The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations

If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state's ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state psi relative to the symmetry group G are completely specified by the characteristic function of the state, defined as chi_psi(g)=where g\in G and U is the unitary representation of interest. For a symmetry described by a compact Lie group G, we show that two pure states can be reversibly interconverted one to the other by symmetric operations if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. Characteristic functions also allow us to easily identify the conditions for one pure state to be converted to another by symmetric operations (in general irreversibly) for the various paradigms of single-copy transformations: deterministic, state-to-ensemble, stochastic and catalyzed.

interest, the dynamics are sufficiently complicated that one cannot hope to characterize their evolution completely, whereas by appealing to the symmetries of the dynamical laws one can easily infer many useful results. One of the best known examples of such a result is Noether's theorem, according to which a differentiable symmetry of the Hamiltonian or action entails a conservation law (See, e.g. [1]). But there are innumerable results of this sort; symmetry arguments have broad applicability across many fields of physics.
We are interested in determining all the consequences of a symmetry of the dynamics in quantum theory. To find these consequences we ask the following question: Given two quantum states, ρ and σ, does there exist a time evolution with the given symmetry such that under this time evolution the first state evolves to the second?
Suppose, for instance, that the symmetry under consideration is rotational symmetry. Clearly, rotationallyinvariant time evolutions cannot take a rotationallysymmetric state to one that breaks the rotational symmetry. So to answer these types of questions we need to know the extent to which each of the two states breaks the rotational symmetry. It is intuitively clear that there are many different ways in which a quantum state may be asymmetric. For instance, consider a spin-1/2 particle with spin in theẑ direction and another with spin in thex direction. Neither is invariant under the full rotation group, but because they point in different directions, they break the rotational symmetry differently. Furthermore, it is intuitively clear that asymmetry must be quantifiable. For instance, the precision with which one can specify a direction in space, a measure of rotational asymmetry, varies with the quantum state one uses to do so.
We will say that two states have exactly the same asymmetry properties (with respect to a given symmetry group) if there exists a symmetric time evolution which transforms the first state to the second and a symmetric time evolution which transforms the second state to the first. Thus, the symmetric operations define equivalence classes of states and the asymmetry properties of a state are precisely those that are necessary and sufficient to determine its equivalence class. If the symmetry in question is associated with a representation of the group G, we call the equivalence relation G-equivalence. We will consider the case of arbitrary compact Lie groups and finite groups.
The above definition of asymmetry properties is based on the intuition that asymmetry is something which cannot be generated by symmetric time evolutions. We call this the constrained-dynamical perspective.
However, one can also take an information-theoretic perspective on how to define the asymmetry properties of a state. Recall that a quantum state breaks a symmetry, say rotational symmetry, if for some non-trivial rotations, the rotated version of the state is not the same as the state itself, i.e. they are distinguishable. In this case, the ensemble of states corresponding to the orbit of the state under rotations can act as an encoding when the message to be encoded is an element of the rotation group.
To understand better the information-theoretic point of view, consider the following scenario: Suppose Alice wants to inform Bob about a direction in space. She must prepare a quantum system specifying the direction and send it to Bob. For example, to send a direction in a plane she may prepare a number of photons polarized in that direction. Clearly to transmit more information, Alice should prepare the quantum system in a state which sharply specifies the chosen direction. Such a state should break the rotational symmetry in the desired direction as much as possible. Again the relevant property of the state which determines its quality as a pointer can be called its asymmetry. This example suggests that the information-theoretic point of view should be relevant for the study of asymmetry. We will show that these two approaches to the notion of asymmetry, the constraineddynamical and the information-theoretic, provide equivalent characterizations of asymmetry. It follows that one can exploit the machinery of information theory for the study of asymmetry and for finding the consequences of symmetry of dynamics. We will present examples of these consequences in future work. In this paper, we will find the characterization of the G-equivalence classes of pure states using both the constrained-dynamical and the information-theoretic approaches and we will show how these two characterizations are in fact equivalent via the Fourier transform.
In the above scenario the quantum system which is sent to Bob to transfer information about direction is called a quantum reference frame (See [2] for a review of this topic). The theory of quantum reference frames deals with the problem of using quantum systems to transfer information, such as a direction in space, which are unspeakable, i.e. cannot be transferred by sending a sequence of 0s and 1s if two agents do not have access to some shared background reference frame. In other words, unspeakable information can only be encoded in particular degrees of freedom. For example, information about a direction in space cannot be encoded in degrees of freedom that transform trivially under rotations.
Therefore this example suggests that the study of asymmetry is not only useful to learn about the consequences of symmetries of dynamics but it is also useful for the study of quantum reference frames. The relevant property of the state which specifies how well it can act as quantum reference frame is the asymmetry of the state. Indeed, in previous work, the asymmetry has been called the frameness of the state [3,4]. Therefore all the results about the manipulation of reference frames and their frameness are in fact results about the asymmetry of states. In particular [3] presents a systematic study of the manipulations of pure state asymmetry for groups U (1) and Z 2 and also presents some partial results for the case of SO(3). In the present paper, using a different approach based on characterizing the equivalence classes of asymmetries of pure states, we are able to generalize the results in [3] significantly and to extend their scope to arbitrary compact Lie groups and finite groups.

The resource theory point of view
We can think of the study of asymmetry as a resource theory. Any resource theory is specified by a convex set of free states and a semi-group of free transformations (which are required to map the set of free states to itself). Any non-free state is called a resource. The resource theory is the study of manipulations of resources under the free transformations. As we will explain, there are several types of questions and arguments which are relevant for all resource theories and so this point of view can help to achieve a better understanding of a specific resource theory by emphasizing its analogies with other resource theories.
A well-known example of a resource theory is the theory of entanglement The free transformations in case are those which can be implemented by local operations and classical communications (LOCC). The set of free states is the set of unentangled states. This set is closed under LOCC, i.e. an unentangled state cannot be transformed to an entangled one via LOCC [5]. More generally, given two quantum states one cannot necessarily transform the first one to the second with LOCC. Here the relevant properties of the states which determine whether such a transformation is possible or not are their entanglement properties. In the case of pure bipartite states it is a wellknown fact that the entanglement properties of a state are uniquely specified by its Schmidt coefficients [5]. For example, the Nielsen theorem provides the necessary and sufficient condition for the existence of LOCC operations which transform one given state to another in terms of their Schmidt coefficients [6]. Entangled states are also a resource in the sense that they can be used to implement tasks that re impossible by LOCC and unentangled states alone. For example, one can use entangled states for teleportation, which can be interpreted as consuming a resource (entanglement) to simulate a non-free transformation (a quantum channel) via free transformations (LOCC).
Similarly, we can think of the study of asymmetry relative to a given group G as a resource theory. In this resource theory the time evolutions which respect the symmetry G (G-covariant time evolutions) are free transformations and the states which do not break the symmetry (G-invariant states) are the free states. This is a consistent choice because G-covariant time evolutions form a semi-group and the set of G-invariant states is mapped to itself under G-covariant time evolutions. Similarly to entanglement theory, a resource (an asymmetric state) can be used to simulate a non-free transformation (non-G-covariant time evolution) via a free transformation (Gcovariant time evolution).
In the resource theory of asymmetry, we seek to classify different types of resources and to find the rules governing their manipulations. For every question in entanglement theory, it is useful to ask whether there is an analogous question in the resource theory of asymmetry. For example in this paper we will show that all the asymmetry properties of a pure state ψ relative to the group G and the unitary representation {U (g), g ∈ G} are specified by its characteristic function χ ψ (g) ≡ ψ|U (g)|ψ . This is analogous to how all the entanglement properties of a pure bipartite state are specified by its Schmidt coefficients.

