Quantum communication using a bounded-size quantum reference frame

Typical quantum communication schemes are such that to achieve perfect decoding the receiver must share a reference frame (RF) with the sender. Indeed, if the receiver only possesses a bounded-size quantum token of the sender's RF, then the decoding is imperfect, and we can describe this effect as a noisy quantum channel. We seek here to characterize the performance of such schemes, or equivalently, to determine the effective decoherence induced by having a bounded-size RF. We assume that the token is prepared in a special state that has particularly nice group-theoretic properties and that is near-optimal for transmitting information about the sender's frame. We present a decoding operation, which can be proven to be near-optimal in this case, and we demonstrate that there are two distinct ways of implementing it (corresponding to two distinct Kraus decompositions). In one, the receiver measures the orientation of the RF token and reorients the system appropriately. In the other, the receiver extracts the encoded information from the virtual subsystems that describe the relational degrees of freedom of the system and token. Finally, we provide explicit characterizations of these decoding schemes when the system is a single qubit and for three standard kinds of RF: a phase reference, a Cartesian frame (representing an orthogonal triad of spatial directions), and a reference direction (representing a single spatial direction).


I. INTRODUCTION
Many communication protocols implicitly require that the communicating parties share a reference frame [1].For instance, if one party, Alice, transmits qubits to another party, Bob, using spin-1/2 particles, the quantum state of the collection can only be recovered by Bob if he and Alice share a reference frame for orientation.Lacking such a shared reference frame (for instance, by lacking knowledge of the relation between their local reference frames) is equivalent to having a noisy channel; the density operator relative to Bob's local frame is the average over rotations of the density operator relative to Alice's local frame.For a single qubit, such an average over rotations yields complete decoherence -no information about the quantum state survives.Nonetheless, Alice and Bob can still achieve perfect classical and quantum communication by encoding the information into the rotationallyinvariant degrees of freedom of many qubits [2].Indeed, in the limit of large numbers, the cost of not sharing a reference frame is only logarithmic in the number of systems.Similarly, if Alice and Bob lack a phase reference, they can still encode classical and quantum information in phase-invariant states of composite systems [1].However, such communication schemes are technically challenging to implement because they make use of highly entangled states of many qubits [3,4].Such schemes will be referred to here as "calibration-free".
A more straightforward strategy for coping with the lack of a shared reference frame is for Alice and Bob to begin their communication protocol by setting up a shared reference frame, that is, they begin by calibrating or aligning their local reference frames.Thereafter Alice transmits her quantum systems normally.This strategy is illustrated in figure 1.The problem of aligning reference frames using finite communication resources has been well studied [1,5,6,7,8,9].If only finite communication resources are devoted to the task, then the token of Alice's reference frame that is transmitted to Bob will be of bounded size.Assuming the reference frame in question is associated with a continuous degree of freedom, this bound leads to a nonzero probability of error in the decoding of messages.Nonetheless, this scheme has two advantages over its calibration-free counterpart: (i) Alice does not need to implement any entangling operations to encode classical bits or logical qubits into the physical qubits, nor does Bob require such operations to decode (They may require entangling operations to prepare and measure the reference frame token, but the preparation and measurement they require is always the same and their effort does not scale with the size of the message); (ii) If Alice wishes to communicate a classical bit string or a string of logical qubit states, she can encode one logical bit or qubit per physical qubit, and Bob can decode one logical bit or qubit per physical qubit (in other words, the blocks of physical qubits into which they encode and decode their logical bits and qubits can be as small as one, unlike the calibration-free scheme).By virtue of (ii), Alice does not need to know the entire message string at the outset to achieve her optimal communication rate, nor does Bob need to store all of the b) S R FIG. 1: a) Alice describes the reference frame token R by the state |e e| and the system S by the state ρ.Bob, who is not correlated with Alice's reference frame, describes the joint state of R and S as the twirling of Alice's description, namely G(|e e| ⊗ ρ).b) We can consider Alice's act of adjoining the reference frame token to the system together with the twirling as an encoding operation E .Composing this with Bob's decoding operation R, the resulting channel can be described as an effective R • E .
systems coherently until he has received the entire sequence of physical qubits, whereas such capabilities are required to achieve the optimal rate of communication in the calibration-free scheme.We are therefore motivated to explore how well Alice can communicate quantum information to Bob after supplying him with a boundedsize token of her reference frame.
The general case we consider is that of a reference frame associated with a compact Lie group G.If the system is prepared in the state ρ and the reference frame token in the state |e relative to Alice's local reference frame, then relative to Bob's local reference frame, the pair is in the state where G averages over the collective action of the group and is termed the G-twirling operation (see figure 1).This state encodes ρ.We consider the case where the reference frame token is prepared in a particular state |e , suggested by previous investigations [9,10], which is near-optimal for transmitting information about the group element and which makes the mathematics particularly simple.We then determine how well Bob can reconstruct the state ρ.It turns out that the recovery (or decoding) operation R that is equal to the Hilbert-Schmidt adjoint of E, normalized to be trace-preserving, is provably near-optimal (in a sense we will define later).The composition of encoding and decoding yields an effective decoherence of the form (R • E)(ρ) = dg p(g)U S (g)ρU † S (g) , where is a probability distribution over the group with dg the group-invariant measure, U S and U R are unitary representations of G on the system S and the reference frame token R respectively.With our particular choice of reference state |e , we find that p(hgh −1 ) = p(g) for all h ∈ G, ensuring that R • E is a G-invariant map (it commutes with the action of every g ∈ G).The figure of merit relative to which the decoding operation is judged to be near-optimal is the entanglement fidelity.
We also demonstrate two distinct ways of implementing this decoding operation.The first is an obvious scheme: Bob estimates the orientation of the token relative to his local frame and then re-orients the system appropriately.We call this the "measure and re-orient" scheme.The second is less intuitive, but reveals more about the structure of the problem: Bob projects into the virtual subsystems that support the representation of the group induced by relative transformations of the system and token and implements an isometry that maps these onto a single Hilbert space.We call this the "extract from the relational subsystems" scheme.These relational subsystems are the places in the Hilbert space of the combined token and system where the quantum information associated with ρ is to be found.Their characterization is the key technical result of the article.The second scheme is also particularly interesting because, with a slight modification, it can yield a decoding that is probabilistically perfect, that is, one which sometimes fails but which yields a perfect decoding when it succeeds.
We work out the explicit form of the recovery operation R and the effective decoherence R•E for several interesting examples: (i) a phase reference; (ii) a Cartesian frame (representing an orthogonal triad of spatial directions) 1 ; (iii) a reference direction (representing a single spatial direction).In each case, we consider a system consisting of a single qubit.The explicit form of the decoherence map is actually quite simple in this case.Because R • E is a G-invariant map, it follows from the results of Ref. [11] that it is a sum of irreducible G-invariant maps called moments.But in the case of a qubit, there is only a single nontrivial moment -the G-twirling map G. Thus, we have where I is the identity map.We show that for our examples, p is inversely proportional to the size of the reference frame token, where the size of the reference frame is given by the quantum number of the highest irreducible representation ap-pearing in the state of the token.It is also shown that in probabilistically perfect decoding schemes, the probability of failure is inversely proportional to the size of the reference frame token.These results are specific to the special form of reference frame state that we consider here.
The idea that bounded-size quantum reference frames induce an effective decoherence is not new.The effect of a bounded-size clock, which is a kind of phase reference, has been considered previously by many authors [12,13,14,15] and that of a bounded-size directional frame has been considered by Poulin [16].Our results go beyond this work in several ways.For one, the case of a Cartesian frame, which is particularly significant given that it is representative of the general non-Abelian case, has not been examined before.More significantly, we consider many different sorts of reference frames within a unified framework and we provide insight into the structure of the problem.It should be noted that although our method can be applied to a system of arbitrary dimension, we here obtain explicit expressions for the effective decoherence only in the case of the simplest possible system: a qubit.We hope to provide a discussion of the foundational implications of modeling bounded-size reference frames by effective decoherence in a subsequent paper.
Finally, it is worth pointing out that the problem of communication in the presence of a bounded-size reference frame has interesting connections with a disparate set of topics in quantum information theory and the theory of quantum reference frames: Partially correlated reference frames.When some information is known about the relative orientation of Alice and Bob's local reference frames, they are said to share partially correlated reference frames.This is a resource that interpolates between having and lacking a shared reference frame.Its quality can be characterized by the probability distribution over the relative orientation -the more peaked the distribution, the better the correlation.What can be achieved with this resource is an interesting question that has only begun to be addressed.We gain some insight into the question in this article because the "measure and re-orient" implementation of the recovery operation begins with a reference frame alignment protocol [1,5,6,7,8,9] that leaves Alice and Bob holding partially correlated reference frames.
Programmable operations.There have been many investigations into the possibility of encoding an operation into a quantum state such that the state can subsequently be used to implement the operation on another system.If the system into which the operation is encoded is bounded in size, then it is known that one can only achieve an approximate version of the operation or a perfect version with non-unit probability [17,18,19].The token of the quantum reference frame in our communication protocol is an instance of a program that encodes the unitary that relates Alice's local reference frame to Bob's.Bob subsequently uses it to implement the in-verse of this unitary on the system.Our results therefore provide interesting examples of both approximate and unambiguous programmable operations.
Measures of the quality of a quantum reference frame.Our results also contribute to the project of quantifying the extent to which a quantum reference frame of bounded size can stand in for one of unbounded size [20,21,22], an important element of a resource theory of quantum reference frames.For instance, a measure of the strength of the effective decoherence associated with the bounded-size reference frame may serve as an operational measure of its quality.
Private channels.If no party besides Bob has a sample of Alice's RF, then Alice and Bob are said to possess a private shared reference frame.These have been shown to constitute a novel kind of key that is useful for private communication schemes [23].Our results establish a lower bound on the fidelity between input and output in private communication schemes that rely on Bob possessing a bounded-size token of Alice's reference frame.They also establish a bound on the probability of achieving perfect fidelity by post-selection.This cryptographic application provides a particularly useful perspective on Bob's recovery operation: the message is encoded in the relative orientation between the system and the token of Alice's reference frame in the same way that the plaintext in classical cryptography is encoded in the bit-wise parity of the cypher-text message and the key.Bob decodes by using the token of Alice's frame as a key.
Dense coding.As noted above in point (ii), the calibration-free communication scheme requires Alice to know the entire message to be sent prior to transmitting any systems to Bob if she is to achieve the maximum rate.In the communication scheme that first sets up a bounded-size reference frame, Alice can transmit to Bob the quantum token of her local reference frame prior to knowing anything about the message.Consequently, Alice can use the quantum channel at an early time, when it is perhaps cheaper, to enhance the communication capacity at a future time, when it is known what message is to be sent.Sending the frame token is therefore akin to establishing entanglement in a dense coding scheme [24].One difference, however, is that the fidelity of communication is never perfect for a bounded-size token whereas a single maximally-entangled state allows for perfect communication of one qubit.

