarded as an "information gap" between them; closing the information gap, i.e. negotiating on a map T , corresponds to setting up a common frame of reference (regarding the physical property that is intended to be sent).
Sometimes, instead of preprocessing her classical description via the map T = ϕ B •ϕ −1 A , Alice may equivalently postprocess the physical system that she obtains by applying the physical transformation T phys : S phys → S phys , given by T phys := ϕ −1 A • ϕ B , cf. Figure 2. If this is possible, we call T implementable. Note that T and T phys are different kinds of objects: T is a transformation on classical descriptions, and thus something that can always be accomplished by doing computations on representations on a piece of paper (or in a computer memory). The map T phys , in contrast, is an actual physical transformation; in many cases, physics may disallow the actual implementation of T phys . We will soon discuss an example below.
If there were no further restrictions on the maps ϕ A , ϕ B , then the coordinate transformation functions T = ϕ B • ϕ −1 A could exhaust all possible bijective maps from S to itself. However, in many cases, it is physically impossible to actually use every map ϕ as a description map, in the sense that ϕ(x) may be physically impossible to determine, given the physical object x. This is the case when there are physical restrictions on the possible measurement devices that agents may access or implement. As a simple example, if we have again Alice in three-dimensional classical mechanics, then only continuous maps ϕ A will be physically relevant, simply because of the possible measurement procedures that Alice may implement: she can build devices that A B physical answer request (description) Figure 2: Sometimes, instead of preprocessing the classical request via T , Alice may equivalently postprocess the physical answer via T phys . In this case, we call T implementable. measure velocity to arbitrary accuracy, but there will always be unavoidable inaccuracies. Alice may achieve many digits of accuracy, but never perfect accuracy. But then, the coordinate transformation maps T = ϕ A • ϕ −1 B (and T −1 = ϕ B • ϕ −1 A ) must be continuous, too. In general, we obtain a group G max of physically meaningful transformations, G max = {T : S → S | T transforms between physically accessible/allowed descriptions} .
In the example of classical physics and billard ball velocity, this would be the group of homeomorphisms of R 3 into itself. In general, one would expect that G max is the subgroup of all bijections that preserve some key physical structure (which, in this example, is the topology). Now recall the scenario from Figure 1. Suppose hat Alice and Bob are both cooperative and would like to agree on a transformation T in order to succeed with the communication task, and do so with as little effort as possible. For this, they are willing to adjust their own local descriptions as long as it helps to agree. Then one way to reduce the effort is to stop using arbitrary possible encodings ϕ A and ϕ B , and to draw instead from a physically distinguished subset of encodings.
Definition 2.1. A subgroup G ⊂ G max is called achievable if there is a subset of "distinguished encodings" Φ such that every observer A can locally ensure by physical means that she is using a distinguished encoding ϕ A ∈ Φ, and An achievable subgroup G is called minimal if there is no proper subgroup G G which is achievable, too.
Why is the set G always a group? This follows from a natural assumption on any subset Φ of physically distinguished encodings; namely that Intuitively this is plausible: if ϕ 1 , ϕ 2 , ϕ 3 are "equally natural" encodings of physical objects, then an agent can obtain another natural encoding by first using encoding ϕ 1 , then determining the physical object that would have the same description via ϕ 2 , and finally encode this object via ϕ 3 , and the resulting encoding should not be outside of the set of natural encodings. More formally, we can argue as follows. In our communication task, Alice and Bob will try to use a common set of encodings Φ that is as small as possible. If (2.1) was violated, then they could use a smaller setΦ of encodings, which would facilitate their task, rendering the choice of Φ inefficient. To see this, suppose that there are ϕ 1 , ϕ 2 , 2 , which is a fixed map from S to S that Alice can describe to Bob in terms of classical information, and setΦ := {ϕ ∈ Φ | f • ϕ ∈ Φ}. Since ϕ 1 ∈Φ but ϕ 2 ∈Φ, the setΦ is a non-empty strict subset of Φ which Alice and Bob could use instead to encode physical objects. Thus, it makes sense to assume (2.1), from which it easily follows that G is closed with respect to composition and inversion, that is, G is a group. As a simple example, recall the example of billard ball velocity in classical mechanics. Instead of using an arbitrary continuous encoding map ϕ A , Alice may ensure to use an inertial frame to describe the velocity. Classical mechanics teaches us how to achieve this: an inertial frame is one in which Newton's laws are valid; this is something that Alice can check by physical means.
Different inertial frames are related by a Galilean transformation. Since we are not interested in the spatial location of the billard ball (it will be part of the protocol to make sure it is located in Alice's lab after being sent by Bob), only the action of this group on the ball's momentum is relevant. Restricting the Galilean group to the ball's velocity, we obtain the Euclidean group E(3), describing combinations of rotations and translations in momentum space which relate Alice's and Bob's laboratory. Moreover, this group is also minimal : there is clearly no way in which a proper subgroup of E(3) would be sufficient to relate the velocity descriptions of observers who have never met. This follows from the fact every element of E(3) defines a distinct inertial frame for velocity, and all inertial frames are physically equivalent.
What would be examples of non-minimal achievable groups in this case? The simplest example is G max , which in this situation is the group of all homeomorphisms of R 3 itself. We obtain it by allowing all continuous encodings ϕ; every observer can make sure to use a continuous encoding by simply being forced to do this by physics, as discussed before.
A less trivial example would be given by allowing all affine-linear invertible encodings ϕ; that is, encodings that preserve the vector space structure, but not necessarily the distance between points in momentum space. The corresponding group G would be Aff(3, R), the affine group. Having Alice and Bob agree on an element of this group to establish a common frame of reference is more efficient than referring to the full group G max ; it would describe a situation where Alice and Bob are not using arbitrary encodings, but those that they can find by probing the affine vector space structure of momentum space (and building their measurement devices accordingly). If Alice and Bob are for some reason not capable of measuring angles or lengths (notions that are only computable from an inner product), this would be the best they can do.
However, we know that in principle they can do better, which is reflected by the fact that G is achievable, but not minimal. It contains a proper subgroup which is also achievable -which is the Euclidean group. The Euclidean group corresponds to the best possible strategy to agree on a frame of reference, in the sense that different observers can individually commit to using inertial frame encodings, and by doing so, minimize the information gap between them.
Is there always a unique minimal group? Taken literally, the answer is "no", as can be seen in our standard example. Let Φ be the set of all encodings of velocities into vectors in R 3 which correspond to choices of inertial frames, such that the corresponding group G = , which is a bijective map from R 3 to itself. Then Φ := {f • ϕ | ϕ ∈ Φ} is also a physically distinguished encoding. It is an encoding where an observer chooses an arbitrary inertial frame, determines the velocity in the corresponding coordinates, and then takes the third power of all entries. The corresponding group G is then achievable and even minimal according to Definition 2.1, and it is different from the Euclidean group.
However, G is isomorphic to the Euclidean group; in fact, we have Taking any element f ∈ G and mapping it to g = f • g • f −1 is a bijective group homomorphism. That is, G is "exactly the same" as G except for a relabeling. As an abstract group, the Euclidean group is the unique minimal group for the scenario we described. Interestingly, this turns out to be true in all scenarios: Lemma 2.2. For any given scenario, all achievable minimal groups are isomorphic. That is, for any given scenario, the group G min which describes the minimal effort that two observers have to spend to agree on a common frame of reference is unique as an abstract group.
Proof. Fix any scenario. Suppose that G and H are both achievable and minimal according to Definition 2.1. Denote by Φ G and Φ H the corresponding sets of physically distinguished encodings. Choose ϕ G ∈ Φ G and ϕ H ∈ Φ H arbitrarily.
Then Φ H is a set of physically distinguished encodings. The corresponding group is Thus, the group H is isomorphic to the group H. Furthermore, H is achievable and minimal, because H is.
Furthermore, it is a set of physically distinguished encodings (observers can physically ensure that an encoding is in Φ G and also that it is in Φ H , hence they can ensure that it is in both). Therefore, the corresponding group is achievable, and we have K ⊂ G as well as K ⊂ H . Since G and H are both minimal, we must have K = G and K = H , hence G = H , and so G and H are isomorphic.
Proper Euclidean transformations (in the connected component of the identity) are implementable: instead of postprocessing her classical description, Alice can simply rotate and accelerate the physical objects she obtains from Bob. The question whether a spatial reflection is implementable or not depends on the detailed assumptions on the physics. For example, we can think of a machine that measures the velocity v of an object, and then accelerates the object exactly to velocity −v. If the physical background assumptions allow this machine, then reflections will be implementable, too. In the following, we will consider transformations on quantum states, where implementability is more severely restricted by the probabilistic structure.
In general, there may be scenarios in which no minimal achievable group exists at all; however, all examples of this that we can think of at present are rather unphysical, for example in the sense that they refer to groups which are not topologically closed. In all remaining examples of this paper, a minimal group G min exists.