Outline
We now summarize the structure of the paper. In section II we review some elementary concepts. We also formally define G-equivalence classes of states. Appendix A includes a discussion about the situations where the input and output Hilbert space of a time evolution are different. In section III, we introduce the idea of two dual points of view to asymmetry, constrained-dynamical and information-theoretic. We also show how these two dual points of view arise naturally in the study of quantum reference frames. In section IV, we define the notion of unitary G-equivalence, another equivalence relation over states that is slightly stronger than G-equivalence. Using the constrained-dynamical and information-theoretic perspectives, we find two different ways to characterize the unitary G-equivalence classes of states: the characteristic function and the reduction to the irreps. In section V, we show that these two different characterization are in fact two different representations of the same object, the reduction of the state to the associative algebra and that they can be transformed one to the other via Fourier and inverse Fourier transforms.
We also show that both the amplitude and the phase of the characteristic function are important for specifying the asymmetry of a state. We outline several nice mathematical properties of the characteristic function of a state, properties which make it particularly useful for the study of asymmetry of pure states. We also explain more about characteristic functions and their connection with the classical characteristic function of probability distributions in appendix B. In section VI, we present our main result, the characterization of the G-equivalence classes. Specifically, we show that for compact Lie groups, the G-equivalence class of a state is uniquely specified by its characteristic function up to a 1-dimensional representation of the group. In the important case of semisimple Lie groups, we show that it is uniquely specified by the characteristic function alone. Finally, in section VII we study the problem of approximate transformations in which one state should be transformed to a state that is close to (but not necessarily exactly equal to) a second. The proofs of this section are presented in appendix C.
In part II, we apply the techniques developed in part I to find the complete set of selection rules for pure states under deterministic and stochastic single-copy operations.That is, we determine the necessary and sufficient conditions under which one pure state can be converted to another by a deterministic G-covariant time evolution, and we determine the necessary and sufficient conditions for this conversion to be achieved by a stochastic G-covariant time evolution. We also determine the necessary and sufficient conditions for asymptotic interconversion, i.e. for an arbitrary number of copies of one pure state to be converted to an arbitrary number of copies of another, and we determine the maximum rate at which this can be achieved. This rate provides a useful measure of asymmetry. These results also represent a significant generalization of previous work [3] and underline the significance of our characterization of the Gequivalence classes.

A. Symmetric operations
According to a well-known theorem by Wigner, any symmetry transformation of a system is represented by a unitary or an anti-unitary operator on the Hilbert space of the system [7]. In this paper, we do not consider symmetry transformations, such as time-reversal, that are represented by anti-unitary operators. Therefore, we assume that for every group of symmetry transformations of interest and every physical system, there is an associated unitary representation of the group on the Hilbert space of the system. This representation is given as part of the specification of the physical system.
We denote a specific representation of G by the set of unitaries {U (g), ∀g ∈ G}. The representation of the symmetry group on composite systems is the collective representation: if the unitary representations of a symmetry transformation on the systems with the Hilbert spaces H A and H B are U A and U B respectively, then the unitary representation of that symmetry transformation on the Hilbert space of the composite system We say that a time evolution is G-covariant if it commutes with all symmetry transformations in the group G, that is, for any initial state and any symmetry transformation, the final state is independent of the order in which the symmetry transformation and the time evolution are applied. We will sometimes refer to an operation that is G-covariant as a symmetric operation. (It is important not to confuse symmetry transformations, which correspond to a particular group action, with symmetric transformations, which commute with all group actions.) We provide the rigorous form of the notion of G-covariance first for closed system evolutions and then for open system evolutions.
Closed system dynamics are described by unitary operators over the Hilbert space. However, noting that the global phase of a vector in Hilbert space has no physi-cal significance, it is useful to describe the dynamics in terms of its effect on density operators (every parameter of which has physical significance). Closed system dynamics are then described by linear maps V on the operator space that are of the form V[ρ] = V ρV † , where V is a unitary operator. A closed system dynamics associated with the unitary V is G-covariant if In other words, the map V commutes with every element of the (superoperator) representation of the group {U(g) : g ∈ G}. This implies that where ω(g) is a phase factor that can easily be shown to be a 1-dimensional representation of the group. In the case of finite dimensional Hilbert spaces (which is the case under consideration in this paper), we can argue that ω(g) = 1 if the closed system dynamics is required to be continuous and symmetric at all times (in contrast to requiring only that the effective operation from initial to final time be symmetric) [8]. This argument justifies the common definition in the literature of when a closed system dynamics respects the symmetry, namely, when ∀g ∈ G : V U (g) = U (g)V. (2.4) We call any unitary V which satisfies this property a G-invariant unitary because ∀g ∈ G : U (g) † V U (g) = V . Clearly, if a Hamiltonian is G-invariant then all the unitaries it generates are G-invariant. Finally, note that if V is an isometry rather than a unitary, then it is said to be G-invariant if ∀g ∈ G : V U in (g) = U out (g)V , where U in (g) and U out (g) are the representations of the group on the input and output spaces of the isometry. In general, a system might be open, i.e., it may interact with an environment. In this case, the time evolution cannot be described by the Hamiltonian of the system alone. Rather, to describe the time evolution we need the Hamiltonian of system and environment together. In the study of open systems we usually restrict our attention to the situations where the initial state of system and environment are uncorrelated. (When the system and environment are initially correlated the final state of the system is not even necessarily a function of its initial state.) To describe the most general form of time evolution in which the system and environment are initially uncorrelated we use quantum operations. Mathematically, a deterministic quantum operation E is a completely positive, trace preserving, linear map from B(H in ) to B(H out ), respectively, the bounded operators on the input Hilbert space H in and the output Hilbert space H out 1 . After a time evolution described by quantum operation E, the initial state ρ evolves to the final state E(ρ). Note that a general quantum operation may have input and output spaces that are distinct. This possibility is useful for describing transformations wherein the system of interest may grow (by incorporating into its definition parts of the environment) or shrink (by having some of its parts incorporated into the environment).
We now state the conditions for a general quantum operation (which may represent open or closed system dynamics) to be G-covariant.
5) where {U in (g)} and {U out (g)} are the representations of G on the input and output Hilbert spaces of E.
We use the notation ρ G−cov − −−− → σ to show that state ρ can be transformed to state σ under a G-covariant time evolution.
If the input and output spaces are equivalent, so that E is an automorphism, then the condition of G-covariance can be expressed as (2.6) or equivalently, where U(g)[·] = U (g)(·)U † (g). As we demonstrate in appendix A, any G-covariant operation for which the input and output Hilbert spaces are different can always be modeled by one wherein the input and output Hilbert spaces are the same. The reason is that the input and output Hilbert spaces can always be taken to be two different sectors of a single larger Hilbert space, H in H out , and any operation from B(H in ) to B(H out ) that is G-covariant relative to the representations {U in (g)} and {U out (g)} can always be extended to an operation on B(H in H out ) that is G-covariant relative to the representation {U in (g) U out (g)}. Similarly, any G-invariant isometry (a reversible operation where the input and output Hilbert spaces may differ) can always be modeled by a G-invariant unitary (where the input and output Hilbert spaces are the same). Again, this is shown in appendix A. It follows therefore that without loss of generality, we can restrict our attention in the rest of this paper to G-covariant operations where the input and output spaces are the same.
Clearly, G-covariant quantum operations include those induced by G-invariant unitaries, that is, operations of the form V(·) = V (·)V † where ∀g ∈ G : [V, U (g)] = 0. As another example, consider an operation of the form K ≡ K dhU(k), where K is a subgroup of G and dk is the uniform measure over K. We refer to this as the uniform twirling over K 2 . The uniform twirling over any normal subgroup of G is a G-covariant operation. First, recall that if K is a normal subgroup of G then ∀g ∈ G : (2.8) and consequently that K is G-covariant. In particular any group is the normal subgroup of itself, therefore uniform twirling over any group G is a G-covariant time evolution.
Furthermore, if we couple the object system with an environment using a Hamiltonian which has the symmetry G and if the environment is initially uncorrelated with the system and prepared in a state that is G-invariant, and finally some proper subsystem is discarded, then the total effect of this time evolution is described by a Gcovariant quantum operation. (Intuitively this is clear, because there is nothing in such a dynamics that can break the symmetry.) As it turns out, every G-covariant quantum operation can in fact be realized in this way, i.e. by first coupling the system to an uncorrelated environment in a G-invariant state via a G-invariant unitary and secondly discarding a proper subsystem of the total system. This is a consequence of a version of Stinespring's dilation theorem applied to G-covariant operations [9]. This result provides an operational prescription for realizing every such operation. U(1) example: For concreteness, it is worth examining a specific example of symmetric operations, namely, those that are covariant under a unitary representation of the U(1) group. U (1)-covariant quantum operations are particularly interesting because of their relevance in quantum optics and the problem of synchronizing clocks (For more discussion see [2] ). Consider a Harmonic oscillator whose Hilbert space is spanned by the orthonormal basis {|n, α : 0 ≤ n} with the number operatorN such thatN |n, α = n|n, α where n is integer and α labels possible degeneracies. Then the operator which shifts this oscillator in its cycle by phase θ is exp (iθN ). For example this operator transforms the coherent state |γ to |e iθ γ . Now clearly the set of unitaries {exp (iθN ) : θ ∈ (0, 2π]} forms a representation of U(1). U(1)-invariant states are those which commute with all elements of the set {exp (iθN ) : θ ∈ (0, 2π]} and so they should commute withN . Therefore in general the only pure invariant states are eigenvectors ofN . So in the particular case where there is no degeneracy (and so we can drop index α) invariant states are diagonal in the {|n : 0 ≤ n} basis and pure invariant states are all the elements of the basis {|n : 0 ≤ n}. In this particular case, all G-invariant unitaries should also be diagonal in {|n : 0 ≤ n} basis and so they are of the form These U(1)-invariant unitaries all commute with each other. This happens because we have assumed there is no multiplicity. In general when eigenvalues of number operator are degenerate U(1)-invariant unitaries do not necessarily commute with each other and can have more complicated structure. Clearly using these U(1)invariant unitaries we cannot transform one arbitrary state to another. For example we cannot transform |0 to (|0 + |1 )/ √ 2: The first state is a symmetric state while the second has some asymmetry.