A. Mathematical preliminaries
In this section, we present some formal mathematical tools that are useful for describing classical and quantum reference frames.We follow the notation of [1], to which we refer the reader for further details.Suppose Alice and Bob are considering a single quantum system described by a Hilbert space H. Let this system transform via a representation of a group G relative to some reference frame.We will restrict our attention to Lie groups that are compact, so that they possess a groupinvariant (Haar) measure dg, and act on H via a unitary representation U , ensuring that they are completely reducible [25].
Let g ∈ G be the group element that describes the passive transformation from Alice's to Bob's reference frame.Furthermore, suppose that g is completely unknown, i.e., that Alice's reference frame and Bob's are uncorrelated.It follows that if Alice prepares a system in the state ρ on H relative to her frame, then it is represented relative to Bob's frame by the state The action of the representation U of the (compact Lie) group G on H yields a very useful structure.It allows for a decomposition of the Hilbert space into a direct sum of charge sectors, labeled by an index q, where each charge sector carries an inequivalent representation U (q) of G.Each sector can be further decomposed into a tensor product of a subsystem M (q) carrying an irreducible representation (irrep) of G and a subsystem N (q) carrying a trivial representation of G.That is, Note that this tensor product does not correspond to the standard tensor product obtained by combining multiple qubits: it is virtual [26].The spaces M (q) and N (q) are therefore virtual subsystems.
Expressed in terms of this decomposition of the Hilbert space, the map G takes a particularly simple form, given by where Π (q) is the projection into the charge sector q, D M denotes the trace-preserving operation that takes every operator on the Hilbert space M to a constant times the identity operator on that space, and I N denotes the identity map over operators in the space N .A proof of this result is provided in Ref. [1].
Note that the operation G has the general form of decoherence.Whereas decoherence typically describes correlation with an environment to which one does not have access, in this case the decoherence describes correlation with a reference frame to which one does not have access [27].