We finish this section with an example meant to elucidate in more detail the rules of the communication task that we have in mind. Suppose that the physical objects to be transmitted are pairs of vectors in classical mechanics; say, the velocities of two billiard balls. Since the Euclidean group E(3) is a (minimal) achievable group for one billiard ball, the group G = E(3) × E(3) is achievable -two copies of the group, one for each billiard ball. However, Alice and Bob can just agree to use the same encoding (resp. inertial frame) for the description of both billiard balls, and thus achieve G min = E(3) also for pairs of billiard balls. This is possible whenever it is physically clear that if Bob sees two billiard balls next to each other and acknowledges that they have the same velocity, then Alice will agree with this fact even after the balls have been transported to her laboratory.
This example illustrates that objective relations between given physical objects represent useful physical structure that can be used to simplify the communication task. This will become important in Subsections 3.2 and 4.4.

Transmitting finite-dimensional quantum states
We now consider the special case that the objects Alice and Bob are intending to describe and to send are abstract finite-dimensional quantum states 1 . In the communication scenario in Figure 1, Alice sends a request to Bob, asking him to prepare a physical system in a specific N -level quantum state; Bob in turn sends a system in this state to Alice. We are only interested in the quantum states themselves, not in the question whether these quantum systems are correlated with any other system. In particular, if Bob is supposed to prepare and send a mixed quantum state, it does not matter whether this is a proper or improper mixture, and whether there is any other system that serves as a purifying system of the mixed state. All that matters for the communication task to succeed is the pure or mixed state that will in the end arrive in Alice's laboratory, which should match Alice's request.
The "information gap" in this scenario is that Alice and Bob will in general not have agreed beforehand on a common orthonormal basis in the Hilbert space that is supposed to carry the quantum state. Since different bases are related by unitaries, the following result is not surprising: Lemma 3.1. The projective unitary antiunitary [14] group PUA(N ) is achievable in the setup described above -that is, the group of conjugations ρ → U ρU −1 with U either unitary or antiunitary. The subgroup of implementable transformations is the projective unitary group PU(N ).
Note that PUA(N ) can also be characterized as the set of transformations of the form ρ → U ρU † and ρ → U ρ T U † , with U unitary. The transposition can be achieved by conjugation with an antiunitary map.
The proof of this lemma is simple. First, we may assume that Alice and Bob have both agreed to represent quantum states by N × N unit trace Hermitian matrices (that is, density matrices), which thus constitute the set S of mathematical objects used as descriptions of the physical systems (cf. the notation in Section 2). Moreover, it is a physically distinguished choice to encode quantum states such that statistical mixtures of quantum states are mapped to convex mixtures of the corresponding density matrices, as it is standard in quantum mechanics. The probabilistic interpretation of mixtures implies that different observers cannot disagree on the question whether a given state is a statistical mixture of other given states.
Thus, every group element T = ϕ A • ϕ −1 B must be a convex-linear symmetry of the set of N × N density matrices. According to Wigner's Theorem [16], this is equivalent to T being either conjugation by a unitary or antiunitary matrix.
Recalling the definition of implementability as explained in Figure 2, we also see that only the subgroup of unitary conjugations is (physically) implementable; antiunitary conjugations are not. They correspond to maps which are not completely positive.
The group PUA(N ) is achievable, but is it minimal? The answer to this question clearly depends on the detailed physical background assumptions, in particular on the question whether the physical carrier system (which Alice asks Bob to use) carries any "natural" choice of Hilbert space basis. For example, if N = 2, a carrier system could be given by a ground state and an excited state (say, of an atom) that span a two-dimensional Hilbert space. In this case, Alice and Bob would under many circumstances agree on this orthonormal Hilbert space basis, and then G min = {1} or G min = SO(2), depending on whether Alice and Bob would also agree on the relative phase between the states. Thus, the question of G min for transmitting quantum states only becomes interesting when we consider suitable additional physical background assumptions. A crucial property of quantum systems in actual physics is that different systems can interact. In the following subsection, we will explore in detail the implications of this simple fact for the communication scenario of two observers in local quantum laboratories.

How to relate complete local laboratories: interaction graph
From a physical point of view, a natural question is whether we can describe the relation between the two laboratories of Alice and Bob, including all local quantum physics, by a single group. If we consider Alice's local laboratory as a collection of (many) finite-dimensional quantum systems S, S , S , . . ., then Lemma 3.1 only tells us that we can achieve PUA(N ) for every system S with dim S = N , and the product over all these groups, with one factor per system. This is a huge group that seems highly inefficient for the communication task. Can we somehow use a more efficient subgroup? Maybe by exploiting the fact that some systems S interact with other systems S in natural ways?
A crucial piece of structure turns out to be something that we may call "universal measurement devices" -devices that measure a given observableM for two systems S and S (of possibly different Hilbert space dimensionalities) universally. In other words, we have a measurement device which accepts as inputs both systems S and S , and will in the former case measure an observableM (S), and in the latter another observableM (S ). An example is given by an (idealized) Stern-Gerlach device which can measure, say, the spin in z-direction, S z , both for systems S of spins 1/2 and systems S of spin 1 with the same given magnetic field gradient. The operatorsŜ z (S) andŜ z (S ) represent the same physical quantity, but correspond to observables on different Hilbert spaces.
Why should universal measurement devices exist at all, and what does it mean in general thatM (S) andM (S ) correspond to "the same" observableM on the different Hilbert spaces S and S ? In the following, we will not answer these questions in any detail, but simply assume that such observables and devices exist. However, one motivating idea is thatM represents something that may be called a "conserved quantity"; that is, a physical quantity whose total value is typically preserved in closed systems, even in interactions between different kinds of quantum systems, which in the end allows to compare the value ofM on S and S . Another motivation is that quantum systems of some kind can be used to build devices in which they interact in "canonical ways" with systems of another kind. For example, we can use the spin of many electrons (which are qubits) to build a magnet which in turn can be used in a Stern-Gerlach device, defining a quantization axis for particles of higher spin.
Let us now argue more formally. Suppose thatM is a physical quantity such that for two given quantum systems S and S , there is a natural way of defining "that very same quantity" on both Hilbert spaces. That is, we have two observablesM (S) andM (S ), and we think of a measurement device that accepts both kinds of systems S and S , measuring the corresponding observable. When we writeM (S) resp.M (S ), we do not mean a concrete matrix representation, but rather a physical observable, being measured in a device on physical quantum states which are elements of S phys (S) or S phys (S ), the physical states on S resp. S (reusing some notation from Section 2). Let us furthermore assume thatM (S) uniquely definesM (S ) and vice versa 2 , and that this interdependence is continuous in both directions. ThenM will be called (S, S )co-measurable. The following observation will be crucial.
Suppose that a set of observables M := {M i } i∈I is (S, S )-co-measurable on two quantum systems S and S , and suppose this set is large enough to be tomographically complete on S . That is, the set of outcome probabilities tr(ρ π i the eigenprojectors ofM i (S ), determines the physical state ρ ∈ S phys (S ) uniquely 3 . Then there is an algorithm which assigns to every encoding ϕ (S) : S phys (S) → S(S) a corresponding encoding ϕ (S ) : S phys (S ) → S(S ).
Moreover, if all observables of S are (S, S )-co-measurable, then the map ϕ (S) → ϕ (S ) is continuous. In the other case where M(S) is a strict subset of the observables on S, we have the following weaker statement: call two encodings ϕ (S) and ϕ (S) (S, S )-co-measurability equivalent if their image 4 of the set of (S, S )-co-measurable observables agrees, i.e. if ϕ (S) (M(S)) = ϕ (S) (M(S)). Then continuously changing ϕ (S) within one (S, S )-co-measurability equivalence class continuously changes ϕ (S ) .
In other words: if S and S have enough observables in common (even if they have different Hilbert space dimensionalities), then observers agreeing on a description of S obtain an unambiguous method to agree on a description of S .
The most intuitive understanding of this lemma comes from the communication scenario of Figure 1. Suppose that we have a situation as described in Lemma 3.2, and Alice and Bob have agreed on a common encoding ϕ (S) of quantum system S. In order to agree with Bob on an arbitrary encoding ϕ ( The resulting encoding ϕ (S ) will then be shared between Alice and Bob. It is not unique, but depends on the choice of matrix encodings that Alice suggests. These constitute classical data that can be communicated between Alice and Bob. Note however that this protocol fails if theM i (S) 0 do not encode (S, S )-co-measurable observables; in other words, the protocol resp. algorithm is tailored to the set of matrices ϕ (S) (M(S)).
