On the classification of two-qubit group orbits and the use of coarse-grained 'shape' as a superselection property

Recently a complete set of entropic conditions has been derived for the interconversion structure of states under quantum operations that respect a specified symmetry action, however the core structure of these conditions is still only partially understood. Here we develop a coarse-grained description with the aim of shedding light on both the structure and the complexity of this general problem. Specifically, we consider the degree to which one can associate a basic `shape' property to a quantum state or channel that captures coarse-grained data either for state interconversion or for the use of a state within a simulation protocol. We provide a complete solution for the two-qubit case under the rotation group, give analysis for the more general case and discuss possible extensions of the approach.


I. INTRODUCTION
Symmetry principles are ubiquitous in both the classical and quantum realms. They constrain dynamics, connect with conservation laws and simplify computations of physical properties. In crystallography the crystal structure is described by a symmetry group and largely determines the properties of the material [1].
Symmetry considerations have been extremely successful in the field of quantum information theory [2][3][4]. Moreover a substantial component of this work has addressed the regime in which symmetry principles disconnect from conservation laws [5], and which has found application in the study of quantum features of thermodynamics [6][7][8]. Quantum states that break the symmetry can also be used to circumvent limitations on the precision of measurements imposed by conservation laws, as described by the WAY theorem [9][10][11][12][13][14], while metrology can be viewed as using broken symmetries to distinguish different group transformations [15]. Symmetry principles have also provided new insights into quantum speed limits and how quickly quantum operations can be performed [16]. Finally, symmetry groups provide an invaluable toolkit in the context of quantum computing [17][18][19][20].
In this work we look at how general quantum states ρ can be classified by groups that break a given symmetry constraint to an equal degree. A concrete example makes this clearer: both a chair and an arrow break rotational symmetry, however an arrow has a residual (axial) symmetry group to it, and so breaks rotations to a weaker degree than the chair. For a fixed quantum system of dimension d, and symmetry action G, we can ask what residual symmetries does the system admit, and how do these residual symmetries transform under symmetric dynamics. We also address the degree to which this aspect of a quantum system constitutes a property in the 'resource-theoretic' sense.
The structure of the paper is as follows. In the next section we provide notation and background motivation.
In section III we provide the coarse-graining scheme and some basic relations involved. This demonstrates the complexity of the topic, and highlights some subtleties if one wishes to associate the scheme to a quantum resource. In section IV we provide a complete classification for two-qubit systems, and which illustrates how quantum state spaces admit a coarse-grained partial order.
FIG. 1. Residual symmetry-types in two-qubit systems. We associate to each state a 'shape' that describes the residual symmetries of that state. As an example, in the above figure we show the symmetry properties of the set of all mixtures of Bell states under SU (2). This is determined solely by the triplet component of the state. The only possibilities are for the state to have a residual symmetry of (a) a cylinder, (b) a double-sided rectangle, or (c) a sphere. This 'shape' can be used to provide superselection rules when using the two-qubit state to simulate quantum operations. Details are given in section IV. arXiv:1804.09967v1 [quant-ph] 26 Apr 2018

II. BACKGROUND AND MOTIVATION
In this paper we shall use the following notation. For any quantum system A we denote by H A its associated Hilbert space, and B(H A ) the set of linear operators on H A . We assume that a symmetry action of G is defined on H A through the unitary representation g → U (g) ∈ B(H A ). The group action on a quantum state ρ ∈ B(H A ) is given by the adjoint action U g (ρ) := U (g) ρ U † (g).
For any quantum channel E : B(H A ) → B(H A ), the action of the group on E is U g (E) := U g • E • U g −1 (where • denotes the concatenation of channels). This can be interpreted as moving to a different frame, performing the channel and returning to the original frame. If U g (E) = E for all g ∈ G then E is called a symmetric operation. This includes certain preparation procedures, and the resulting states are known as symmetric states because they are invariant under the group action: U g (ρ) = ρ for all g ∈ G.
Our concern here is the study of general quantum states or quantum operations under a symmetry action. In particular we would like to study the degree to which any given quantum state breaks the symmetry. One way of comparing states in this manner is to say that ρ σ if and only if σ = E(ρ) for some symmetric quantum operation E, which defines a partial order on quantum states [21,22]. A measure of how much a state breaks the symmetry is then a function M that respects the partial order -namely, if ρ σ then M (ρ) ≥ M (σ) [23,24]. A central question is whether a complete set of measures exist that fully capture this ordering of states.
Very recently [7] this general problem was solved in terms of single-shot entropies of the form H min (R|A) that quantify the amount of information in a system A about an external quantum reference frame R. However this complete set of measures is highly complex to use, and applying it to general systems requires sophisticated techniques in single-shot information theory. Here we wish to tackle a more modest goal, and instead of describing the complete symmetry properties of a quantum state, wish to coarse-grain states into groups that break the symmetry in the same way. Notably, this itself turns out to be a highly non-trivial problem and so sheds light on the structure of the more general set of measures given in [7].