II. ENCODING
Consider a communication scheme wherein Alice prepares a system R in a pure quantum state |e and sends it to Bob as a quantum sample of her reference frame, together with a system S (a collection of qubits for example) that is described by a quantum state ρ relative to her reference frame.Let R transform via the unitary representation U R of G, and S via the unitary representation U S .The lack of a shared reference frame between Alice and Bob implies that the transmitted composite RS is described relative to Bob's reference frame by the G-invariant state where G RS is the G-twirling operation of ( 6) for the representation This map E will be referred to as the encoding map.Note that its input space is B(H S ) (the bounded operators on H S ), while its output space is B(H R ⊗ H S ).It maps ρ to a G-invariant state of the composite RS.
It is useful to define the set of states which form the orbit under the representation U R of G of the fiducial state |e (associated with the identity element of the group).By expressing the G-twirling operation explicitly and making use of these, we can express the encoding operation as The encoding map clearly depends on the choice of the state |e for the reference frame.We turn to this choice now.

A. Quantum reference frames
We begin by considering what properties of a quantum state make it a good representative of a reference frame for the group G (the case wherein the reference frame is associated with a coset space will be considered in Section VI); for a more complete discussion, see [1,10].
States |g corresponding to different orientations of the reference frame must be distinct, so at the very least one requires that the the fiducial state |e is not invariant with respect to G or any subgroup thereof.To emulate a perfect reference frame for G, these states must in fact be perfectly distinguishable, where δ(g) is the delta-function on G defined by dg δ(g)f (g) = f (e) for any continuous function f of G, where e is the identity element in G.If the states {|g } of Eq. ( 10) satisfy these requirements, then U R is the left regular representation of G.In the case of a Lie group, the dimensionality of any system H R that carries the regular representation must necessarily be infinite.We refer to such an infinite-dimensional quantum RF as unbounded ; such systems and states were considered in Ref. [10].
If the Hilbert space dimensionality of the system R serving as a quantum RF is finite, then we say that the quantum RF is of bounded size.If the RF is associated with a Lie group, having a continuum of elements, then a bound on the size of the RF implies that the condition (12) cannot be satisfied precisely.In this case, a key question is: what state on R is the best approximation to a perfect reference frame?The answer will depend on the figure of merit for the task at hand, but we will make use of a generic construction [9,10] that illustrates the key features.
Suppose the representation U R reduces to a set of irreps where the tensor product is the one appearing in the decomposition ( 7) of H R and where I is the identity on R .We are interested in a special subset of these irreps, namely, the U (q) R that appear in the decomposition of U R a number of times greater than or equal to their dimension d q , i.e. those for which We denote the set of q that label such irreps by Q R and, in what follows, we will be restricting our attention to only these irreps.Also, for irreps q ∈ Q R , choose an arbitrary subspace R with dimension d q , i.e., with dimension equal to that of M (q) R .We now define a new Hilbert space HR as which is of dimension The state of R that we will use for our quantum RF is where {|q, m } is an arbitrary basis for M (q) , and {|φ q,m } an arbitrary basis for N (q) .Note that the orbit of |e under G has support in HR .The embedding R provides a way of embedding |e in the original Hilbert space H R , and in addition U R acts on the Hilbert space HR in the obvious way.If Q R contained all irreps of G, then U R would be the (left) regular representation, and for a Lie group the Hilbert space HR would be infinite dimensional.If the quantum RF is of bounded size, then a limited set of irreps appear in Q R .
We note that, for the problem of optimally encoding a reference frame relative to a maximum likelihood figure of merit, given a general Hilbert space H R the optimal states will not have this precise form [1,9,28,29].However, such optimal states do take the form of Eq. ( 17) when restricted to HR .These are the states of interest here.
The restriction to irreps having the special property of Eq. ( 14) is in fact critical for our analysis, because only in this case can we define a useful right action of G on the Hilbert space [10].Consider the representation V R of G defined by its action on the covariant set (10) as To obtain an explicit form for this right action in terms of the decomposition of Eq. ( 15), we make use of the fact that the state |e is maximally entangled across the virtual tensor products R .Thus, we have for any transformation where T denotes the transpose, * the complex conjugate, and we have made use of the fact that U R is unitary.Given that the complex conjugate of a representation U (q) R of G is also a representation of G (called the conjugate representation and denoted by U In contrast to U R , the representation V R acts on the subsystems N (q) R irreducibly according to the conjugate representation q * , and leaves the subsystems M (q) R invariant.Clearly, the two actions U R and V R commute.Furthermore, it is easy to verify that V R satisfies Eq. (18).
As the states of the reference frame are restricted to the Hilbert space HR , it is useful to consider our encoding operation E of Eq. ( 11) with fiducial state |e of Eq. ( 17) as a map from B(H S ) to B( HR ⊗ H S ).For the remainder of this paper, we consider the encoding map to be defined thus.
Finally, we note that the map E with the fiducial state |e chosen to be of the form ( 17) is unital.(Because the input and output spaces of E are of differing dimension, we define unital for such a trace-preserving map as one which maps the (normalized) completely mixed state to the completely mixed state.)This result is seen as where d S is the dimension of H S .Here, we have used the fact that the maximally-entangled states dq m=1 |q, m ⊗ |φ q,m in Eq. ( 17) have reduced density matrices on N (q) R that are proportional to the identity.