Proof. LetM i (S ) 0 be an arbitrary set of concrete matrices that encodes the physical observableŝ M i (S ); it is sufficient to specify this for all i in a finite subset I 0 ⊂ I such that {M i (S )} i∈I 0 is still tomographically complete on S , and then all other matrices are fixed automatically. This is the part where Alice and Bob have a choice: they have to agree on a set of matrices that represent those physical observables. This set of matrices can also be seen a classical specification of the algorithm, corresponding to an identificationM i (S) 0 ↔M i (S ) 0 between matrix representations of the co-measurable observables. In other words, this defines a map T S→S such that T S→S M i (S) 0 =M i (S ) 0 , and T S →S := (T S→S ) −1 , both maps defined on the (representations of) the co-measurable observables. Furthermore, by the assumption of comeasurability, nature itself provides a map T phys S→S with T phys Since T phys S→S is a homeomorphism by assumption, the same must be true for T S→S .
Then, for every ϕ (S) , define a corresponding encoding ϕ (S ) via In this equation, consider replacing ϕ (S) by ϕ (S) . The resulting expression will still be welldefined if and only if ϕ (S) (M(S)) agrees with the domain of definition of T S→S , which is ϕ (S) (M(S)). If this is the case, the resulting map is clearly continuous in ϕ (S) .
We will give a concrete example how this algorithm works in our physical world for spin systems in Example 3.5 below.
Now we formalize the idea of the beginning of this section: if there is a two-level system that "interacts naturally" indirectly with all other systems, then a choice of reference frame for the two-level system implies a choice of reference frame for the full laboratory.

Assumptions 3.3 (Interaction graph).
Consider the set of all finite-dimensional quantum systems S, S , S , . . .; we regard them as vertices of a graph. Draw a directed edge from S to S if and only of the situation of Lemma 3.2 holds, i.e. if there is a set of (S, S )-co-measurable observables {M i } i∈I which is tomographically complete on S .
Assumption: there exists a two-level system S that is a "root" of this graph, in the sense that every vertex can be reached from S by following directed edges. Furthermore, we assume that no quantum system with a partially preferred choice of basis or encoding is a root of this graph.
Suppose that S is any quantum system such that there is a directed edge directly from the two-level system S to S . By assumption, the qubit S does not carry any natural choice of Hilbert space basis, in the sense that there is no distinguished subset of encodings ϕ (S) at all. We know how to parametrize all possible encodings ϕ (S) : given an arbitrary fixed encoding ϕ, all others can be written in the form ϕ (S) (ρ) = T (ϕ(ρ)) for all ρ ∈ S phys , where T ∈ PUA(2) O(3). In this case, we write ϕ (S) = ϕ (S) T . Due to Lemma 3.2, the directed edge induces a corresponding set of encodings ϕ for the two-level system S. If S is a system such that all observables of S are (S, S )-comeasurable, we get due to (3.1) (dropping some "•") which is a linear operator on S , then we get for all V, W ∈ PUA(2) W is continuous, it is a group representation of PUA(2) within PUA(S ). By considering the connected component at the identity, we thus get a projective representation of PU (2) SO(3) on S . The same conclusion holds for quantum systems S that are not directly connected to S, but can be reached from S by a path of several directed edges in the graph. 5 Now suppose that S is a system such that the set M(S) of (S, S )-co-measurable observables is a strict subset of all physical observables of the qubit S. According to (3.1), the calculation in (3.2) continuous to make sense if ϕ  (2) that preserve the image of the (S, S )-co-measurable observables. In other words, we obtain a projective representation of this subgroup.
Let C be the set of all systems S such that all qubit observables are (S, S )-co-measurable. The formal product defines a physically distinguished set of encodings of all these systems at once. Suppose there was a smaller subset of distinguished encodingsΦ Φ, thenΦ = {ϕ T | T ∈ G PUA(2)}, with G a strict subset of PUA (2). Operationally, being able to encode all laboratory quantum systems via some ϕ T implies in particular that one can encode the two-level system S via ϕ T | T ∈ G} would constitute a subset of distinguished encodings of the qubit, which contradicts our assumption that there is no such subset. Hence Φ must be a minimal set of encodings.
But then, in particular (2.1) must hold, such that for every choice of T, V, W ∈ PUA(2), there is some X ∈ PUA (2) for the two-level system S, it follows that X = W V −1 T . This equation must be consistent with the other factors of ϕ T in accordance with (3.4), which is only possible if an analogous equation holds for all systems S ∈ C simultaneously, yielding an independent proof of (3.2). Most importantly, the group associated to Φ, that is G : . This is the minimal group that translates between Alice's and Bob's encodings, if they describe the totality of all systems S ∈ C in their laboratories.
It is tempting to generalize (3.4), and to define the product to range over all finite-dimensional quantum systemsC, including those S for which not all qubit observables are (S, S )-comeasurable. This does not cause any problems for systems S that carry no (S, S )-co-measurable 5 If there are several paths leading from the qubit S to S , then a choice has to be made as to which path to take to assign a resulting encoding map ϕ (S ) T . However, this is also a choice that can be communicated by classical information between observers. If S → S → S in the interaction graph, then co-measurability will be transitive, in the sense that if the set of all S-observables M is (S, S )-co-measurable, and also (S , S )-co-measurable, then this set is also (S, S )-co-measurable. However, this set need not be tomographically complete on S , which is why there need not be an edge in the interaction graph going from S to S directly. observable at all (or only trivial co-measurable observables λ · 1 with λ ∈ R). The conclusions will be the same as above, namely that these systems carry projective representations of SO(3) (which will typically be trivial representations), and the set of encoding S ϕ (S ) T , where the product is over all those S and all S ∈ C, will be a minimal set of encodings.
However, a subtle difficulty arises with quantum systems S that carry a non-trivial strict subset M of (S, S )-co-measurable observables (among those observables of S). Our picture of coding and encoding of quantum states on S rests on a tacit assumption: namely that the agent (or Alice and Bob in the communication scenario) have perfect knowledge on the choice of quantum system S , which in this case includes the specification of the set of (S, S )-comeasurable observables M. If this was the case, then this knowledge could help Alice and Bob to encode qubits, inducing a strict subset of physically distinguished encodings ϕ (S) of the qubit S among all encodings ϕ (S) T with T ∈ PUA(2). Thus, taking the formal product in (3.4) over all (possibly continuously many) quantum systems S ∈C would yield a set of encodingΦ which is not minimal. Not only would this contradict our assumptions, but it would also ignore the difficulty of setting up the agreement of the choice of S between Alice and Bob.
In order to deal with this situation, we have to treat quantum systems S with non-trivial sets of (S, S )-co-measurable observables M differently: suppose that S andS are physically equivalent except for the corresponding sets of co-measurable observables M andM, and that those sets are related by unitary conjugation and/or transposition on S, i.e. that there is T ∈ PUA(2) withM(S) = T (M(S)). Then treat them as two instances 6 of the same system S , which is however in two different states (M(S), ρ S ) resp. (M(S), ρS ). That is, we consider the set of co-measurable observables as part of the specification of the state.
This prevents the difficulty just mentioned. We can define an encoding of S via the map (M(S), ρ phys ) → (ϕ (S) (M(S)), ϕ (S ) (ρ phys )), and define the analog of encodings (3.4) by taking the product over all S , obtaining a setΦ of encodings ϕ T with T ∈ PUA(2) of the totality of all quantum systems. Simply agreeing on some S does not allow to simplify agreement on the qubit S, andΦ will be a minimal set of encodings. Arguing as above, we get G min = PUA(2) = SO(3) as abstract minimal group. Furthermore, if S is the "root qubit" defined in Assumptions 3.3, and S any other quantum system such that all observables of S are (S, S )-co-measurable, then S carries a projective representation of SO(3). All other quantum systems S carry a projective representation of the subgroup of SO(3) which preserves the (S, S )-co-measurable observables.
Thus, our assumptions have reconstructed an important property of quantum theory in our universe: that many systems come with a representation of the rotation group in three dimensions. The usual point of view is that this is a consequence of the three dimensions of space. However, here we argue the other way around: the fundamental theory is quantum mechanics, and the emergence of an SO(3)-symmetry can be understood on this basis alone. This insight is very much in spirit of von Weizsäcker's "ur theory" [27], but goes far beyond it by putting the argumentation on firm operational grounds.
In the next subsection, we will explore in more detail how our abstract assumptions and Theorem 3.4) are concretely realized in our actual physical world.