B. Symmetries of states
A complete specification of the asymmetry properties of a state includes a specification of the state's symmetries (indeed, this can be considered to be a condition that must be satisfied by any proposed specification of the asymmetry properties). To be precise, the symmetries of a state are simply the elements of its symmetry subgroup, which is defined as follows: The symmetry subgroup of a state ρ relative to the group G, denoted Sym G (ρ), is the subgroup of G under which ρ is invariant, (2.11) The set Sym G (ρ) is clearly a subgroup of G because if g 1 ∈ Sym G (ρ) and g 2 ∈ Sym G (ρ), then U(g 2 g 1 )[ρ] = U(g 2 ) • U(g 1 )[ρ] = ρ and consequently g 2 g 1 ∈ Sym G (ρ). If the symmetry subgroup contains only the identity element, it is said to be trivial. In this case, it is often said that the state has no symmetries (meaning no nontrivial symmetries). If the symmetry subgroup of a state ρ is the entire group G, so that it is invariant under all symmetry transformations g ∈ G, then we say that the state is G-invariant. U(1) example. In the case considered earlier of the unitary representation e −iθN of the group U(1), any eigenstate of the number operatorN is U(1)-invariant, while the coherent superposition |0 + |1 has no symmetries. The state |0 + |2 has a nontrivial symmetry subgroup because it is invariant under a π phase shift.
Using the concept of a state's symmetries, we can already express a consequence of symmetric time evolutions, namely, that every symmetry of the initial state is a symmetry of the final state.
In particular, therefore, one cannot generate an asymmetric state starting from a symmetric one, and one cannot transform a state with one kind of asymmetry to a state with another. For instance, rotationally-invariant time evolutions cannot transform a spin pointing alonĝ z to one pointing alongx. This result can be understood as a cognate of Curie's principle, which states that symmetric causes cannot have asymmetric effects [10].

C. G-equivalence classes of states under symmetric operations
The first step in characterizing asymmetry is to specify when two states have the same asymmetry. We stipulate that this is the case when there exists both a symmetric time evolution that transforms the first state to the second and one that transforms the second state to the first. In this case, we say that the pair of states can be reversibly interconverted one to the other by symmetric operations. This defines an equivalence relation among states.
Definition 4 (G-equivalence of states) Two states, ρ and σ are said to be G-equivalent if and only if they are reversibly interconvertible by G-covariant operations, i.e., there exists a quantum operation E such that ∀g ∈ G : [E, U(g)] = 0, and E[ρ] = σ, (2.13) and there exists a quantum operation F such that ∀g ∈ G : [F, U(g)] = 0, and F[σ] = ρ. (2.14) (Using the notation we introduced in section II A, ρ and σ are G-equivalent iff ρ A complete specification of the G-asymmetry properties of a state is achieved by specifying its G-equivalence class. For instance, by Proposition 3 it is clear that if two states are G-equivalent then they have all the same symmetries. As another example, if we want to know whether there exists a one-way deterministic or stochastic symmetric transformation from one given state to another, all we need to know is the G-equivalence class of the two states; if there exists a symmetric transformation from one member of class I to one member of class II, then there exists a symmetric transformation from every member of class I to every member of class II.

III. TWO POINTS OF VIEW TO THE MANIPULATION OF ASYMMETRY
We pointed out in the introduction that there are two different perspectives on the manipulation of asymmetry: the constrained-dynamical perspective and the information-theoretic perspective. In this section, we discuss their interpretation in more detail.

A. Contrasting the constrained-dynamical and information-theoretic perspectives
In the constrained-dynamical point of view, we characterized the asymmetry properties of a state as those features that are required to determine whether any pair of states are reversibly interconvertible by symmetric operations.
It seems natural in this point of view, to use dynamical concepts to describe and study asymmetry. For example if the symmetry group under consideration is the rotation group, then we may use angular momentum to describe asymmetry: we know that if the expectation value of any component of the angular momentum is nonzero then the state necessarily breaks the rotational symmetry and so is asymmetric. Moreover according to Noether's theorem, in an isotropic closed time evolution every component of the angular momentum is conserved. We can generalize this result to symmetric reversible transformations on open systems using a Carnot style of argument -in a reversible transformation the environment cannot be a source of angular momentum and therefore if a transformation can be achieved reversibly on the system alone, then it must conserve all components of angular momentum (on pain of allowing a cycle that generates arbitrary amounts of angular momentum). It follows that the angular momentum is a function of the G-equivalence class. Clearly, the dynamical concepts provide a useful framework for describing asymmetry.
On the other hand, as we pointed out in the introduction, information-theoretic concepts are also useful for the study of asymmetry. Consider the example in which one of two distant parties is trying to communicate a secretly chosen direction to the other party by sending a quantum system. The quantum system is prepared such that the receiver can learn about the chosen direction by performing some measurement on it. This is in fact a specific case of a more general set of communication protocols in which one chooses a message g ∈ G according to a measure over the group and then sends the state U(g)[ρ] where ρ is some fixed state. We call any set of states of the form {U(g)[ρ] | g ∈ G} a covariant set.
We now argue that one can reasonably define the asymmetry properties of a state ρ as those properties that determine the effectiveness of using the covariant set {U(g)[ρ] : g ∈ G} to communicate a message g ∈ G.
First note that if ρ is invariant under the effect of some specific group element h then the state used for encoding h would be the same as the state used for encoding the identity element e, (U(h)[ρ] = U(e)[ρ] = ρ), such that the message h cannot be distinguished from e. In the extreme case where ρ is invariant under all group elements this encoding does not transfer any information. On the other hand, in the case of Lie groups, if ρ is such that even for those g which are close to the identity of the group, U(g)[ρ] and ρ can be distinguished from each other with high probability, then it is natural to describe ρ as having a high degree of asymmetry.
So from this point of view, the asymmetry properties of ρ can be inferred from the information-theoretic properties of the encoding {U(g)[ρ] : g ∈ G}. To compare the asymmetry properties of two arbitrary states ρ and σ, we have to compare two different encodings: {U(g)[ρ] : g ∈ G} (encoding I) and {U(g)[σ] : g ∈ G} (encoding II). If for all g ∈ G the receiver can get more information from a signal received in encoding I than a signal received from encoding II, then the state ρ has more asymmetry than the state σ. Furthermore, if for any prior distribution over g ∈ G, each state U(g)[ρ] can be converted to U(g)[σ] for all g ∈ G, then encoding I encodes as much or more information about g than encoding II. If the opposite conversion can also be made, so that the two sets are reversibly interconvertible, then the two encodings have precisely the same information about g. Consequently, in an information-theoretic characterization of the asymmetry properties, it is the reversible interconvertability of the sets (defined by the two states) that defines equivalence of their asymmetry properties.
Definition 5 (G-equivalence of states) Two states, ρ and σ, are said to be G-equivalent if and only if the covariant sets {U(g)[ρ] : g ∈ G} and {U(g)[σ] : g ∈ G} are reversibly interconvertible, i.e., there exists a quantum operation E such that and there exists a quantum operation F such that The equivalence of our two different approaches to defining the asymmetry properties of a state (definitions 4 and 5) follows from some simple lemmas.