B. Relational subsystems
It is illustrative to investigate the action of the encoding map (11) (the fiducial state |e will be assumed to be given by Eq. ( 17) except in the final section of the article) and to explicitly identify the subsystems of HRS ≡ HR ⊗ H S into which the system's state is encoded.The details of this section require extensive use of the virtual tensor product structure of HRS induced by the unitary representation U RS of G, given in Eq. ( 7), as well as an application of the Stinespring theorem for covariant maps [31]; this section may be skipped on first reading.To facilitate this, we first state the main result of this section prior to our detailed investigation of the encoding map.
Main result: According to Eq. ( 7), the joint Hilbert space HRS can be decomposed under the representation where Q RS are the set of irreps q of G that appear in the decomposition of U RS .The encoding map (11) yields G-invariant density operators which, in terms of the decomposition (22), are block-diagonal in the irreps q ∈ Q RS and, within each block, have the form of a tensor product of the completely mixed state on the subsystem M (q) RS and some non-trivial state on the subsystem N (q) RS .Thus, the action of the encoding can be expressed as where RS is the identity operator on M (q) RS , and E (q) is a trace-decreasing map from states on H S to states on N (q) RS .We show below that, under the assumption that H S is an irrep of G, each of these encodings E (q) takes the form where I K (q) is the identity operator on a Hilbert space K (q) carrying an irrep q * of G, and A (q) : N (q) RS → K (q) ⊗ H S is a linear map satisfying A (q) † A (q) = I N (q) RS .In addition, each map A (q) takes a very simple form, which depends on the irrep q.
Specifically, there is a subset of irreps Q ok RS ⊂ Q RS such that, for q ∈ Q ok RS , the map A (q) is a bijective isometry, that is, a unitary; in these instances, the map E (q) can be inverted and ρ can be recovered perfectly.For q not in Q ok RS , the map A (q) is an isometry that is not surjective, i.e., it maps onto a proper subspace of K (q) ⊗ H S .The map E (q) is not invertible in these cases.
We can identify the relational degrees of freedom in which the message is encoded by investigating how relational transformations act on the Hilbert space HRS .The subsystems M (q) RS carry an irreducible representation of G corresponding to the collective action U RS and describe collective degrees of freedom.In contrast, the subsystems N (q) RS are relational.However, not all degrees of freedom in N (q) RS describe relations of the system S to the reference frame R; some of these describe relations among the parts of R (or among the parts of S if the latter are composite systems).We seek to identify, for each irrep q, the precise subsystem of N (q) RS into which the message state ρ is encoded.
The system Hilbert space H S carries a representation U S of G.If we act with G on the system but not on the RF, this will induce a relative transformation of the two; however, this action alone is not G-invariant.While it is possible to construct a G-invariant action of U S by using the techniques of Refs.[1,10], it is much more straightforward to make use of the right action V R of G defined in Eq. ( 18).This action commutes with the left action U R , and thus also commutes with the collective action U RS of G.By acting with V R (h) for h ∈ G on a state ρ RS = E(ρ) of the form (9), we have where we have used the invariance of the Haar measure.
The action of V R (h) on ρ RS yields another invariant state, but one which is now an encoding of the transformed state U S (h)ρU † S (h).Thus, V R (h) acts as a transformation of the relation between S and R. A map E satisfying Eq. ( 25) is called G-covariant.
As V R is a relational action, it acts on the subsystems N (q) RS in Eq. ( 22); we now decompose these subsystems according to the irreps of G under the action of V R , and in doing so identify the subsystems in which we find the image of ρ under the encoding map.
At this stage, we restrict our attention to the case where U S is an irrep of G, labelled q S .It appears straightforward (although with substantially more burdensome notation) to extend our results to the general case wherein this restriction is relaxed.Indeed, the U(1) example considered in Sec.IV provides evidence of the generality of our theorem.However, we do not consider the general case here.
Recall that the reference frame R has a Hilbert space given by Eq. ( 15), and the system's Hilbert space is H S = M (qS ) S .Thus, where (q ′ , q S ) → q denotes that the irreps q ′ and q S couple to the irrep q, V q ′ ,qS q is the multiplicity space for the irrep q in tensor representation U ⊗ U S , and where Q RS is the set of all irreps that are obtained by coupling some irrep q ′ ∈ Q R to q S .Comparing the expression above with Eq. ( 22), the subsystems N (q) RS are given by N (q) RS = q ′ ∈QR|(q ′ ,qS )→q We use the fact that if (q ′ , q S ) → q, then (q * , q S ) → (q ′ ) * [31], and that V q ′ ,qS q ≃ V q * ,qS (q ′ ) * [10].Thus, where each subsystem N (q ′ ) R on the right-hand side carries an irrep (q ′ ) * of G under the action V R .(We note that in the examples presented in the latter sections, with groups U (1) and SU (2), the subsystems V are trivial and can be ignored.) At this stage, we will make use of the G-covariance of the encoding map E, given by Eq. ( 25), to determine how the message is encoded into the relational subsystems N (q) RS .As the state E(ρ) is G-invariant under the action of U RS for any ρ, it can be expressed according to the Hilbert space decomposition (22) as where RS is the identity operator on M (q) RS and E (q) (ρ) is an (unnormalized) density operator on N (q) RS .This expression defines a set of trace-decreasing superoperators E (q) : B(H S ) → B( N (q) RS ); note that the latter can be naturally embedded in the full multiplicity spaces N (q) RS of the combined system.From Eq. ( 21), we see that the maps E (q) are also unital in that E (q) (I S ) is proportional (because E (q) is trace-decreasing) to the identity on N (q) RS .Also, as V R commutes with U RS , we have that each term . We now make use of the Stinespring theorem for covariant CP maps [30]; in particular, we use a form due to Keyl and Werner [31] for unital covariant CP maps.There exists another unitary representation W (q * ) of G on a space K (q) and an intertwiner (linear map) satisfying A (q) † A (q) = I N (q) RS with such that The form of Eq. ( 28) allows us to identify a suitable Stinespring extension.The representation V (q * ) R