Concrete realization in our universe
In order to give a concrete physical interpretation of the abstract argumentation above, we will in the following consider three kinds of finite-dimensional quantum systems that exist in our universe: • The spin-1/2 qubit encoded into an electron spin (call this quantum system S); • a spin-1 degree of freedom S -either elementary (as in a W or Z boson), or as an effective degree of freedom (such as the nuclear spin in orthohydrogen); • photon polarization qubits S . Figure 3: Part of the interaction graph of our universe (relativistic quantum mechanics). Massive spin-1/2 particles S (like an electron) and massive spin-1 particles S have a set of co-measurable observables which is tomographically complete on both, namely spin in any direction. For massive spin-1 particles S and photon polarization S , however, only the set of observables which is supported in the span of | + 1 and | − 1 of spin in the direction of the photon momentum is co-measurable, which is tomographically complete on S , but not on S . Thus, we can indirectly lift an encoding map ϕ (S) of the electron spin qubit to an encoding map ϕ (S ) of the photon polarization qubit, but not vice versa. 7 Figure 3 shows part of the interaction graph of our universe as defined in Assumptions 3.3, namely the vertices S, S and S . Arrows are drawn according to co-measurability as defined in Assumptions 3.3. In particular, there is an arrow from S to S , and another arrow from S to S, since both systems can be measured in common Stern-Gerlach devices. More generally, let I be the set of all unit vectors in R 3 , then the set of spin observables {S n } n∈I is (S, S )-co-measurable: by constructing a Stern-Gerlach device with magnetic field (and inhomogeneity) in direction n, we can measure these observables on both systems universally.
We will now give a concrete example how a possible algorithm as in the proof of Lemma 3.2 works in this case. This allows to construct an encoding of spin-1 quantum states from an encoding of spin-1/2 quantum states.
Example 3.5 (Encoding: from spin-1/2 to spin-1). According to [32], there are five spatial directions n 1 , . . . n 5 ∈ R 3 with | n i | = 1 such that {Ŝ n i } i=1,...,5 is tomographically complete on the spin-1 state space. Concretely, we can choose The actual data to start with is the set of physical spin-1/2 observablesŜ n i (S), represented mathematically as n i · σ, with σ = (σ x , σ y , σ z ) the Pauli matrices. By using universal Stern-Gerlach devices, this also defines the physical observablesŜ n i (S ). Now we have to construct an arbitrary matrix representationŜ n i (S ) 0 of these observables. All possible choices are related by unitary conjugation and possibly transposition; we can choose one arbitrarily. To this end, for every unit vector n ∈ R 3 , setŜ n (S ) 0 := n · S, where S := (Ŝ x ,Ŝ y ,Ŝ z ), and From this, we obtain a unique physical encoding ϕ (S ) of quantum states. For example, the pure state ρ phys = |ψ ψ| phys that gives unit probability of "spin-up" in z-direction will have This algorithm of constructing ϕ (S ) from ϕ (S) had one arbitrary choice, namely how to describê S n (S ) in terms of concrete 3×3 matrices. This is an arbitrary choice among all encodings (they are all related by unitary conjugation and possibly a transpose; other attempts of encoding will be in conflict with the observed measurement statistics), and part of the specification of the algorithm (which can, for example, be communicated in terms of classical information between Alice and Bob). Fixing this data, the resulting encoding map ϕ (S ) depends on ϕ (S) : choosing another ϕ (S) in the first place will select another physical device as the appropriateŜ z (S)-measurement, for example, which leads to another physical observableŜ z (S ) and to another pure physical quantum state |ψ phys that will be considered to be the (+1)-eigenstate of spin in z-direction.
Therefore, changing the encoding of S by a rotation will have an effect on the encoding of S ; in other words, we get a projective spin-1 representation of SO(3) on S (as claimed in Theorem 3.4), and this representation is not trivial.
The spin example uses the background knowledge that there is a notion of underlying 3space, constituting a mechanism to set up these universal Stern-Gerlach devices in the first place. However, there is a way to argue that these universal Stern-Gerlach devices should exist, without directly resorting to spatial degrees of freedom. The idea is that we can use a large number N of electron spin qubits to build a magnet. The ensemble of spins should be in a coherent state | n ⊗N , and then another particle S can interact with that system via a simple interaction Hamiltonian [26]. Thus, the electron spin qubit's quantum state can define a frame of reference which is transferred to other quantum systems via interaction. This suggests, but does not necessitate the interpretation of n as a spatial direction.
This idea is very similar to von Weizsäcker's suggestion that the symmetries of the elementary binary quantum alternative should be identical to the symmetries of space [27], and it resembles the distinguished role of the qubit for the structure of quantum theory [19,20,21,22,23,24]. It has recently been used to argue why space should have three dimensions [25,26].
We now turn to photon polarization S . Given a photon with momentum p, the photon spin must be oriented either parallel or antiparallel to p. Set p 0 := p/| p|, and consider the spin observableŜ p 0 . This observable is well-defined on the photon S , and it can be written S p 0 (S ) = |R R| − |L L|, where |R and |L denote the left-and right-circular polarization states of the photon. Clearly, the same observable can be defined on the spin-1 particle S , which will be a 3 × 3 matrixŜ p 0 (S ).
We claim thatŜ p 0 is (S , S )-co-measurable, as already indicated by the notation. A concrete way to measure both systems in the same device is via photon absorption by an atom in its electronic ground state, as described in [29,30]. Conservation of angular momentum will force the atom's electron from the ground state to an = 1 excited state, with magnetic quantum number m = ±1 corresponding to the photon's spin quantum state, |L or |R . That is, the result of the absorption will effectively be the transfer of the photonic quantum information on S to a quantum system S . Finally, the observableŜ p 0 (S ) can be measured in a Stern-Gerlach-like device, at least in principle.
In fact, transmission of quantum states from photon polarization qubits to energy levels of atoms are currently performed in many concrete experiments, see for example [31]. Since superpositions of |L and |R are preserved, we can use the tomographic completeness of Stern-Gerlach measurements of spin in all directions [18] to effectively measure any observable with support on span{| + 1 , | − 1 } on quantum system S , not onlyŜ p 0 (S ). This yields a set of observables that is (S , S )-co-measurable and tomographically complete on S . Therefore, the interaction graph has an arrow from S to S .
We have assumed in Subsection 3.2 that the observablesM (S ) uniquely define the observ-ablesM (S ); in other words, there is a unique interpretation of these observables in terms of a physical quantityM . This shows that the other observables on S -those that are not fully supported on span{| + 1 , | − 1 } -cannot have counterparts on S . Therefore, we have already found the maximal set of (S , S )-co-measurable observables, and no such set can be tomographically complete on S . Thus, there is no arrow in the interaction graph from S to S . Theorem 3.4 also tells us that there is no representation of the rotation group on the photon polarization qubit, in contrast to the spin-1/2 system S and the spin-1 system S . Instead, one expects to find a representation of the subgroup of SO(3) that preserves the set of (S , S )-comeasurable observables. These are exactly the rotations that stabilize the photon momentum vector p -in other words, the subgroup equivalent to SO(2) which rotates the transversal photon polarization vector.

Two causes of disagreement: active and passive
So far, we have motivated the "information gap" between Alice and Bob by imagining that they reside in different, distant laboratories, and have never met before. Under this assumption, we have argued in Subsection 3.2 that their descriptions of local quantum physics in their laboratories must be related by an element of G min = O(3).
Concretely, we can think of Alice and Bob as talking on the phone, Alice sending a request to Bob in terms of classical information, and Bob responding in terms of a physical system that he sends back, as depicted in Figure 1. Clearly, the classical information that Alice sends to Bob must be encoded into some physical system as well, that serves as the signal carrier. However, we can imagine that they are using a physical (quantum) system S that carries at least a partial natural choice of Hilbert space basis. For example, Alice can send information bitwise, encoding a zero into the ground state, and a one into an excited state of a two-level system; or encoding it into the relation between the orthogonal basis elements of two successive quantum systems.
In addition to this "passive" picture, we can also imagine an "active" scenario: suppose that Alice and Bob have actually met before, and have used this encounter to calibrate their measurement devices and synchronize their frames of reference. But imagine that their laboratories have become separated afterwards, and subjected to all kinds of physical influences. Concretely, think of Alice and Bob and their laboratories as traveling in spaceships to distant parts of the galaxy, under the influence of all kinds of gravitational fields.
In this case, their descriptions of local physics of each others' laboratories will have become desynchronized, and they may recognize it once they try to implement the information-theoretic scenario of Figure 1. However, in this case, there is a difference to the earlier "passive" scenario: namely, one would expect that their "information gap" is characterized by an element of the implementable subgroup of G min -that is, in this case, of SO(3). This is because the universe has actually implemented the corresponding transformation, by acting on their laboratories.
Indeed, this is consistent with our result: different observers may become twisted relative to each other (described by a rotation R ∈ SO(3)), but not reflected relative to each other.

Physical quantum states and the Lorentz group 4.1 Types of quantum systems and physical measurement outcomes
In the previous section, we have taken an abstract operational point of view on quantum states inspired by quantum information theory. In this picture, a state of a quantum N -level system is merely a concise catalogue of probabilities of the outcomes of all possible measurements that can be performed on the quantum system. For example, if a state ρ = N i=1 λ i |i i| (assuming non-degenerate spectrum) is measured in its eigenbasis, then the probability to obtain outcome i is λ i , but the outcome itself is not considered to have any particular physical meaning. Any observable of the formM = N i=1 m i |i i| for any choice of m 1 , . . . , m N can be measured by the same physical device with outcomes that merely differ by their classical labels m i . This point of view is particularly common in quantum information theory, and it seems especially appropriate in the case of destructive measurements where the physical system is annihilated on detection.