Lemma 6
The following statements are equivalent: (3.1) For pure states, we have The following statements are equivalent: There exists a unitary operation V which maps U (g)|ψ to U (g)|φ for all g ∈ G, i.e., Note that in both of these lemmas, the condition A concerns whether it is possible to transform a single state to another under a limited type of dynamics. This condition describes the constrained-dynamical perspective. On the other hand, in the B condition, there is no restriction on the dynamics, but now we are asking whether one can transform a set of states to another set. Because the set of states is an encoding of the group element, this condition describes the information-theoretic perspective. Adopting the latter perspective enables us to use the machinery of quantum information theory and quantum estimation theory to study asymmetry and, via the lemmas, the consequences of symmetric dynamics. We will see examples of this technique in the rest of the article and in part II. Also in the next section we explain how these two different perspectives on asymmetry naturally arise in the study of uncorrelated reference frames. First however, we present the proofs of the lemmas. Proof. (Lemma 6) A can be seen to imply B by taking E = E G-cov . To show the reverse, note that B implies the existence of a quantum operation E which satisfies Eq. (3.1). Now we can define One can then easily check that E is a G-covariant operation and that A can be seen to imply B by taking V = V G-inv . In the following we prove that B also implies A. Assume there exists a unitary V such that ∀g ∈ G, Now suppose Π is the projector to the subspace spanned by all the vectors {U (h)|ψ , ∀h ∈ G}. Note that by this definition it is clear Π commutes with all {U (g)}. Since the above equality holds for all U (h)|ψ . Therefore we conclude V Π unitraily maps a subspace of the Hilbert space to another subspace and it commute with all {U (g)}. Using lemma 22 we conclude that there aways exist a G-inv B. Interpreting the two points of view in terms of uncorrelated reference frames Interestingly these two points of view to asymmetry naturally arise in the study of a communication scenario when the two distant parties lack a shared reference frame for some degree of freedom.
Specifically, consider a degree of freedom that transforms according to the group G. Passive transformations of the reference frame for this degree of freedom will then also be described by the group G, as will the relative orientation of any two such frames. Consider two parties, Alice and Bob, that each have a local reference frame but where these are related by a group element g ∈ G that is unknown to either of them. For instance, they might each have a local Cartesian frame, but do not know their relative orientation. (See Ref. [2] for a discussion.) Now consider the following state interconversion task. Alice prepares a system in the state ρ relative to her local reference frame and sends it, along with a classical description of ρ, to Bob. She also sends him a classical description of a state σ, and asks him to try and implement an operation that leaves the system in the state σ relative to her local frame. In effect, Alice is asking Bob to transform ρ to σ but without the benefit of having a sample of her local reference frame. For instance, she may ask him to transform a spin aligned with herẑ-axis to one that is aligned with herŷ-axis. We consider how the task is described relative to each of their local frames.
Description relative to Alice's frame. In this case, the initial and final states, ρ and σ, are described relative to Alice's frame. If the operation that Bob implements is described as E relative to his frame, then it would be described as U † (g) • E • U(g) relative to Alice's frame by someone who knew which group element g connected their frames. However, since g is unknown to Alice and Bob, they describe the operation relative to Alice's frame by the uniform mixture of such operations, i.e., by dgU(g)• E • U † (g). It is straightforward to check that this quantum operation is G-covariant. So all the operations that Bob can implement are described relative to Alice's frame as G-covariant operations. From this perspective, the interconversion can be achieved only if ρ can be mapped to σ by a G-covariant quantum operation.
Description relative to Bob's frame. The initial state is described as U(g)[ρ] relative to Bob's frame. Bob must implement an operation that transforms this to a state which is described as U(g)[σ] relative to his frame. But the group element g that connects Alice's to Bob's frames is unknown, therefore the transformation is required to succeed regardless of g. Bob can implement any operation relative to his own frame and so the set of operations to which he has access is unrestricted. The question, therefore, is whether there exists an operation E such that ∀g ∈ G : E [U(g)[ρ]] = U(g) [σ]. In other words, from this perspective the interconversion task can be achieved only if every element of the set {U(g)[ρ] | g ∈ G} can be mapped to the corresponding element of {U(g)[σ] | g ∈ G} by some quantum operation.
We see therefore that the constrained-dynamical and information-theoretic points of view to the manipulation of asymmetry arise naturally as Alice's and Bob's points of view respectively. They constitute the descriptions of a single interconversion task relative to two different reference frames.

IV. UNITARY G-EQUIVALENCE
Here we introduce another equivalence relation over states that is slightly stronger than G-equivalence This definition is based on the constrained-dynamical point of view. Alternatively we can define this concept in the information-theoretic point of view in terms of the unitary interconvertability of the covariant sets defined by the two states. The equivalence of these two definitions follows trivially from lemma 7.
As we will see later, it turns out that for compact Lie groups it is a small step from characterizing unitary Gequivalence to characterizing general G-equivalence. In particular in section VI, we will show that for semi-simple Lie groups the unitary G-equivalence classes are the same as the G-equivalence classes.
A. The constrained-dynamical characterization: equality of the reductions onto irreps We here find a characterization of the unitary Gequivalence classes within the restricted-dynamical perspective. We begin by determining the most general form of a G-invariant unitary.
Suppose {U (g) : g ∈ G} is a representation of a finite or compact Lie group G on the Hilbert space H. We can always decompose this representation to a discrete set of finite dimensional irreducible representations (irreps). This suggests the following decomposition of the Hilbert space, where µ labels the irreducible representations and N µ is the subsystem associated to the multiplicities of representation µ (the dimension of N µ is equal to the number of multiplicities of the irrep µ in this representation). Then the effect of U (g) can be written as where U µ (g) acts on M µ irreducibly and where I Nµ is the identity operator on the multiplicity subsystem N µ . We denote by Π µ the projection operator onto the subspace M µ ⊗ N µ , the subspace associated to the irrep µ.
Using decomposition (4.3) and Schur's lemmas, one can show that any arbitrary G-invariant unitary is of the following form [2], where V Nµ acts unitarily on N µ . Now we are ready to characterize the unitary Gequivalence classes: Theorem 9 Two pure states |ψ and |φ are unitarily G-equivalent if and only if ∀µ : tr Nµ (Π µ |ψ ψ| Π µ ) = tr Nµ (Π µ |φ φ|Π µ ) (4.5) Proof. Suppose state |ψ can be transformed to another state |φ by a G-invariant unitary V G-inv . Then given that V G-inv has a decomposition in the form of Eq. (4.4), it follows that for all µ, Eq. (4.5) then follows from the cyclic property of the trace and the unitarity of V Nµ . Now we prove the reverse direction. If Eq. (4.5) holds, then there exists a G-invariant unitary which transforms |ψ to |φ . First note that we can think of the two vectors Π µ |ψ and Π µ |φ as two different purifications of tr Nµ (Π µ |ψ ψ|Π µ ) = tr Nµ (Π µ |φ φ|Π µ ). So Π µ |ψ can be transformed to Π µ |φ by a unitary acting on N µ , denoted by V Nµ , such that (See e.g. [5]). By defining we can easily see that V is a G-invariant unitary and moreover V |ψ = |φ . This completes the proof. For arbitrary state ρ we call the set of operators {tr Nµ (Π µ ρ Π µ )} the reduction onto irreps of ρ. So in the above theorem we have proven that the unitary Gequivalence class of a pure state is totally specified by its reduction onto irreps. Note, however, that as we will see in Sec. V A, this is not true for general mixed states. U(1) example. In our quantum optics example, the irreps are labeled by n, the eigenvalue of the number operator, and the representation spaces M n are 1dimensional. It follows that the reduction onto irreps of a pure state |ψ = n,α ψ n,α |n, α is simply given by p ψ (n) ≡ ψ|Π n |ψ = α |ψ n,α | 2 , that is, the probability distribution over number induced by |ψ . Consequently, two pure states are unitarily U(1)-equivalent if and only if they define the same probability distribution over number 3 .

B. The information-theoretic characterization: equality of characteristic functions
We will show that by taking the information-theoretic point of view, one finds that the unitary G-equivalence class of a pure state is specified entirely by its characteristic function, which is defined as follows.
Definition 10 (Characteristic function) The characteristic function of a pure state |ψ relative to a unitary representation {U (g) : g ∈ G} of a group G is a function χ ψ : G → C of the form (4.9) Specifically, we have Theorem 11 Two pure states |ψ and |φ are unitarily G-equivalent if and only if their characteristic functions are equal, ∀g ∈ G : ψ|U (g)|ψ = φ|U (g)|φ . (4.10) The benefit of trying to characterize the G-equivalence classes using the information-theoretic perspective is that we can make use of known results concerning the unitary interconvertability of sets of pure states. We express the condition for such interconvertability as a lemma, after recalling the definition of the Gram matrix of a set of states.
Definition 12 (Gram matrix) Consider the set of states {|ψ i } where i = 1, ..., n. The Gram matrix assigned to this set is an n × n matrix X such that This result is a simple consequence of linear algebra which is highlighted, for instance, in Ref. [11].
It is now straightforward to prove theorem 11. Proof of theorem 11. By definition IV, |ψ and |φ are unitarily G-equivalent if there exists a unitary transformation V inv which take |ψ to |φ . By lemma (13), the necessary and sufficient condition for the existence of such a unitary is the equality of the Gram matrices of the set {U (g)|ψ : g ∈ G} and the set {U (g)|φ : g ∈ G}.
In the U(1) case, the reduction onto irreps and the characteristic function are related by a Fourier transform. The Fourier transform can also be defined for finite groups and for compact Lie groups that are non-Abelian and in these cases, it also describes the relation between the reduction onto irreps and the characteristic function, as will be shown in the next section.