acts on N (q)
RS through what appears to be (ignoring the limits on the sum q ′ ∈ Q R ) a tensor representation of an irrep q * with an irrep q S .Thus, we can choose our Stinespring extension in a minimal way such that W (q * ) acts on K (q) irreducibly as the irrep q * of G.The operators A (q) intertwine the representation V (q * ) R with the collective representation W (q * ) ⊗ U S on K (q) ⊗ H S .We now consider two cases: Case A: If q ∈ Q RS is such that, for all q ′ obtained via (q * , q S ) → (q ′ ) * then q ′ ∈ Q R , (i.e., Q R contains all of the irreps q ′ that one can obtain by (q * , q S ) → (q ′ ) * ), then the direct sum Hilbert space in Eq. ( 28) is given by where ≃ denotes that these spaces are unitarily equivalent; that is, the map A (q) is a bijective isometry and simply represents the Clebsch-Gordan transformation relating the tensor product of two irreps with the direct sum decomposition of N (q) RS given in (28).Let Q ok RS ⊆ Q RS denote the set of irreps satisfying this condition.
Case B: If, however, q ∈ Q RS is such that the condition of case A fails (i.e.Q R does not contain all of the irreps q ′ that one can obtain by coupling q * and q S ), then the intertwiner is no longer surjective.Rather, the intertwiner maps N (q) RS onto a proper subspace of K (q) ⊗ H S , specifically the subspace defined by the carrier space of the irreps (q ′ ) * , with q ′ ∈ Q R , obtained through the coupling of the irrep q * on K (q) with the irrep q S on H S .(This space is necessarily a proper subspace of K (q) ⊗ H S because, by the conditions of case B, Q R does not contain all irreps obtained in this coupling.)A set of basis states for this subspace can be calculated explicitly in any particular instance using the Clebsch-Gordan coefficients for the group G.
We now turn to the probabilities of each of these cases.We note that p q = Tr[Π q E(ρ)] = Tr[E (q) (ρ)] is the probability that the system is encoded into the irrep q.We now prove that, for the case where H S carries an irrep U S of G, this probability is independent of ρ.Using Eq. ( 32), we have where in the second line we have used the G-invariance of E (q) and in the third line we have used Eq.(31).Because H S is an irrep, it follows that dg U S (g)ρU S (g) −1 = I S /d S where d S = dimH S , and therefore p q is independent of ρ for all q.The SU(2) case, presented in Sec.V, provides an explicit example of this.We note that for the general case, where the system does not carry an irrep of G, then this probability can be state-dependent.For example, if H S is a direct sum of irreps q S , then dg U S (g)ρU S (g) −1 = qS Tr(ρΠ where Π (qS ) S is the projector onto the q S irrep of H S .In this case, for q ∈ Q ok RS (where A q is unitary), we have p q = d 2 q /D R , independent of ρ.However, for q / ∈ Q ok RS , the weight p q can depend on ρ.This occurs for the U(1) case, as seen explicitly in Sec.IV.
With each map E (q) now defined through Eq. ( 32), we can explicitly express E in Kraus operator form as E(ρ) = q,m,µ K q,m,µ ρK † q,m,µ , where and where |q, m is a basis for M (q) RS and |q, µ is a basis for K (q) .Finally, we point out a useful expression for E (q) (which applies regardless of whether H S carries an irrep of G).From Eq. ( 29), it is clear that Combining this with the expression for E in Eq. ( 9) and making use of Eq. ( 8), we obtain This form will be used frequently when working out explicit examples.

III. DECODING
In the communication protocol we are considering, Bob's task is to recover the quantum message ρ by implementing a decoding map R : B( HR ⊗ H S ) → B(H S ).
A useful recovery map to consider is the following where the adjoint for superoperators is defined relative to the Hilbert-Schmidt inner product, Tr(AE † We have therefore verified that R is a valid quantum operation that can be implemented deterministically.
Assuming that there is no prior information about the input state to E, the map D R E † is precisely the "approximate reversal" operation for E proposed by Barnum and Knill [32] which yields an error no more than twice that of the optimal reversal operation.The error here is defined in terms of the deviation from unity of the average entanglement fidelity for an arbitrary input ensemble.

A. "Measure and re-orient" implementation of decoding
Given a superoperator A, the adjoint A † is easily determined through a Kraus decomposition of A. Specifically, if (11) for the encoding map E provides one Kraus decomposition: the covariant set of operators {K(g), g ∈ G}, where K(g) = |g ⊗ U S (g).It follows that a covariant set of Kraus operators for E † is {K † (g), g ∈ G} where K † (g) = g| ⊗ U † S (g) and consequently From this expression, we see that one way in which R can be implemented is as follows: measure the covariant POVM {D R |g g|dg} on R, then implement the unitary U S (g −1 ) on S, and finally discard R and the measurement result g.We refer to this as the "measure and re-orient" implementation of the decoding map.(It is the adjoint of the "prepare and G-twirl" implementation of the encoding map.)So we see that the decoding map we are considering is in fact the most obvious recovery scheme one can imagine!Bob simply estimates the relative orientation between Alice's reference frame and his own by measuring how the sample R of Alice's RF is oriented, then reorients the system appropriately (i.e. in such a way that it is finally oriented relative to his RF in precisely the way that it was initially oriented relative to Alice's RF).One can view the system R as a cryptographic key or calibrating system that contains the information for how to recover the quantum state of S. It is noteworthy that this implementation of Bob's decoding map does not require any entangling operations between R and S. Bob can achieve it with local operations and classical communication (LOCC) between R and S. Because Bob does not need to possess S to implement the appropriate measurement on R, it follows that Alice can subsequently transmit an arbitrary number of systems and Bob can decode these with the same fidelity as the first.

Effective decoherence
Consider the action of the decoding map on states ρ RS = E(ρ), i.e., on states of the form of Eq. ( 11).After a measurement on R having outcome g ′ , followed by a transformation U S (g ′ ) −1 to system S, the reduced density operator on S is where the simplification occurs because dg is invariant.Note that the result is independent of the outcome g ′ .
It is straightforward to check that this state is normalized, as . This is precisely how a state ρ relative to Alice's frame would be redescribed relative to Bob's frame if their relative orientation g was known to be distributed according to the probability distribution p(g) a function of g is highly peaked around the identity group element e, then the only unitary that will contribute significantly in the integral will be the identity operation, and we will have R • E[ρ] ≃ ρ.It is the narrowness of the distribution | g|e | 2 , a measure of the quality of the quantum reference frame, that determines the degree to which one can recover the quantum information.
We see that for bounded-size samples of Alice's reference frame, the decoding map we have described achieves approximate error correction.Further on, we will show that the degree to which it deviates from perfect error correction is inversely proportional to the size of the quantum reference frame.