However, in many situations, measurement outcomes carry concrete physical meaning, in the sense that the outcome describes a specific physical property of the particle after the measurement. In this case, the eigenvalue is not just a classical label, but describes a physical post-measurement property of the system (say, its energy in some units like Joule). We have already seen in Subsection 3.2 that actual physical properties of an observable (for example the property of being measurable by universal devices) can have important structural consequences. Therefore, we should analyze how the conclusion of the previous section is modified if we take this complication into account.
Suppose we have a physical quantity which can in principle take one of infinitely, maybe continuously many values (such as energy). Then even if we have an effectively finite-dimensional quantum system (such as a superposition of only two energy levels in an atom), this quantum system will be a subspace of a much larger, typically infinite-dimensional Hilbert space or operator algebra which describes the laboratory as a whole. We will now argue that no matter which fundamental theory (say, what specific quantum field theory) we assume to hold, this simple fact will have the universal consequence that finite-dimensional quantum subsystems will come in different types, even if they have the same Hilbert space dimensionality. A simple example illustrates this fact.
Imagine a Stern-Gerlach apparatus which performs a spin measurement on spin-1/2 particles. For this, the particles of mass m and velocity v will enter an inhomogeneous magnetic field which defines a quantization axis z, and then spread into two beams corresponding to the two possible values of the spin in z-direction, finally hitting a screen (let us assume for now that v z = 0). The particles will hit the screen either in a small vicinity up-or downwards in direction of the quantization axis, which constitutes the two possible outcomes "spin up" or "spin down". We can associate an observable, ∆z, to this experiment, with eigenvalues corresponding to the two possible spatial displacements (in units of length) along the quantization axis. The two eigenvalues (differing only in sign), corresponding to the two possible deflections 8 , are proportional to 1/(mv 2 ).
Thus, two electrons with different velocities will make the same fixed Stern-Gerlach device perform different kinds of measurements. Even though the same physical degree of freedom is probed in both cases (namely the quantum bit carried by the electron's spin), the actual qubit observable ∆z differs between both cases. Somehow, the qubits come in different types (in this case corresponding to different velocities v), and the measurement device responds differently to the different types of qubits.
Another possible source of having different types of quantum states is that there are different sorts of physical systems that carry them, such as different sorts of particles. For example, we can use other types of neutral atoms with an unpaired electron, or even think of replacing the electron by a muon in principle. Now let us disregard all concrete physics, and give an abstract operational definition of what we mean by different "types" of N -level systems. Suppose we have a set M of measurement devices (for example a set of identical Stern-Gerlach devices, however with different directions and magnitudes of the magnetic field). Every measurement deviceM ∈ M accepts certain kinds of physical systems as input (such as neutral atoms in the Stern-Gerlach case), probes a quantum state ρ encoded in a common degree of freedom of all systems (the spin-1/2 qubit in the Stern-Gerlach case), and measures an N -level observableM (S) that may depend on the system S that is sent in.
We say that two physical systems S and S have the same behavior ifM (S) =M (S ) for all measurement devicesM . The behavior (that is, the corresponding equivalence class of physical systems) will then be called β, and the corresponding measurement that is performed is denoted M (β) ≡M (S) =M (S ).
How are the observablesM (β) andM (β ) for different behaviors β = β related to each other? To this end, we have to specify in more detail what kind of measurement devices we are considering. The intuition is that we would like to talk about one fixed deviceM that blindly performs a test that is in some sense "the same" on all systems S, S , . . . that we send in. We can formalize this intuition by postulating that a set of properties (of the measurements) which is true for one kind of system must remain true also for all other kinds of systems.
For example, suppose we have a sequence of (infinitely many) devicesM 1 ,M 2 , . . . such that the expectation values converge to that of another deviceM in the limit. If this is true for systems of some given behavior β, then it makes sense to demand that this should be true for the systems of any other behavior β as well. This leads tô (4.1) Since we postulate this for all behaviors β, β , it follows immediately thatM 1 (β) =M 2 (β) impliesM 1 (β ) =M 2 (β ). Thus there is a bijective map T β ,β which maps the observableM (β) toM (β ). According to (4.1), this map is continuous, and so is its inverse.
It is now a matter of taste, or physical intuition, which additional properties we should demand to characterize "the same measurement" for different kinds of physical systems. The good news is that several different choices turn out to be equivalent in the end. Here is a possible list of natural properties that one can postulate. The first one is motivated by the fact that comparison of measurement results is an elementary aspect of physics. Hence the fact that one measurement always yields larger expectation values than another measurement should be an objective property that does not depend on the behavior: We introduce one further condition, namelŷ This assumption gives a special meaning to the zero measurementM = 0, namely the interpretation that no measurement is being performed at all. That is, it suggests to view the measurement outcomes as a measure of change of the particle's properties that is affected by the measurement. In the Stern-Gerlach example above, this would mean that we measure the particle's displacement ∆z relative to the position where the particle would hit the screen in the case of zero magnetic field. The measurementM = 0 would correspond to a Stern-Gerlach device where the magnetic field is switched off. In general, (4.3) can then be interpreted as saying that "no measurement being performed" is a behavior-independent concept. Surprisingly, these conditions are very restrictive: Proof. It is easy to check that maps of the form given above satisfy both conditions. For the converse implication, set Y := T β ,β (0), and W (M (β)) := T β ,β (M (β)) − Y . Then W is a continuous bijective order-preserving map on the Hermitian N × N matrices which satisfies W (0) = 0. According to [2,3], this implies that W (M ) equals X †M X or X †M T X.
In the following, we will assume all three conditions (4.1), (4.2) and (4.3). This lemma shows that some other natural properties follow automatically, such as linearity of T β ,β , Anticipating the communication task between Alice and Bob, where both will not agree on a "reference behavior" of the quantum systems, this corresponds to the fact that Alice and Bob however agree on the linearity structure of observables. This is very natural: observers using Figure 4: Physical systems come in different types; together with a choice of orthonormal Hilbert space basis (resp. parity), they yield different behaviors β in measurements. Systems of the same type live on the same Hilbert space. Degrees of freedom corresponding to the same sort of physical system (for example sort of particle, such as electron) may still live on different Hilbert spaces; in the Stern-Gerlach example mentioned in the main text, the Hilbert space carrying the spin degree of freedom depends on the velocity v. If we consider all possible measurements, then it is natural to expect that physical systems of different sorts will make at least some measurement devices respond differently. Thus the types of these systems are always different, which is why the dashed rectangles representing different sorts of physical systems in the picture do not overlap. different physical units (such as "meters" and "miles") will usually agree on whether a value is the sum of two other values (but they will not, for example, agree on the product). In the next section, we will also see that this property becomes even more important when we go from the Heisenberg to the Schrödinger picture: the "dual" property of this equation establishes the type-independence of the probabilistic structure.
If S and S are physical systems with behaviors β and β , we say that they are of the same type if the matrix X in Lemma 4.1 can be chosen unitary; that is,M (S ) = U †M (S)U with some unitary U . Operationally, this situation is distinguished by the property that for every state ρ S of system S, there is a state ρ S of system S that yields exactly the same statistics on all measurementsM (and vice versa). We can interpret this as saying that S and S are actually identical kinds of systems, but with different choices of Hilbert space basis. Thus, our formalism of behaviors naturally contains the situation that different observers will in general assign different Hilbert space bases, generalizing 9 the situation described in Section 3.1.
Therefore, in order to specifyM (S) for a given system S, we need to know two things: first, the type of that system, and second, the choice of basis in the corresponding Hilbert space, or more generally the way that quantum states are described as density matrices. The pair of both is the behavior β. If systems S and S have the same behavior β, thenM (S) =M (S ) ≡M (β) is the exact same N × N matrix. In the case that S and S are of the same type, but have different behaviors β and β , there will be a unitary U such thatM (β ) = U †M (β)U orM (β ) = U †M (β) T U . As we will discuss in the next subsection, this means that the corresponding degrees of freedom live on the same Hilbert space. Degrees of freedom of different types will live on different Hilbert spaces. This is schematically depicted in Figure 4

From Heisenberg to Schrödinger: unnormalized quantum states
So far, we have taken the different types and behaviors of systems into account by saying that a measurementM is described by an observableM (β) which depends on the behavior β. However, we can also use a "dual" formalism, which is similar to the change from the Heisenberg to the Schrödinger picture. Fix an arbitrary reference behavior β 0 . Suppose β is any other behavior for which Lemma 4.1 yields the transformation equation without transpose, such thatM (β) = X †M (β 0 )X. Then, for any quantum state ρ carried by a system with behavior β, tr(ρM (β)) = tr(ρX †M (β 0 )X) = tr(XρX †M (β 0 )). This allows us to use the following alternative formalism: every measurementM is described by a fixed observableM (β 0 ). However, quantum states ρ of systems with behavior β = β 0 are described instead by pseudo quantum states XρX † = T † β,β 0 (ρ), where X is the complex matrix such that T β,β 0 = X † • X. In the case where T β,β 0 = X † • T X, the pseudo quantum state is T † β,β 0 (ρ) =Xρ TX † , where the bar denotes complex conjugation. Note that this "dualization" is impossible in the case where Y = 0 in Lemma 4.1. Thus, condition (4.3) is necessary for translating from type-dependent observables to type-dependent quantum states if we want to keep the Born rule (trace rule) framework.