V. WHAT ARE THE REDUCTION ONTO IRREPS AND THE CHARACTERISTIC FUNCTION?
If we are interested in only some particular degree of freedom of a quantum system then we do not need the full description of the state in order to infer the statistical features (expectation values, variances, correlations between two different observables, etcetera) of that degree of freedom. In particular suppose we are interested in the statistical properties of the set of operators {O i ∈ B(H)}. Closing this set under the operator product and sum yields the associative algebra generated by {O i }, which is the set of all polynomials in {O i }. We denote this associative algebra by Alg{O i }. To specify all the statistical properties of the state ρ ∈ B(H) for the set of observables {O i } it is necessary and sufficient to specify all the expectation value of the state for the operators in Alg{O i }. The object that contains all and only this information is called the reduction of the state to the associative Algebra, denoted ρ| Alg{Oi} .
In this section, we will show that the reduction onto the irreps and the characteristic function are simply two particular representations of the reduction of the state to the associative algebra (for the degree of freedom associated to the symmetry transformation) and that these representations are related to one another by a generalized Fourier transform.
A. Two representations of the reduction to the associative algebra A finite-dimensional Alg{O i } (as a finite-dimensional C * -algebra) has a unique decomposition (up to unitary equivalence) of the form where M m J is the full matrix algebra B(C m J ) and I n J is the identity on C n J [12]. This means that there is a basis in which any element A of the algebra can be written as where A (J) ∈ B(C m J ). Furthermore, if we consider the set of all elements of the algebra, that is, all A ∈ Alg{O i }, and look at the set of corresponding A (J) for fixed J, this set of operators acts irreducibly on C m J and spans B(C m J ). Clearly this decomposition induces the following structure on the Hilbert space where M J is isomorphic to C m J and N J is isomorphic to C n J . Suppose Π J is the projective operator to the subspace M J ⊗ N J . Then to specify all the relevant information about the observables in the Algebra for the given state ρ it is necessary and sufficient to know all of the operators Then for any arbitrary observable A in the Algebra we have and so specifying the set {ρ (J) } we know all the relevant information about the state. In other words, {ρ (J) } uniquely specifies the reduction to the Algebra ρ| Alg{Oi} . The above discussion applies to any arbitrary set of observables. Here, we will be interested in the case where this set describes the degree of freedom associated to some symmetry transformation. If the symmetry transformation is associated with the symmetry group G and unitary representation {U (g) : g ∈ G} on the Hilbert space of system, then the set of observables to consider are all those in the linear span of {U (g) : g ∈ G}. In particular, in the case of Lie groups this set contains the representation of all generators of the Lie Algebra (associated to the group) and all the polynomials formed by these generators. For example in the case of SO(3) the set includes all the observables in the linear span of {U (Ω) : Ω ∈ SO(3)} and so it clearly contains all the generators, which in this case are angular momentum operators, as well as all polynomials of these generators.
Decomposition of this algebra in the form of Eq. (5.1) in fact coincides with the decomposition of the unitary representation {U (g) : g ∈ G} to irreps.
where µ labels the irreps and I Nµ is the identity acting on the multiplicity subsystem associated to irrep µ (Remember that G is by assumption a finite or compact Lie group and so it is completely reducible.). Here we can think of µ playing the same role as J in the decomposition of the arbitrary Algebra in Eq.  4)) is simply the reduction onto the irreps of the state ρ, the generalization to mixed states of the notion defined in the section IV A, and therefore we can conclude that the reduction onto the irreps is a representation of the reduction onto the associative algebra. Another way to specify the reduction of the state onto the associative algebra is to specify the Hilbert-Schmidt inner product of ρ with each of the U (g), namely, tr(ρU (g)) for all g ∈ G. So if we define the characteristic function associated to the state ρ as the function χ ρ : G → C defined by χ ρ (g) ≡ tr(ρU (g)), then the characteristic function is a particular representation of the reduction to the associative algebra. It is clear that this definition constitutes a generalization to mixed states of the notion of characteristic functions introduced in the section IV B.
To summarize, we have Definition 14 For a state ρ ∈ B(H) and a unitary representation U of a group G, the reduction of ρ to the associative algebra Alg({U (g) : g ∈ G}) can be represented either in terms of the reduction onto irreps of ρ, defined as (where the Hilbert space decomposition induced by U is H = µ M µ ⊗ N µ and Π µ projects onto M µ ⊗ N µ ), or in terms of the characteristic function of ρ, defined as χ ρ (g) ≡ tr(ρU (g)). (5.8) Finally, we note that the relationship between these two representations is the Fourier transform over the group.

Proposition 15
The characteristic function and reduction onto irreps can be computed one from the other via and Proof. The expression for χ ρ (g) in terms of {ρ (µ) }, Eq. 5.9, follows directly from Eqs. (5.6) and (5.8). Conversely, to find the {ρ (µ) } in terms of χ ρ (g) we use the Fourier transform over the group. The idea is based on the following orthogonality relations which are part of the Peter-Weyl theorem (See e.g. [13]): where dg is the unique Haar measure on the group, bar denotes complex conjugate and d µ is the dimension of irrep µ. According to this theorem any continuous function on a compact Lie group can be uniformly approximated by linear combinations of matrix elements U (µ) i,j (g). Note that for the finite groups, we can get the same orthogonality relations by replacing the integral with a summation. Furthermore any function over a finite group can be expressed as a linear combination of the matrix elements of irreps. So basically all the properties we use hold for finite groups as well as compact Lie groups.
An arbitrary operator A (µ) in B(M µ ) can be written as a linear combination of elements of {U (µ) (g) : g ∈ G}. The above orthogonality relations implies that this expansion has a simple form as Clearly this can be considered as a completeness relation where we have decomposed the identity map on B(M µ ) as the sum of projections to a basis (which is generally overcomplete). Also note that the orthogonality relations imply that for ν = µ Using these orthogonality relations, we obtain Eq. (5.10).
We should emphasize that the reduction onto the associative algebra, though sufficient for deciding Gequivalence of pure states, is not in general sufficient for deciding G-equivalence of arbitrary states, i.e., mixed and pure. Its sufficiency in the case of pure states follows from its sufficiency for deciding unitary G-equivalence (proven in Sec. IV B) and the fact that the unitary G-equivalence classes are a fine-graining of the G-equivalence classes. Its insufficiency in the case of mixed states can be established by the following simple example of two states (one pure and one mixed) that have the same characteristic function but fall in different G-equivalence classes. The example is for the case of U(1)-covariant operations, and the two states are 1 2 (|0 + |1 )( 0| + 1|) and 1 2 (|0 0| + |1 1|). The second is clearly U(1)-invariant while the first is not and so they must lie in different U(1)-equivalence classes. Nonetheless, the characteristic function for both equals χ(θ) = 1/2(1 + exp(iθ)). We close this section by mentioning another consequence of the orthogonality relations Eq.(5.11) which is useful later. Suppose A, B are arbitrary operators in B(M µ ) and χ A (g) ≡ tr(AU (µ) (g)), χ B (g) ≡ tr(BU (µ) (g)) and χ AB (g) ≡ tr(ABU (µ) (g)) are respectively the characteristic functions of A, B and AB. Then where * is the convolution of two functions (Note that for non-Abelian groups f 1 * f 2 is not necessarily equal to f 2 * f 1 .) In particular, since tr(AB) = χ AB (e) (where e is the identity of group) the above formula can be used to find tr(AB) in terms of the characteristic functions of A and B. Using Eq.(5.14) we get