B. "Extract from the relational subsystems" implementation of decoding
Recall that the "measure and re-orient" implementation of the recovery operation R was inferred from the adjoint of the Kraus decomposition {|g ⊗ U S (g) | g ∈ G} of E. We exhibited a different Kraus decomposition of the encoding operation in Eq. (35).The adjoint of the latter provides a novel Kraus decomposition of R and therefore also a new way of implementing the recovery operation.We will refer to it as the "extract from the relational subsystems" implementation.We find that R = D R E † can be written as where we define as a map from B( N (q) RS ) to B(H S ).Recalling the form of the encoding map E (q) of Eq. ( 24), we see that R (q) = d q E (q) † .
This implementation of the decoding map R differs from the "measure and re-orient" scheme in that it requires joint (i.e.nonseparable) operations on R and S. Specifically, it is implemented via a joint unitary on RS followed by a trace on R.
Finally, we highlight another decomposition of R (q) that will be useful to us further on.It is the one obtained by taking the adjoint of Eq. ( 37), Effective decoherence Given Eqs. ( 23) and ( 41), the composition R • E can be written as Substituting the expressions for E (q) and R (q) in Eqs.(32) and (42), we obtain Tr K (q) A (q) A (q) † I K (q) ⊗ρ A (q) A (q) † .
(45) We now consider the two subsets of Q RS from Sec. II B. In Case A, where q ∈ Q ok RS , the intertwiner A (q) is a bijective isometry, and consequently A (q) A (q) † is the identity and R (q) • E (q) [ρ] = (d 2 q /D R )ρ.Therefore, in this case the quantum information is perfectly recovered by the decoding map.In Case B, however, P (q) = A (q) A (q) † is a non-trivial projection on K (q) ⊗ H S and the recovery is not perfect.We can express Eq.(45) as This is just an alternative Kraus decomposition of the effective decoherence map of Eq. (40).

C. Comparison of implementations
We have shown two very distinct ways of implementing one and the same decoding operation.If we describe the reference frame token R as an ancilla, then what we have is an example wherein a single map can be implemented either by a joint unitary followed by a trace on the ancilla, or by a measurement of the ancilla followed by a unitary rotation on the system that depends on the outcome of the measurement.The existence of many different implementations of an operation is familiar in quantum information theory.For instance, Griffiths and Niu have made use of a similar multiplicity of possibilities for the optimal eavesdropping strategies in quantum cryptography [33].
The multiplicity of ways of implementing a single operation is analogous to the multiplicity of mixtures that lead to the same density operator.Two Kraus decompositions of our G-invariant recovery operation differ in their transformation properties under the group: one is a G-covariant set of operators (a continuous set in the case of a Lie group) and the other is a discrete set of G-invariant operators.Similarly, a G-invariant density operator ρ on a finite-dimensional Hilbert space admits two sorts of convex decompositions: a spectral decomposition with a discrete number of G-invariant elements, and the decomposition induced by ρ-distortion of a Gcovariant POVM (continuous if G is a Lie group) [34].
Recognizing this multiplicity of convex decompositions and the fact that no particular decomposition is preferred has been important for resolving many conceptual confusions [35].Furthermore, each decomposition may yield important insights.In quantum optics, for example, a Poissonian mixture of number eigenstates is equivalent to a uniform mixture over coherent states with the same mean number but differing in phase.The decomposition into states with well-defined phase is particularly useful for making predictions about wave-like phenomena, such as interference experiments, whereas the number state decomposition is best for particle-like phenomena, such as determining number statistics [36].
Similarly, each of the two decompositions we have provided of our decoding operation provides some insight into our problem.The "measure and re-orient" scheme is clearly the most intuitive and demonstrates that joint operations on reference token and system are not necessary to implement our recovery map.The "extract from relational subsystems" scheme demonstrates that if Bob begins by measuring the irrep of the composite of R and S, he learns whether the state was in a "good" irrep or not and consequently whether or not he has achieved a perfect decoding.This sort of post-selectively perfect decoding operation is discussed in the following section.

D. Post-selectively perfect decoding
Thus far we have only judged decoding schemes by their average performance.It is also possible to say something about the best and worst case performance.The "measure and re-orient" scheme is not particularly interesting in this regard: one achieves precisely the same fidelity of recovery regardless of the outcome of the covariant measurement on the RF token, so that the best and worst case recoveries are equivalent to the average.On the other hand, in the "extract from the relational subsystems" scheme, we found that the fidelity of the recovery depends on the irrep of the composite of RF token and system into which the input state was encoded.Furthermore, given that the decoding operation was incoherent over these irreps, it is always possible to make a projective measurement that distinguishes these.Depending on the measurement outcome, one can achieve decodings with fidelities that are sometimes better and sometimes worse than the average.Indeed, by enhancing the "extract from the relational subsystems" scheme with such a measurement, Bob can achieve perfect decoding with some probability.Specifically, if he finds one of the "good" irreps, q ∈ Q ok RS , then E (q) is invertible and the decoding operation R (q) of Eq. ( 42) recovers the quantum message perfectly.(Of course, if he achieves one of the "bad" irreps, q / ∈ Q ok RS , then he achieves a decoding that is worse than the average.)Recalling Eq. ( 46) and making use of Eq. ( 16), the probability of perfect recovery is Such a decoding scheme achieves post-selectively perfect error correction [37].It is akin to achieving unambiguous discrimination of a set of nonorthogonal quantum states [38].
For this implementation of the decoding, note that Bob must be able to store the quantum token of Alice's reference frame coherently until the time when he receives the message systems.Another point worth noting: if Alice is sending a large number of systems and Bob wishes to implement probabilistically perfect error correction on some subset of them, he must wait until he has collected all of the systems in that subset.The reason is that he must perform a joint measurement on the composite of these and the RF token.Furthermore, after his measurement is complete, he has disrupted the state of the RF token and he can no longer achieve perfect error correction for any other systems.The tradeoffs involved in such postselectively perfect decoding schemes are an interesting topic for future research.
Finally, we can consider modifying the encoding operation rather than the decoding in a similar way.Note that if, immediately after implementing her encoding operation, Alice implements a projective measurement of the irrep of the composite of RF token and system, she can come to know whether a subsequent decoding operation will achieve a perfect recovery or not.In addition, if it happens that Alice has a classical description of the quantum message rather than merely having a sample, then she can prepare the quantum state many times and only initiate transmission when her measurement finds one of the "good" irreps.In this case, the quantum message is perfectly encoded into a pure G-invariant state of the composite of system and RF token.Such a scheme therefore achieves a relational encoding akin to the one presented in Bartlett et al. [2].The precise connection of our results to the latter encoding is an interesting topic for future research (as is the application of the mathematical tools developed here to the general problem of calibration-free communication schemes discussed in the introduction).