Pseudo quantum states are positive semidefinite, but not necessarily normalized; their trace can be any positive real number. At first sight, this seems to conflict with the usual postulates of quantum mechanics, but in fact it does not. Note that if ρ is any N -level pseudo quantum state, it is an operator on C N . This vector space, however, becomes a Hilbert space only with a choice of inner product. It turns out that choosing an inner product different from the canonical one turns pseudo quantum states into quantum states, and thus admits the usual probabilistic interpretation of quantum states.
Observation 4.2. By choosing a fixed reference behavior β 0 , we can denote every behavior β in the form β X or β T X ; we use the former notation if T β,β 0 = X † • X, and the latter if For any given behavior β, consider the polar decomposition This decomposition is unique [4]. In fact, denoting behaviors by β X resp. β T X is a slight overparametrization 10 due to the fact that β X = β X (resp. β T X = β T X ) if and only if X = e iθ X for some θ ∈ R (or equivalently if U = e iθ U and R = R ). Pseudo quantum states with behavior β X or β T X are quantum states on the Hilbert space H R , which is the complex vector space C N with inner product ϕ|ψ R := ϕ|R −2 |ψ , and the unitary U signifies the different choice of basis in this Hilbert space, compared to the choice of basis in the Hilbert space which carries the quantum states of behavior β 0 . Two behaviors are of the same type if and only their matrices R agree; that is, if and only if they live on the same Hilbert space, H R . If β = β X (no transposition), then the matrices X, interpreted as maps X : H 1 → H R , are isometries of Hilbert spaces. In the case where β = β T X , the map |ψ →X|ψ , where the bar denotes componentwise complex conjugation, is an antilinear isometry. Normalization of pseudo quantum states ρ is given by tr R (ρ ) := tr(ρ R −2 ) = 1.
Given systems with behavior β X in the pure quantum states |ψ and |ϕ , the pseudo quantum states are |ψ = X|ψ and |ϕ = X|ϕ . With the given choice of inner product, we have Thus, pseudo quantum states are indeed normalized with this choice of inner product, and they yield valid transition probabilities. If we have a system with behavior β T X instead, we have |ψ =X|ψ , and we obtain analogously ϕ |ψ R = ψ|ϕ = ϕ|ψ .
Since transition probabilities are given by the modulus square of an inner product, this still reproduces the probabilistic predictions of the standard quantum states in the Heisenberg picture of the previous subsection. More generally, for systems with behavior β X , we can check that the trace of the pseudo quantum state ρ = XρX † , evaluated in the new inner product 11 , equals the usual trace of ρ: for systems with behavior β X , we have tr R ρ = tr(XρX † R −2 ) = tr(ρX † R −2 X) = trρ = 1, (4.5) and the same result holds for systems with behavior β T X , using that tr(ρ T ) = trρ.

The communication scenario with different types of systems
We now return to the communication scenario of the previous sections. Again we have Alice and Bob as in Figure 1. Alice sends Bob a classical request, asking him to prepare a system with behavior β in a quantum state ρ. Bob then tries to send a physical system with the given behavior in the corresponding quantum state back to Alice. For the time being, let us describe the situation in the Heisenberg picture. The main communication problem -if Alice and Bob have never met before -is that both may describe the different types of systems in different ways. In more detail, Alice and Bob may have chosen two different "reference behaviors" β A 0 and β B 0 respectively, which means that they assign different invertible matrices X A and X B to the same behavior β. To distinguish between both cases with and without transpose (as introduced in Observation 4.2), we introduce another variable P ∈ {−1, +1}, "parity", which is set to +1 if there is no transpose, and −1 otherwise. Again, Alice and Bob may differ in the assignment of parity, in which case P A = P B .
Note that the communication scenario in the previous section is a special case of this one: it applies to the case where Alice and Bob agree on the type of system to be sent, such that the only source of disagreement is the choice of Hilbert space basis, i.e.M (β B 0 ) = U †M (β A 0 )U for some unitary U and all measurement devicesM (possibly with an additional transpose). With this additional knowledge, Alice's and Bob's descriptions differ by a unitary or antiunitary map, and we recover the result that the projective unitary antinuitary group is achievable. In other words, the complication in this section comes from the fact that Alice and Bob do not know 11 We can describe the situation alternatively as follows. Given C N as a vector space, the choice of inner product ·, · R defines a set of Hermitian matrices, i.e. matrices M with x, M y R = M x, y R. These are the matrices for which (R −2 M ) † = R −2 M , where the conjugate transpose is defined with respect to the standard inner product and its canonical orthonormal basis. Similarly, the cone of positive semidefinite matrices becomes the set of matrices M for which R −2 M ≥ 0 with respect to the canonical inner product, and the normalization condition becomes tr R (M ) ≡ tr(R −2 M ) = 1. All this is consistent with the Heisenberg picture: consider a quantum system with behavior β in state |ψ . The expectation value of an observableM is ψ|M (β)|ψ , whereM (β) =M (β) † . If β = βX , this equals ϕ|R 2M (β0)|ϕ R, where |ϕ = X|ψ , and R 2M (β0) is Hermitian in the Hilbert space HR.
beforehand how the finite-dimensional degree of freedom is embedded in the infinite-dimensional total Hilbert space or operator algebra.
In the current more general communication scenario, where Alice and Bob have not calibrated their descriptions of the different types, Alice sends out a classical description of X A and P A and ρ. Since Bob uses a different reference behavior, he will not know which system and state he should send to Alice. Thus, as in the previous sections, there should be a transformation T (depicted in Figure 1) that corrects this description before it arrives in Bob's lab. There is no need to correct the description of ρ -once Bob knows the behavior, he also knows the choice of Hilbert space basis and knows how to prepare ρ. We will now describe a group of transformations T that achieves the task; these transformations act on X and P only.
First, consider the case that Alice's and Bob's reference behaviors have the same parity, in the sense thatM (β B 0 ) = Y †M (β A 0 )Y (no transposition) for some fixed invertible matrix Y . Furthermore, suppose that β is a behavior that Bob describes as β X B (no transposition). Then Thus, we get X B = Y −1 X A and P B = P A = +1. On the other hand, if β is a behavior that Bob describes as β T X B , then where the bar denotes componentwise complex conjugation. Thus, we get X B =Ȳ −1 X A and P A = P B = −1, and we obtain the following table defining a correcting transformation T : Second, consider the case that Alice's and Bob's reference behaviors have opposite parities. Analogous calculations as above show that the correcting transformation T has to be chosen slightly differently: If we denote the transformations T in (4.6) by T + Y −1 and those in (4.7) by T − Y −1 , we obtain the multiplication table As expected, these transformations form a group. Since matrices X B and X B that differ only by a complex phase describe the same type, we have to identify Y and e iθ Y for θ ∈ R, which yields a quotient group. If we define the maps G + Y (M ) := YM Y † and G − Y (M ) := YM T Y † on the self-adjoint matricesM , then these maps satisfy the same multiplication rules (with T replaced by G), so the group we just obtained is isomorphic to the group G containing the G + Y and G − Y . We have just seen that we can always choose an element T ∈ G of this group, and apply it as in Figure 1 to ensure that Alice's and Bob's communication task succeeds. Thus, we obtain the following statement: Lemma 4.3. In the communication scenario described above, the group is achievable. This is the group of linear automorphisms of the cone of positive semidefinite complex N ×N matrices. The subgroup of implementable transformations is its connected component at the identity, which can also be written GL(N, C)/U(1) = R + × PSL(N, C).

Relating full laboratories: emergence of O + (3, 1)
As in Section 3, we can ask whether all local quantum physics of Alice's and Bob's laboratories can be related by a single finite-dimensional group element, in the more general setting that quantum systems can display different types of behaviors in measurement devices. Indeed, it turns out that this is possible, as long as we make assumptions analogous to those in Subsection 3.2.
Let us call two quantum systems S and S similar if they are the same kinds of systems, but possibly of different type or different behavior β. This is the case if we can measure them in the same devices, and the corresponding observables are related as stated in Lemma 4.1; it is necessary but not sufficient that dim S = dim S . In our physical world, spin qubits of electrons with different velocities are similar, but an electron spin and a photon polarization qubit are not. The set of all quantum systems that are similar to a given system S will be denoted [S], and called the similarity class of S.