B. Characteristic functions and pairwise distinguishability
Any measure of the distinguishability of a pair of pure states, |α 1 and |α 2 , depends only on the absolute value of their inner product, | α 1 |α 2 |. This is a consequence of the fact that for two pairs of states, {|α 1 α 1 |, |α 2 α 2 |} and {|β 1 β 1 |, |β 2 β 2 |}, the condition | α 1 |α 2 | = | β 1 |β 2 | implies that it is possible, via a unitary dynamics, to reversibly interconvert between the two pairs, which in turn implies (on the grounds that no processing can increase the distinguishability of a pair of states) that they have the same distinguishability. Moreover using the same type of argument we can easily see that any measure of distinguishability should be monotonically nonincreasing in this overlap. Therefore, for any pair of states U (g 1 )|ψ and U (g 2 )|ψ , the distinguishability is specified by | ψ|U † (g 1 )U (g 2 )|ψ | = |χ ψ (g −1 1 g 2 )|. At first glance, therefore, one might think that the Gram matrix for any set of pure states merely encodes the distinguishability of every pair of these states, and therefore, that the characteristic function of a state merely encodes the pairwise distinguishability of the state and every group-transformed version thereof. This is not the case however. Although it is true that if two (covariant) sets are reversibly interconvertible [i.e. they have the same Gram matrix (characteristic function)], then every pair from the first has the same distinguishability as the corresponding pair from the second, the opposite implication fails. In other words, the information content of the set (in particular its entropy for different probability measures) is not specified by the pairwise distinguishabilities of its elements. This phenomenon is highlighted by the results of Jozsa and Schlienz [11]. To understand it, it is useful to consider the corresponding classical result: that two sets of classical probability distributions which have the same pairwise distinguishabilities may not be reversibly interconvertible. A simple example (attributed to Peter Shor in Ref. [11]) illustrates the point. Consider a discrete sample space with four elements, and the following two sets of probability distri One can easily generate a quantum example from this by using coherent superposition in the place of classical mixing. Specifically, consider a four-dimensional Hilbert space, and the following two sets of pure states: {|1 +|2 , |1 +|3 , |2 +|3 } and {|1 +|2 , |1 +|3 , |1 +|4 } (the normalization factor 1/ √ 2 has been left implicit). Clearly, despite the equality of the pairwise overlaps, these two sets are not reversibly interconvertible.
It is similarly straightforward to exhibit this phenomenon in the context of covariant sets. A particularly nice example is provided by a result of Gisin and Popescu concerning the optimal state of two spin half systems to use for sending a direction in space [15]. Define |↑n and |↓n to be the eigenstates of spin along the +n direction, that is,n · σ|↑n = |↑n and n · σ|↓n = −|↓n . Then it is shown in [15] that the state {|↑n |↓n } is better than {|↑n |↑n } for this task when the figure of merit is the fidelity of the estimated direction with the actual sent direction. In other words, they showed that the set {(U (Ω) ⊗ U (Ω))|↑ẑ |↓ẑ , Ω ∈ SO(3)} provides more information about Ωẑ than the set {(U (Ω) ⊗ U (Ω))|↑ẑ |↑ẑ , Ω ∈ SO(3)}. On the other hand, one can easily check that the amplitudes of the characteristic functions for these two states, which encode the pairwise distinguishability of elements of the sets, are exactly the same. This follows from the fact that The insufficiency of the pairwise overlaps within a set of states for specifying the information contained in the set implies that the relevant global properties of the set are encoded in the phases of the components of the Gram matrix, or, in the case of the covariant set of pure states, in the phase of the characteristic function.

C. Properties of characteristic functions
The characteristic functions introduced here are quantum analogues of those used in classical probability theory. The connection is discussed in detail in Appendix B. Here we simply summarize some useful properties of characteristic functions.
1. A function φ(g) from the finite or compact Lie group G to complex numbers is characteristic function of a physical state iff it is (continuous in the case of Lie groups) positive definite(as defined in appendix B) and normalized (i.e. φ(e) = 1 where e is the identity of group.).
5. If |χ ρ (g s )| = 1 for g s ∈ G then g s is a symmetry of ρ. If ρ is a pure state, then |χ ρ (g s )| = 1 if and only if g s is a symmetry of ρ.
6. So |χ ρ (g)| = 1 for all g ∈ G implies that the state is invariant; in this case χ ρ (g) is a 1-d representation of group.
7. Suppose L is the representation of a generator of the Lie group on the Hilbert space of system such that for arbitrary θ, e iθL is the representation of an elements of the group. Then we can find all moments of L using the characteristic function.
tr(ρL k ) = i −k ∂ k ∂θ k χ ρ (e iθL ) | θ=0 (5.17) (Note that by χ ρ (e iθL ) we really mean χ ρ (g) for the group element g ∈ G which is represented by e iθL .) 8. A group action on the state corresponds to the conjugate action on the characteristic function, If the group G is Abelian then group actions are themselves G-covariant operations and the characteristic function is invariant under a group action on the state. 9. For a state ρ that has reduction onto irreps {ρ (J) }, Proof. Item 1 is proven in Appendix B 2.All the rest of these properties can simply be proved by using the definition of the characteristic function, χ ρ (g) = tr(ρU (g)), and group representation properties. For example to prove 8 we note that To prove 5 we note that if |χ ρ (g s )| = 1 for g s ∈ G then all eigenvectors of ρ are eigenvectors of U (g s ) with the same eigenvalue. As a result we get [ρ, U (g s )] = 0 and so the state has the symmetry g s . On the other hand, [ρ, U (g s )] = 0 does not imply that |χ ρ (g s )| = 1. For instance, the state 1 2 |0 0| + 1 2 |1 1| where |n is a number eigenstate is U(1)-invariant, but nonetheless, for φ = 0, |χ ρ (φ)| = 1. Therefore the points for which the amplitude of the characteristic function is one are a subset of the symmetries of the state. Meanwhile, if a pure state |ψ has symmetry g s , such that U (g s )|ψ = e iθ |ψ for some θ, then obviously |χ ψ (g s )| = 1. So for pure states the points for which the amplitude of the characteristic function is one are exactly the state's symmetries.
To prove 6, we first note that if |χ ρ (g)| = 1 for all g ∈ G, then the symmetry subgroup of ρ is the entire group G, which is the definition of ρ being G-invariant. Furthermore, for each g, the eigenvectors of ρ all live in the same eigenspace of U (g). Since the eigenvalue of a unitary is a phase factor, each such eigenvector |ν must satisfy U (g)|ν = e iθ(g) |ν for some phase e iθ(g) . It is then clear that χ ρ (g) = e iθ(g) and is a 1-dimensional representation of the group.
Among the above properties, the fact that the tensor product of states is represented by the product of their characteristic functions (property 3) turns out to be particularly useful. This is because the alternative representation, in terms of reductions onto irreps, does not provide a simple expression for the reduction of ρ ⊗ σ in terms of the reduction of ρ and the reduction of σ. It involves Clebsch-Gordon coefficients and is generally quite complicated for non-Abelian groups.
For this and other reasons, the characteristic function is generally our preferred way of respresenting the reduction of the state onto the algebra, and consequently we will make heavy use of it in this article and its successor to answer various questions about the manipulation of asymmetry of pure states.

VI. G-EQUIVALENCE CLASSES
We have seen that the characteristic function of a pure state uniquely specifies its unitary G-equivalence class. However, it is G-equivalence rather than unitary G-equivalence that implies that two states have the same asymmetry properties, so we must ultimately characterize the former. Fortunately, for compact Lie groups, the conditions under which two states are G-equivalent can also be stated simply in terms of their characteristic functions, as is shown presently.
Theorem 16 For G a compact Lie group, two pure states |ψ and |φ are G-equivalent (i.e. they can be reversibly interconverted one to the other by G-covariant operations) iff there exists a 1-dimensional representation of G, e iΘ(g) , such that ∀g ∈ G : ψ|U (g)|ψ = e iΘ(g) φ|U (g)|φ . (6.1) Since the Semi-simple compact Lie groups do not have a non-trivial 1-dimensional representation the above theorem implies