IV. EXAMPLE: PHASE REFERENCE
The quantum state of a harmonic oscillator is always referred to some phase reference [35].In this example, we consider using one quantum harmonic oscillator (a single mode) as a phase reference for another, and investigate the effect of bounding the maximum number N R of excitations in the phase reference.Specifically, consider the single-mode RF to be prepared in the bounded-size phase eigenstate where |n is the Fock state with n excitations.This state is of the form of our general state (17) for the case of G = U(1).For the system, we consider a qubit encoded in the two-dimensional subspace spanned by |0 and |1 .Note that the system we consider does not carry an irrep of U(1), and in fact U(1) has only one-dimensional irreps.
Because our main result concerning the representation of the encoding map, presented in Sec.II B, was only proven under the assumption that H S is an irrep, the U(1) example cannot be presented as a special case of this result.Nonetheless, we find the U(1) example to be in accord with the general result, suggesting that our theorem applies more generally.
For simplicity, we consider a system prepared in an arbitrary pure state Our results will directly extend to the mixed-state case via the linearity of convex combination.

A. Effective decoherence
The overlap of the RF state |e NR with its rotated version is Rotations in U(1) act on the qubit system state as Evaluating Eq. ( 40) then gives which also gives the normalization, we have where G here denotes the U(1)-twirling operation (the dephasing map).It follows that in the "measure and reorient" scheme for decoding, regardless of the outcome of the measurement, the reduced density operator is with probability N R /(N R +1) the state α|0 +β|1 , while with probability 1/(N R + 1) it is completely dephased in the |0 , |1 basis.The overall effect of encoding and decoding is to implement a partial dephasing.

B. Relational subsystems
Because the irreps of U( 1) are all one-dimensional, we have dim M (N ) RS = 1 and consequently, by Eq. ( 23), the encoding operation E may be expressed simply as E(ρ) = N E (N ) (ρ).By virtue of Eq. ( 37), each operation E (N ) (ρ) may in turn be expressed as which evaluates for different values of N as: The decoding operation has the form R = N R (N ) where R (N ) ∝ E (N ) † .One easily verifies that R (N ) maps |N, 0 to |0 and |N −1, 1 to |1 , so that (58) The probability of the outcome N = 0 is |α|2 /(N R + 1), of N = N R + 1 is |β| 2 /(N R + 1), and of each of the other outcomes is 1/(N R + 1).Weighting the decoded states R (N ) • E (N ) (ρ) by these probabilities, we can verify that Eq. ( 55) is recovered.
In this particular example, taking the adjoint of the encoding operation as one's recovery operation is actually optimal.The proof is as follows.For 0 < N < N R +1, the recovery operation is perfect and consequently optimal.Otherwise, the action of the encoding map is to measure the system in the {|0 , |1 } basis and update it to one of two orthogonal states [as can be inferred from Eq. ( 58)].It is a well-known result that the update map that maximizes the entanglement fidelity is simply the Lüders rule (or projection postulate) [39], and this is precisely what the composition of the encoding with the recovery operation achieves.
The fact that the optimal recovery operation can be achieved using a "measure and reorient" scheme shows that having the classical resource of partially correlated reference frames that is obtained by this scheme is just as good as having the quantum reference frame token, at least for the purpose of optimizing average-case performance in decoding.This is a surprising result because one might have expected the quantum resource to always do better.
Finally, by implementing a projective measurement of the total number and post-selecting on finding N = 0, N R + 1, it is clear that Bob can achieve perfect decoding.This occurs with probability V. EXAMPLE: CARTESIAN FRAME For a Cartesian frame, the relevant group is the rotation group. 2 The charge sectors (irreps) are labeled by a nonnegative integer or half-integer j, and the irreps are (2j + 1)-dimensional with the standard basis {|j, m , m = −j, . . ., j}.We bound the size of our reference frame token by bounding j.Recall that the fiducial state for the frame, Eq. ( 17), requires us to work in a subspace H ′ R ⊆ H R satisfying Eq. ( 14).In the Cartesian case, we are confined to j values such that dim N (j) R ≥ 2j + 1.We denote the largest such value by j R .(As an example, for an even number N of spin-1/2 particles, only the highest irrep, j = N/2, fails to satisfy Eq. ( 14), and consequently j R = N/2 − 1. See [1,9].)For simplicity, we restrict our attention to integer values of j R (similar results can be obtained if one also allows non-integer values).The fiducial state of the RF token is where The system is taken to be a spin-1 2 particle.Because this is an irrep of SU(2), the general results of Sec.II B apply.