We will now repeat the construction of Subsection 3.2, and explain which steps are different. First, an observableM will be called (S, S )-co-measurable on two (possibly not similar) quantum systems S and S of possibly different Hilbert space dimensionalities if there is a natural device that measures the quantityM for both systems universally, resulting in observablesM (S) and M (S ) which mutually determine each other. In particular, this means that we can also input any other system that is similar to S (or similar to S ) into this device; henceM is co-measurable for all elements of [S] and all elements of [S ], and we can say thatM is ([S], [S ])-co-measurable.
In Section 3, there was an encoding map ϕ (S) for every system S; now, however, the encodings for similar quantum systems are related to each other by conjugation. Therefore, we have an encoding map ϕ [S] for every similarity class [S]. Instead of encoding quantum states, we can also encode observables by using a corresponding map ϕ [S] † . It is defined such that it yields the correct expectation values, tr(ρ physMphys ) = tr ϕ [S] (ρ phys ) ϕ [S] † (M phys ) , and so † symbol can indeed be interpreted as the adjoint with respect to the Hilbert-Schmidt inner product. In contrast to Section 3, we now have ϕ = ϕ † ; indeed, if ϕ(ρ) = XρX † (with possibly a transpose on the ρ), then ϕ † (M ) = X † −1M X −1 (with possibly a transpose on theM ). Nevertheless, we will in the following drop the † symbol, and slightly abuse notation by using the symbol ϕ [S] also for encodings of observables.
It is easy to see that an analogue of Lemma 3.2 remains true in the more general setting of quantum systems with different behaviors: Note that it would make no sense to demand that both physical state and behavior are uniquely determined by the measurement statistics: a system with behavior β in state ρ is physically identical to a system with behavior β in state ρ = U ρU † , if β is the behavior witĥ M (β ) = UM (β)U † (we can also add a transpose to both). However, in this case, the two systems would be of the same type.
Like Lemma 3.2, this result also has a simple interpretation in terms of the communication scenario depicted in Figure 1. Suppose Alice and Bob have agreed on a common encoding of states on similarity class [S]. Then Alice can send Bob a classical request of the following form: "Please send me the quantum state on a quantum system in similarity class [S ] which has the following probabilities of the following measurement outcomes if it is sent into the deviceŝ M 1 , . . . ,M n : [list of numbers]." In this request, the devicesM i can be described by sending the matricesM i (β S 0 ), where β S 0 is any reference behavior in [S]. From this, Bob will know what devices to build or to look at, in accordance with their agreement on how to encode [S]. Due to tomographic completeness, he can then determine a physical system of uniquely specified type and state, and send it to Alice.
In more detail, for everyM i , there is a set of eigenvalues λ n ofM i (β) which depend only on the type. Suppose β and β are two different behaviors that predict exactly the same eigenvalues for all observablesM i . Since (in the Heisenberg picture)M i (β ) = X †M i (β)X for some invertible matrix X (and possibly an additional transpose), the map X † • X (or X † • T X) must be spectrum-preserving. However, it is well-known [28] that this implies X † = X −1 ; in other words, X must be unitary, and thus β and β must be of the same type.
Therefore, if we choose a large enough set of observables {M i } i∈I , we can learn the type of a state from the collection of all eigenvalues 12 (possible measurement outcomes) of theM i . Together with the probabilities to obtain the corresponding outcomes, this allows Alice to encode the type and state classically and send it to Bob, who can send back a correct physical answer.
The In a similar way, we can generalize Assumptions 3.3, and define interaction graphs in this setting. Assumption: there exists a two-level class [S] that is a "root" of this graph, in the sense that every vertex can be reached from [S] by following directed edges. Furthermore, we assume that no quantum system with a partially preferred choice of encoding is a root of this graph.
We can argue further along the lines of Subsection 3.2, if we keep our notation in the Schrödinger picture. If ϕ is an arbitrary fixed encoding of the qubit class [S], then every other encoding is related to this by an element of the group G 2 , the group of all maps of the form X • X † and X • T X † , with X an invertible complex 2 × 2-matrix. That is, for every encoding ϕ [S] , there is a T ∈ G 2 such that ϕ [S] (ρ) = T (ϕ(ρ)). In this case, we write ϕ [S] = ϕ [S] T . All further steps of the proof of Theorem 3.4 hold also in this more general case of different types, if every occurrence of PUA(2) is replaced by G 2 (and the reference to the special observables λ · 1 with λ ∈ R is removed), yielding that G min = G 2 .
The group G 2 has a simpler description. First, every conjugation X •X † with X an invertible complex matrix can also be written in the form λZ • Z † , with λ > 0 and det Z = 1. Writing 2×2 Hermitian matrices formally as 4-vectors, ρ = , it is well-known that the conjugation ρ → ZρZ † acts like an element of the proper orthochronous Lorentz group [5]. Furthermore, the transposition acts as a space inversion. Therefore, we obtain the following: In the following subsections, we discuss and interpret this result.

Interpretation of the minimal group: scaling and spacetime
If Alice and Bob reside in 'distant' laboratories, and have never met before to synchronize their frames of reference, then their descriptions of local quantum physics will be related by an element (λ, T ) of the group G min = R + × O + (3, 1), as we have learned in Theorem 4.6.
How can we interpret the scalar λ > 0? If we set T = 1 for a moment, and argue in the Heisenberg picture, then Alice's and Bob's observables will be related byM A = λM B . This means that if Alice and Bob measureM on the same physical state, then the probabilities of the different measurement outcomes will be the same in both cases, but they will describe the outcomes (which are physical quantities) differently: Bob will say that the possible outcomes are m N . This does not come as a great surprise because the condition (4.2), comparing sizes of measurement outcomes, introduces a scale into the game on which Alice and Bob have to agree. But the description of a scale requires units and Alice and Bob may adhere to different conventions and thus use different units for the outcomes of observableM , like "kilometers" versus "miles". Due to (4.3), both Alice and Bob will agree on the case that the measured value is exactly zero, but for all other values, there will be a scaling factor between their descriptions due to different choices of units; this is the scalar λ > 0.
Even if T = 1, then the form of G min as a direct product of groups ensures that there is always a unique consistent way of assigning such a scaling factor between any two parties. That is, Alice and Bob can split off a scalar factor λ B→A such that the resulting transformation T becomes a Lorentz transformation (which leads to λ B→A = λ in the example above), and if Bob and Charlie act likewise to obtain a factor λ C→B , then we will have λ C→B λ B→A = λ C→A , which is the consistency condition that one would expect to hold in the case of differing choices of units. Thus, in a world as described in this section, it is consistent for observers to interpret the relation between laboratories in the following way: "The actual relation between any pair of laboratories is given by a Lorentz transformation. Furthermore, parties can of course choose different units, which gives an unimportant and physically irrelevant factor that relates the descriptions." So what about the Lorentz transformations T ∈ O + (3, 1)? It is tempting to interpret those as actual spacetime transformations, and in the following subsection, we will show that they indeed correspond to spacetime symmetries in the concrete example of relativistic Stern-Gerlach devices. But is there any apriori reason to attach this interpretation to our abstractly obtained O + (3, 1)? Arguments by Wald [5] suggest an affirmative answer to this question. The condition that different observers residing in some spacetime should be able to translate their respective operational descriptions of physical systems (and, more generally, fields) into one another, requires the isometries of this spacetime to have a representation on the set of states (classical or quantum) of the physical theory defined on that spacetime. Therefore, if we look at the quantum transformations that relate different observers' descriptions of physics, then this set of transformations should contain a representation of the spacetime isometries. Consequently, if we assume that there is some underlying spacetime, our result tells us that we should not be surprised to find that this spacetime has a symmetry group which is related to O + (3, 1).
Of course, the present status of our thought experiment is not sufficient to permit Alice and Bob to identify their ambient spatio-temporal structure uniquely as a Minkowski spacetime. After all, this would require a representation of the larger Poincaré group involving spacetime translations. Our present scheme is therefore, at this stage, still remote from a successful operational spacetime reconstruction. The fact that we have only found the Lorentz subgroup of the Poincaré symmetry group of special relativity may be related to our assumption that Alice knows "where and when" the physical system that Bob sends is arriving in her laboratory. Technically, the fact that translations are missing is related to our assumption (4.3) which we needed to make sense of a "Schrödinger picture" description in Subsection 4.2. It is interesting to see that (4.4) seems to indicate that we might obtain the full Poincaré group if we dropped this assumption.
Nevertheless, the spacetime structure is ultimately encoded in the relations among the entirety of observers contained in it (or, rather, it is the set of all those relations). Indeed, following von Weizsäcker's ideas [27], we may even reverse the present argumentation. One may argue that the very operational definition of a spacetime symmetry is to translate between two equivalent but distinct descriptions of the same physics, as long as these are descriptions of large physical scenarios (full laboratories) given by distant observers. In this light, the extraction of the Lorentz group from the informational relation of observers is at the very least suggestive of a deeper connection of our communication game and the ambient spatio-temporal structure.