Corollary 17
For G a semi-simple Lie group, two pure states |ψ and |φ are G-equivalent iff their characteristic functions are equal i.e.
∀g ∈ G : ψ|U (g)|ψ = φ|U (g)|φ . (6.2) The above theorem applies only to compact Lie groups. Putting a restriction on the states we can prove the following theorem which applies to both compact Lie groups and Finite groups Theorem 18 Two pure states |ψ and |φ for which ψ|U (g)|ψ and φ|U (g)|φ are nonzero for all g ∈ G are G-equivalent (i.e. they can be reversibly interconverted one to the other by G-covariant operations) iff there exists a 1-dimensional representation of G, e iΘ(g) , such that ∀g ∈ G : ψ|U (g)|ψ = e iΘ(g) φ|U (g)|φ . (6.3) Proof. (Theorems 16 and 18) First we prove that Eq. (6.1) implies that |ψ and |φ are G-equivalent. Suppose |ν is a state with characteristic function e iΘ(g) where by assumption e iΘ(g) is a 1-dimensional representation of the group. Then according to Eq. (6.1) and property (3) of characteristic functions, the characteristic function of |ψ is the same as the characteristic function of |φ ⊗ |ν . It follows from Theorem 11 that there exists a G-invariant isometry which maps |ψ to |φ ⊗|ν .
So by performing this G-invariant isometry and then discarding |ν we can transform |ψ to |φ . Note that such a transformation clearly would be a G-covariant operation. (Alternatively, let |ν * be the state with characteristic function e −iΘ(g) . Note that since e −iΘ(g) is also a 1-d representation of the group then there exists a state |ν * whose characteristic function is e −iΘ(g) . Then since |ψ ⊗ |ν * , and |φ have the same characteristic function, by Theorem 11 there exists a G-invariant isometry which transforms one to the other. Because |ν * is a G-invariant state and because the isometry is G-invariant, the overall operation is G-covariant.) Using an analogous argument, we can easily deduce that there also exists a G-covariant operation which maps |φ to |ψ . Therefore |ψ and |φ are G-equivalent.
We now prove the other direction of the theorem. If |ψ and |φ are G-equivalent, then there exists a G-covariant operation from |ψ to |φ and vice versa. It then follows from Stinespring's dilation theorem that there exists a G-invariant unitary V and a G-invariant pure state |η 1 such that V |ψ |η 1 = |φ |η 2 (6.4) for some pure state |η 2 , and there exists a G-invariant unitary V and a G-invariant pure state |η 1 such that V |φ |η 1 = |ψ |η 2 for some pure state |η 2 . These two equations together imply that V V |ψ |η 1 |η 1 = |ψ |η 2 |η 2 (6.5) Since V and V are both G-invariant we can deduce that the characteristic functions of |ψ |η 1 |η 1 and |ψ |η 2 |η 2 are equal. i.e. χ ψ χ η1 χ η 1 = χ ψ χ η2 χ η 2 (6.6) Since |η 1 and |η 1 are both G-invariant states the amplitude of their characteristic functions are always one and so Now suppose G is a compact Lie group. Then for any state ψ in a finite dimensional Hilbert space, |χ ψ | is 1 at identity and is non-vanishing for a neighbourhood around identity in any direction. This implies that |χ η2 χ η 2 | has value 1 for a neighbourhood around identity in any direction. By analyticity of the characteristic functions, this implies that |χ η2 χ η 2 | is 1 everywhere. Therefore |η 2 |η 2 is an invariant state. Note that it is this step of the proof which necessitates the restriction to compact Lie groups. Since |η 2 |η 2 is G-invariant then |η 2 is also Ginvariant. Therefore Eq. (6.4) implies that χ ψ (g) = χ φ (g)e i[Θ2(g)−Θ1(g)] (6.8) where e iΘ1(g) and e iΘ2(g) are respectively the characteristic functions of |η 1 and |η 2 . Finally, because e iΘ1(g) and e iΘ2(g) are 1-dimensional representations of G, it follows that e i[Θ2(g)−Θ1(g)] is as well. This completes the proof of Theorem 16. As we mentioned above, there is only one point in the proof in which we use the assumption that the group is a Lie group: the fact that |χ ψ | = |χ ψ ||χ η2 χ η 2 | implies |χ η2 χ η 2 | = 1. This follows from continuity of the characteristic function for Lie groups. For finite groups, where we cannot appeal to continuity, if |χ ψ | is zero at some g ∈ G then |χ ψ | = |χ ψ ||χ η2 χ η 2 | does not imply |χ η2 χ η 2 | = 1 at that point. However, if we assume the function χ ψ is nonzero for all g ∈ G then we can again deduce |χ η2 χ η 2 | = 1 and the rest of the argument goes through as before. This completes the proof of theorem 18.
Note that in the above proof V and V together generate a Carnot-type cycle described by Eq. (6.5): we start with state |ψ (the resource) and use invariant states |η 1 |η 1 (non-resources). After a cycle generated by Ginvariant unitaries (free dynamics) we get back |ψ and some residue states |η 2 |η 2 . In the above proof using the properties of characteristic functions, we showed that the residue states should be invariant (non-resource). However this property can be derived from more general considerations. Suppose |η 2 |η 2 is not invariant. This implies that by going through this cycle we have generated some additional resource without consuming any. We can repeat this cycle and generate an infinite number of copies of |η 2 |η 2 and so generate an infinite amount of the resource. This should be impossible if the state |ψ contains only a finite amount of the resource, which is indeed the case for any state on a finite-dimensional Hilbert space if the group is not finite. U(1) example: In our quantum optics example, the criterion of U(1)-equivalence of pure states has a simple form in terms of reductions onto irreps. Suppose that the probability distributions {p ψ (n)} and {p φ (n)} are the reductions onto the irreps of ψ and φ respectively, so that the characteristic functions are the Fourier transforms of these. Theorem 16 implies that ψ and φ are U(1)equivalent if and only if there exists an integer ∆ such that n p ψ (n)e inθ = e i∆θ n p φ (n)e inθ , or equivalently, using the Fourier transform, such that p ψ (n) = p φ (n + ∆), (6.9) which is precisely the condition found in Ref. [3].

VII. APPROXIMATE NOTIONS OF UNITARY G-EQUIVALENCE
We have found the necessary and sufficient condition for the existence of a G-covariant transformation which transforms a pure state ψ to another pure state φ. These are the conditions for exact transformation. But there might be situations in which we cannot transform ψ to φ but we can transform it to some state close to φ.
In the following we demonstrate that if the reductions onto irreps (or equivalently the characteristic functions) of two pure states ψ and φ are close then there exists a G-invariant unitary which transform ψ to a state close to φ. We present the results here and leave the proofs to appendix C.
Recall that the Fidelity of two positive operators A 1 and A 2 is defined as Fid( where || · || 1 denotes the trace norm. According to this theorem if the fidelities of the reductions onto irreps is high then there exists a G-invariant unitary which transforms one of the states to a state very close to the other. On the other hand, if these fidelities are low we can never transform one of the states close to the other via G-invariant unitaries.
Using the standard bounds between fidelity and trace distance of two operators [5] we can express this result in terms of trace distance of the reductions. For future applications here we present the condition which guarantees the existence of a G-invariant unitary for transforming states to each other in terms of trace distance of reductions.

Corollary 20 Suppose {F
The following corollary presents a similar bound in terms of the distance between characteristic functions of states χ ψ1,2 (g) and another bound in terms of the distance between the components of characteristic functions {χ (µ) ψ1,2 (g)} where the µ component of χ ψ1,2 (g) is defined as where ϕ µ (g) = tr(U (µ) (g)) is the character of irrep µ and * (convolution) is defined in Eq.(5.15).

Corollary 21
Suppose χ ψ1 and χ ψ2 are respectively the characteristic functions of states ψ 1 and ψ 2 . Then there exists a G-invariant unitary V such that ψ2 (g)| (7.4) where the summation is over all irreps in which ψ 1 and ψ 2 have nonzero components.