A. Effective decoherence
We choose the following parametrization of SU(2), describing a rotation by angle ω about the unit vector n = (sin θ cos φ, sin θ sin φ, cos θ).Let where |e jR is the fiducial RF state of Eq. (60).Then where U (j) is the spin-j irrep of SU(2), and χ (j) (ω, θ, φ) = Tr[U (j) (ω, θ, φ)] are the characters of SU(2).These characters are independent of θ and φ.They are given by Using the following identity sin we find that For J < j R + 1/2, we note that the multiplicity space N (J) RS is unitarily equivalent to the tensor product of a spin-J and a spin-1/2 system coupled to total angular momentum J ± 1/2.That is, where K (J) carries an irrep J of SU(2) and ≃ denotes unitary equivalence.We explicitly define the bijective isometry A (J) : N (J) RS → K (J) ⊗ H S via its adjoint action on a basis for K (J) ⊗ H S as In terms of this new subsystem structure for the multiplicity spaces, we can express (82) as where |ψ HS is defined by Eq. ( 76).It follows that the encoded state for an irrep J where J Because A (J) is bijective, (specifically, because the set of states {A (J) † |J, M K (J) | 1 2 , s HS ; M = −J, . . ., J , s = ± 1 2 } are orthogonal) we find the qubit faithfully encoded into the relational subsystem whenever J < j R + 1 2 .However, for the result J = j R + 1 2 , the multiplicity space N (J) RS is exceptional; see (75).We cannot factorize N (J) RS as in Eq. ( 83).Nonetheless, we can still introduce a Hilbert space K (jR+ 1 2 ) which carries an irrep j R + 1 2 of SU (2) and in terms of it we can define an isometry 2 ) ⊗ H S , by modifying Eq. (84) to include only the b = − 1 2 term in the sum.This isometry is simply not surjective.It follows that the set of states } are no longer orthogonal.Therefore, the map is no longer invertible.The action of A (jR+ 1 2 ) † can be viewed as a projection of uncoupled states on K (jR+ 1 2 ) ⊗ H S onto the subspace of states which couple to total angular momentum j R .
The probability assigned to each irrep J is We note that these probabilities satisfy The decoding map within each irrep J takes the form For J < j R + 1 2 , Eq. ( 86) gives For J = j R + 1 2 , we make use of Eq. ( 87) and find where G is the SU(2)-twirling operation (complete decoherence).Averaging over the irreps with the weights given in Eq. ( 88), we find the decoded state to be thereby verifying that Eq. ( 71) is recovered.We note that our recovery operation is perfect in the non-exceptional irreps and therefore optimal there.In the exceptional irrep, the error incurred in recovery is at most twice that of the optimal recovery, as described in Sec.III, by virtue of being of the form of Barnum and Knill's approximate reversal.We leave it as an open problem to to prove whether this recovery is in fact optimal, or if not, to identify what form an optimal recovery map would take.
Post-selectively perfect decoding is achieved if Bob implements a projective measurement that distinguishes the irreps and obtains J = j R + 1 2 .By Eq. (88) and Eq.(61) (or by Eq. (47) directly), we see that the probability for this to occur is which is of the same form as Eq.(37).To see that this decomposition exists, simply express Eq. ( 97) as E(ρ) = G(|e jR e jR |⊗ρ) where G is the SU(2)-twirling operation.Then, using Eq. ( 8) and Eq. ( 106), we have RS , but in this case N (J) RS = C, so that it maps density operators to scalars.Specifically, We see that the encoding operation in this case can be described as follows: after adjoining the RF token to the system, destructively measure the total angular momentum J 2 of the composite and upon obtaining quantum number J, reprepare the system in the associated SU(2)invariant state d −1 J I M (J)
The decoding operation defined in Eq. ( 98) is clearly proportional to the Hilbert-Schmidt adjoint of the encoding operation.Consequently, it admits a decomposition into irreps via the adjoints of Eq. (107), namely, R(ρ RS ) = where R (J) ∝ E (J) † is a map from C to B(H S ).This is of the same form as Eq.(41).To determine R (J) , we calculate the adjoint of Eq. ( 108) and determine the normalization by requiring that Tr[R (J) (1)] = 1 for all J.We obtain where p ∈ C. Except for the constant of proportionality, this has the form of Eq. (43).It evaluates to [which one could also infer by taking the adjoint of Eq. ( 110)].Consequently, the decoding operation may be described as follows: destructively measure the total angular momentum (squared), J 2 , on the composite of RF token and system and upon obtaining quantum number j R ± 1 2 , reprepare the system in one of the two states in Eq. ( 113).
The composition of encoding and decoding yields in agreement with Eq. ( 105).We note that this coincides with the result obtained by Poulin [16].This recovery operation is of the form of the approximate reversal operation described by Barnum and Knill [32] and is therefore near-optimal in the sense described in Sec.III.Although it is not itself the optimal recovery map, the latter is easy to find in this example and we do so presently.
Given that the only pure states of the system that one can hope to reconstruct in the limit of an unbounded RF token are the J z eigenstates, denoted here by |0 0| and |1 1|, it is natural to take as our figure of merit the average input-output fidelity equally weighted over these two input states (because then one can achieve fidelity 1 in the limit of an unbounded RF token).The only information about the state of the system that is encoded in E(ρ) is encoded in the relative weights of its two SU(2)-invariant terms.Consequently, the optimal decoding operation must consist of a determination of J followed by a repreparation of the system state.
We make use of previous work on the optimal estimation of the relative angle between a spin-1/2 system and a spin-j R reference frame [23].These results show that a measurement of J 2 on the composite is in fact optimal for estimating whether the system state was |0 0| or |1 1| given a uniform prior over the two.It is also shown there that the posterior probabilities one ought to assign to these two states upon obtaining outcomes j R + 1 2 and j R − In order to optimize the fidelity, one must reprepare the state that is most likely given the outcome, so that one should reprepare |0 0| given the '+' outcome and |1 1| given the '−' outcome.R falls short of this optimal recovery because, by Eq. (113), it does not reprepare |0 0| given the '+' outcome; instead, it prepares a mixed state reflecting Bob's knowledge of the state given the measurement outcome.Finally, we note that there is no possibility for postselectively perfect recovery of U(ρ) from E(ρ).Both irreps, j R ±1/2, encode the state of the system imperfectly.
[B]) = Tr(E[A]B).The map R is completely positive and linear, because the superoperator adjoint preserves these features.It is also trace-preserving.To see this fact, observe that E(I S ) = (I R /D R ) ⊗ I S , where I R = D R dg|g g| is the identity operator on HR ; consequently, for ρ RS ∈ B( HR ⊗ H S ), we have Tr[R(ρ RS )] = Tr[ρ RS R † (I S )] = Tr[ρ RS ].