Relativistic Stern-Gerlach measurements
Given our purely abstract discussion up to this point, one may naturally wonder whether there exists a concrete relativistic example in which observables (or states) transform precisely aŝ M B = SM A S † for S ∈ SL(2, C). In the light of our setup, a relativistic formulation of the Stern-Gerlach experiment naturally comes to mind and, indeed, it is a matching example, albeit only in the WKB-limit of the Dirac theory [6,7]. We shall only provide a brief schematic description of this example here, as an explicit understanding within our language requires a proper SL(2, C)-spinorial formulation of qubits, which we shall deliver in a separate article [8] together with a more detailed analysis of the Stern-Gerlach case.
In the WKB approximation of the Dirac equation one can derive a velocity dependent inner product for SL(2, C)-spinorial qubits from the Dirac current [6,7] ψ|φ p which is linear in the four-momentum p µ of the particle representing the qubit. In this manner one obtains fourmomentum dependent Hilbert spaces H p C 2 , ∀ p µ . The classical state of motion p µ thus represents the type of the qubit in this example. The set of all inner products, and thus Hilbert spaces H p , is related linearly by the set of all spacetime Lorentz boosts and is invariant under local spatial rotations within the reference frame of a given particle.
It turns out that in the WKB-limit every observable on any H p can be written as [8,6,7] where σ µ is the Pauli-4-vector which acts as the operator and M µ is a classical Minkowski vector. For instance, a relativistic spin operator arises naturally on each H p from the Pauli-Lubanski vector [6,7]. This classical treatment of the vector arises because we treat the different types of qubits -and thus the associated scale -classically by construction. In particular, our framework does not include superpositions of different types and thus, in this case, of different states of motion. But while different observers will employ the same representation of the Pauli-4-vectorσ µ , they will agree on the description of M µ only up the Lorentz transformation Λ translating between their inertial frames. One can check that this yields precisely [8] where S is an SL(2, C) transformation corresponding to the Lorentz transformation Λ. This permits to consider concrete observables associated to the Stern-Gerlach device. A spin operator alone does not encode the different deviations which qubits in distinct states of motion may experience when running through the inhomogeneous magnetic field as it does not involve a scale. It is therefore not useful to describe the different reactions of different types of qubits to the detector. By contrast, the gradient of the magnetic field B RF µ , as seen in the rest frame of the particle, constitutes a vector which does describe the different reactions to the Stern-Gerlach measurement. It represents the four-acceleration that the particle experiences when running through the magnetic field. The corresponding operatorĜ = G µ ·σ µ can be interpreted as a hybrid observable measuring the acceleration of the particle. It is a hybrid observable because it contains a classical and a quantum part: the quantum part determines the orientation relative to the magnetic field as usual, while the classical part translates this information into a corresponding spacetime acceleration. (Clearly, we are not quantizing the electromagnetic field here such that the operator G must be of such a hybrid classical and quantum nature.) But thisĜ is seen differently by different systems. Since the four-acceleration is always orthogonal to the four-momentum [15], p µ · ∂ µ (B RF ν B ν RF ) = 0, the gradient is of the form G RF = (0, G) in the rest frame of the particle. As one can easily check, this implies that the eigenvalues of the operatorĜ = G µ ·σ µ are precisely ±| G| in the rest frame of the particle, corresponding precisely to the two opposite accelerations, as seen from the rest frame of the qubit. An expectation value ofĜ RF as seen from the qubit rest frame would correspond to the average acceleration experienced by the qubit ensemble.
However, Alice and Bob, who observe the deflection of the qubits in the Stern-Gerlach device from different states of motion (see Figure 5 for a schematic illustration), will see the resulting acceleration differently, namely as proportional to Λ ν µ G RF µ , where Λ is the Lorentz transformation between the qubit's and Alice's or Bob's frame. (Alice and Bob could measure the acceleration with some detector commonly used in particle physics.) Thus, Alice and Bob ∇B qubit Figure 5: Alice and Bob, being in different states of relative motion as seen by the injected qubits, see the qubits' acceleration and deflection differently. Bob, having a non-vanishing component along the Stern-Gerlach magnetic field direction will see 'spin up' and 'spin down' qubits deflected asymmetrically.
will see the qubit acceleration in the form G A,B = (G A,B 0 , G A,B ), i.e. not as a four-vector with only spatial components. And this is precisely expressed by the transformation (4.9),Ĝ RF → G A,B = S · G RF · S † , which, as can be verified easily, entails the transformed asymmetric eigenvalues G A,B ±| G A,B | as seen by Alice or Bob. But these two eigenvalues uniquely determine the acceleration as seen by either Alice or Bob. Similarly, a corresponding expectation value measures the average acceleration of the qubit ensemble as seen by either Alice or Bob.
The asymmetry corresponds to the fact that the surfaces of simultaneity will be different for Alice, Bob and the qubits. For instance, if Bob's state of motion has non-zero components in the direction of the qubit deflection, one can convince oneself that this necessarily implies that Bob will see the deflection of the qubits asymmetrically.
Finally, we note that the change of normalization (or non-unitarity of transformations) between different observers corresponds to the fact that what is a purely spatial rotation to one, may be seen as a combination of boost and rotation by another. For example, a rotation R ∈ SO(3) of the field gradient G µ within the rest frame of the qubit particle amounts to G RF → G RF = (0, R · G) which does not affect the eigenvalues ±| G|. However, if Alice and Bob move relative to the qubits, this transformation will not appear to them as G A,B → G A,B = (G A,B 0 , R · G A,B ) because of the relativity of simultaneity. Accordingly, for Alice and Bob the eigenvalues of the corresponding operator would not be preserved under this transformation. Consequently, what appears as a norm or eigenvalue preserving transformation to one reference frame, may constitute a norm and eigenvalue changing transformation for another.
The unusual 'non-unitary' behaviour and the multitude of Hilbert spaces is a consequence of the classical treatment of the type (i.e., state of motion) of the qubit. Nevertheless, within this framework, everything is consistent and has a clear physical interpretation within a welldefined physical approximation, namely the WKB limit of the Dirac theory. Clearly, for a more fundamental treatment one would have to consider more sophisticated quantum field theoretic tools where all 'types' of systems can be accommodated within one infinite dimensional Hilbert (or Fock) space. However, the symmetry group does not change under the approximation.
This example will be more thoroughly discussed in [8].

Summary and conclusions
We have shown that the Lorentz transformations emerge naturally in an information-theoretic scenario where two agents in distant laboratories try to synchronize their descriptions of local quantum physics by communication with physical objects. Our derivation does not assume any specific properties or even the existence of an underlying spacetime manifold (except for very basic assumptions like the possibility to send systems between observers, or a time order in the local laboratories) and arises from the informational relations between observers. But it rests on some physical background assumptions. Most importantly, our result relies on the existence of devices that measure different types of quantum systems universally, while satisfying physically meaningful consistency conditions on the physical measurement outcomes. The main idea is simple: if we have qubits that interact naturally, directly or indirectly, with all other quantum systems (as formalized in the "interaction graph", cf. Figure 3), then we can use these qubits to build measurement devices (like we can use electron spins to build a magnet in a Stern-Gerlach device). This allows to lift any description of qubit states to descriptions of all quantum systems, promoting the symmetry group of the qubit to the full symmetry group that relates different observers' descriptions. In the simplified case of stand-alone abstract quantum system, this yields the orthogonal group O(3); in the more realistic case where finitedimensional quantum systems are embedded into higher-dimensional operator algebras, this yields the orthochronous Lorentz group O + (3, 1). The resulting formalism correctly describes actual relativistic Stern-Gerlach measurements [7].
Our result raises several interesting questions. Can we obtain the full Poincaré group if we relax assumption (4.3), as suggested by (4.4)? Is there a more direct interpretation of the Lorentz transformations, acting on quantum observables, in terms of relativistic effects like time dilation etc. without assuming special relativity in the first place? What if we have a specific quantum field theory in more than 3 + 1 dimensions -do our operational assumptions or conclusions still apply to observers in such a universe? Finally, is this approach suitable for information-theoretic descriptions of other geometric properties like curvature?
Our operational approach is in spirit close to Einstein's original belief, viewing relations among observers as a primary and spacetime as a secondary concept. This view is also compatible with a recently developed geometric formulation of special and general relativity in terms of an observer space which highlights the observers' relations as fundamental [17]. The question then arises how far we can push this kind of reasoning towards a deeper information-theoretic understanding of spacetime. In this context, the present paper is supposed to provide a first step towards this goal: it shows that there is more than just peaceful coexistence [33] between quantum theory and relativity. Instead, the structure of one of these theories tightly constrains the structure of the other.