VIII. DISCUSSION
In this paper we have examined the problem of how to classify quantum states according to their asymmetry properties relative to a given symmetry group (and its unitary representation). We have done so at a level of generality that incorporates all compact Lie groups and finite groups. Our main motivation for this study has been to find the appropriate concepts and framework for a systematic study of the consequences of symmetry of time evolutions for open and closed systems. In particular we are interested to find the maximal constraints imposed by the symmetry of time evolution on the final state of a time evolution according to the initial state. From this perspective the concepts of asymmetry of state and G-equivalence classes are the most natural tools to use.
We started by defining the asymmetry of a state as the property which specifies whether the state ρ can evolve to the state σ via a G-covariant time evolution or not (we called it the constrained-dynamical approach). Interestingly, as we showed in the paper, there is another approach (the information-theoretic approach) for characterizing the asymmetry of states which does not emphasize G-covariant dynamics. Instead in this approach the asymmetry of a state ρ is specified by the properties of the covariant set {U(g)[ρ] : g ∈ G}. In this paper we showed that two points of views yield the same notion of asymmetry: Using these two approaches we found two characterizations of unitary G-equivalence classes of pure states (the reduction of the state to the irreps. and the characteristic function of the state). We observed that these two characterizations are equivalent via Fourier and inverse Fourier transform and that they are in fact two different representations of one object namely the reduction of the state to the associative algebra generated by the representation of group.
We studied the problem of characterizing the Gequivalence classes of pure states. We found that the reduction of the state to the associative algebra generated by the representation of the group encodes all the information required to specify the G-equivalence class of a pure state. We showed by example that this is not the case for mixed states. We focused on the characteristic function (χ ψ (g) = ψ|U (g)|ψ ) as the preferred representation of the reduction of state to the associative algebra. For the case of compact Lie groups we proved that two pure states are G-equivalent (i.e. have exactly the same asymmetry properties relative to the group G) iff their characteristic functions are equal up to a 1-dimensional representation of group. In a subsequent article (part II of this paper) we use characteristic functions to answer different interesting questions about the manipulation of the asymmetry of pure states.
We have also mentioned other applications of the study of the asymmetry of states and in particular to the study of reference frames. In fact, the asymmetry of a state relative to a symmetry group is also exactly the property of the state which specifies how well it can act as a quantum reference frame for that degree of freedom. This might be more clear in the information-theoretic perspective in which to specify the asymmetry of a state ρ, we look at the covariant set associated to the state (i.e. {U(g)[ρ] : g ∈ G}) as an encoding to send information about group elements. These covariant sets are also relevant in some other estimation and metrology problems and so asymmetry properties are relevant to these problems as well.
There is much more to say about the consequences of the information-theoretic point of view to asymmetry. It implies that the powerful machinery of information theory can be brought to hear on the study of the asymmetry of quantum states and the consequences of symmetric dynamics. For instance, there are many exotic phenomena in quantum information theory that do not have any counterpart in classical information theory and therefore one expects that quantum asymmetry may have interesting features not found for classical asymmetry. In particular, a symmetry of the quantum dynamics may have consequences which have no counterpart in the case of classical dynamics. We will present examples of these phenomena in future work.
Note that many of the notions we have introduced hold not just in quantum theory, but in any theory wherein we have a representation of symmetry transformations. We need only replace quantum states and quantum operations with states and state-transformations in the theory in question in order to generalize the notions of Gcovariant operations, and G-equivalence classes. In particular, we can consider operational probabilistic theories [16][17][18]. It would be interesting to study which features of the quantum theory of asymmetry are generic to operational theories of asymmetry and which are specific to quantum theory.

IX. ACKNOWLEDGEMENT
We thank Gilad Gour for helpful discussions. We also thank Sarah Croke for a helpful discussion about Gram matrices and Giulio Chiribella for a discussion about Noether's theorem. Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation. I. M. is supported by Mike and Ophelia Lazaridis fellowship. .

Appendix B: Comparison of classical and quantum characteristic functions
The characteristic function of a quantum state can be understood as a generalization of the characteristic function of a probability distribution. In fact, this generalization was the first motivation for introducing the notion of a characteristic function for a quantum state by Gu [20]. W first review some properties of classical characteristic functions and then we talk about their analogues in the case of quantum states and non-Abelian groups. We also review positive definiteness as the main criterion for a complex function over the group to be the characteristic function of a valid quantum state. Almost all the materials of this appendix are borrowed from [20][21][22].

Review of classical characteristic functions
For a real random variable x with the distribution function F (x) the characteristic function is defined as the expectation value of the random variable e itx i.e.
The distribution function is uniquely determined by its characteristic function. Moreover if the probability density exists then it will be equal to the inverse Fourier transform of the characteristic function. One particularly useful property of the acteristic function is the multiplicative property according to which the characteristic function of the sum of two independent random variables is equal to the product of their characteristic functions.
There exists a remarkably simple proof of the central limit theorem using this multiplicative property of characteristic functions. The derivative of characteristic functions at the origin determines the moments of the random variable.
Sometimes it is more favourable to use cumulants of the random variable instead where the n-th order cumulant is defined as the n-th order derivative of the logarithm of the characteristic function at the point 0, multiplied by i −n .
The first and second cumulants are mean and variance of the random variable. By this definition, it turns out that cumulants of a sum of independent random variables is equal to the sum of the cumulants of the individual terms for all orders of cumulants. The set of all classical characteristic functions is determined by Bochner's theorem, according to which a complex function f (t) is the characteristic function of a random variable if and only if 1) f (0) = 1, 2) f (t) is continuous at the origin, and 3) it is positive definite. Recall that a function f (t) is positive definite if for any integer n and for any string of real numbers t 1 , ..., t n the matrix a i,j ≡ f (t i −t j ) is a positive definite matrix. Positive definiteness of a function guarantees that the inverse Fourier transform of this function is positive for all values of the random variable, which is clearly a necessary condition for a function to be a probability density.
For more discussion about the properties of characteristic functions of probability distributions, see e.g. [19].

Quantum characteristic functions
As the characteristic function of a probability distribution determines all of its statistical properties, the characteristic function of a quantum state over the group G uniquely specifies all the statistical properties of observables in the algebra of observables which generates the unitary representation of G. For example suppose L is the representation of a generator of the Lie group G then we have tr(ρL k ) = i −k ∂ k ∂θ k χ ρ (e iθL ) | θ=0 (B5) In particular the first derivative (k = 1) determines the expectation value of the generator.This is just property 7 of characteristic functions from Section V C. Similarly we can define cumulants of the observable L, where the n-th order cumulant is defined as the nth order derivative of the logarithm of the characteristic function at the identity element multiplied by i −n . κ (n) The first and second cumulants are mean and variance of the observable. By this definition, it turns out that the cumulants of the tensor product of two states is equal to the sum of the cumulants of the individual states for all orders of cumulants.
In the rest of this appendix, we are interested to find the generalization of Bochner's theorem i.e. the set of necessary and sufficient conditions for φ(g) a complex function over group to be the characteristic function of some quantum state. We see that such a generalization can be found via both non-commutative Fourier transform and the Gelfand-Naimark-Segal (GNS) construction theorem. As in the rest of the paper, we focus on the finite groups and compact Lie groups.
As the first necessary condition we note that tr(ρ) = 1 implies that χ(e) = 1 (where e is the identity of group). We call the functions which satisfy this condition normalized functions. In the case of compact Lie groups φ(g) should also be a continuous function. We also need a condition on φ(g) equivalent to the positivity of density operators. As we just saw in the case of probability distributions the condition of positivity of probabilities is equivalent to the positive definiteness of characteristic function of the probability distribution. Similarly it turns out that the relevant condition on φ(g) to be the characteristic function of a positive operator is the natural generalization of positive definiteness for the functions defined on the group: for any f ∈ L 1 (G). Now using the Fourier transform, one can easily prove a theorem similar to the Bochner's theorem [20,21]: Theorem 25 A complex function φ(g) on the finite or compact Lie group G is the characteristic function of a quantum state iff φ(e) = 1, φ(g) is positive definite and continuous (in the case of Lie groups).
Proof. We present the proof assuming that the group G is a compact Lie group (The same argument works for a finite group by replacing integrals with summation.). We use the inverse Fourier transform. Suppose B (µ) ≡ d µ dgU (µ) (g −1 )φ(g). Then the set of operators {B (µ) } is the reduction onto irreps of a valid quantum state iff (1) µ tr(B (µ) ) = 1 and (2) all operators {B (µ) } are positive definite. The first condition expresses the fact that the trace of the state is one and is guaranteed by φ(e) = 1. On the other hand, B (µ) is positive iff tr(F F † B (µ) ) ≥ 0 for all operators F acting on M µ (the subsystem on which U µ acts irreducibly). Note that tr(F F † B (µ) ) is equal to the Fourier transform of the operator F F † B (µ) at point e. So using the convolution property of characteristic functions, Eq.(5.14), we get is positive definite and therefore satisfies Eq.(B8) then all B (µ) 's are positive. We can prove the other direction of the theorem similarly.
Therefore the set of normalized positive definite functions (also continuous in the case of Lie groups) are exactly the set of characteristic functions of states.
We can also get this result using a more fundamental theorem in the representation theory of C * algebras, called the GNS construction after Gel'fand, Naimark and Segal. A specific form of this theorem states Theorem 26 (GNS construction) With every (continuous) positive definite function φ(g) we can associate a Hilbert space H, a unitary representation {U (g)} of G in H and a vector ψ, cyclic for {U (g)}, such that Moreover the representation {U (g)} is unique up to a unitary equivalence.
Note that a vector |ξ is cyclic for the representation {U (g)} on the space H if the set of vectors ∀g ∈ G : U (g)|ξ is a dense subset of the space H.
Therefore the GNS construction theorem guarantees that for any given (continuous) normalized positive definite function there exists a corresponding pure cyclic state with that characteristic function. Note that for any arbitrary mixed or pure state there exists a pure state which is cyclic (for the representation on its Hilbert space) with exactly the same characteristic function. So the set of all (continuous) normalized, positive definite function is exactly the same as the set of all characteristic functions of states.
The second bound on µ ||F 2 || 1 is obtained as follows.