A. Simulation of local quantum channels
One very grounded way of probing the degree to which a quantum state breaks a symmetry is to see how useful it is when we wish to perform tasks, such as induce a channel on another system. More generally, if we have some target quantum operation E on a system, we would like to know what kind of resource states and interactions are required to realise that operation. The degree to which a resource σ can do this under symmetric dynamics therefore constitutes a measure of its properties.
In [25], the symmetry properties of quantum channels were analysed, with a focus on the orbit of a channel E, defined as A similar definition applies to the orbit of states under the group, so M(ρ) = {U g (ρ) : g ∈ G}. Now the type of orbit one obtains provides a natural context in which to understand both how the channel breaks a symmetry constraint, and moreover how one might simulate such a channel using an auxiliary quantum system B prepared in a non-symmetric state σ B . Specifically, when using a state σ B to induce a channel E locally on a quantum system A, a distinguished protocol is a measure-and-prepare protocol in which a POVM {M x } is performed on σ B to estimate coordinate data x for M(E) and then conditioned on this, a completely positive map E x is applied to system A.
This protocol is distinguished in that it takes an encoding in σ of a target point z on M(E), performs a measurement to obtain a random classical output x that is then transmitted to an action on A. The label z is the classical data needed to induce E. Therefore σ B functions well as a resource if it can encode a location z on M(E) with sufficient resolution (see [25] for details). A necessary prerequisite for the state to function well is that the orbit of the state itself must have sufficient structure, in the sense of breaking the symmetry to a sufficient degree.
Therefore the usefulness of a state σ to simulate a channel E can be analysed under two considerations: 1. ('Shape') The type of orbit M(σ) the state has under the group action compared to M(E), independent of metrical aspects.
2. ('Geometry') The ability of σ to encode classical coordinate data for x in M(E).
What is a necessary relation between M(E) and M(σ)? It is clear that we need σ to break the symmetry to a 'larger' degree than E, however in this paper we would like to make this statement more precise. As mentioned the conditions in [7] turn out to be highly non-trivial to analyse. Given the complexity of the general problem, our motivation in this work is to start with the above separation into 'shape' and 'geometry', and develop an organisational setting in which we provide a coarse-graining over states of the same 'shape', with the remaining task being to analyse the 'geometry' aspect of the state. This work is restricted to the first of these aims. In the process, we address the degree to which we can sensibly define such a 'shape' property for a quantum system and how it compares with other resource-theoretic approaches.

B. Related topics in quantum state tomography and quantum computation
Beyond the abstract problem of simulating an arbitrary quantum channel using a resource state, there are specific contexts of importance where such a coarsegrained division naturally arises. In [26], quantum state tomography was studied in the context of prior information that restricts the state to a lower-dimensional submanifold of the state space. The analysis shows that the topological genus of the manifold can be used to bound the number of measurements needed to discriminate states on the submanifold. The orbit of a state (or channel) is one such submanifold, and so the study of what kinds of orbits can exist and how they transform among themselves is of potential relevance to quantum tomography under symmetry constraints.
A wholly practical direction where such analysis might be of relevance is in the computational power of gate-sets in quantum computation, and how such gate-sets interact with states that increase the computational power of the gate-set (such as magic states for the stabilizers [27]). For example, a recent work provides a classification of all Clifford gate-sets [17] in a lattice hierarchy, and so aspects of the present work might be applicable in studying how noisy quantum states increase the computational power of easily realisable gate-sets.

A. Basic ingredients
A basic classification of states and channels under a symmetry can be done by studying the subgroup of residual symmetries for the state or channel, called the isotropy subgroup or stabilizer of E and defined as It specifies the residual symmetry that the channel has under the group action. Symmetric channels are invariant under all group transformations, and simply have Iso(E) = G. For continuous Lie groups, we find that the orbit of a channel is always a homogeneous space [28,29], specified by both the group G and the particular isotropy subgroup H of the channel. More precisely the orbit of the channel is M(E) and coincides with the quotient G/H up to diffeomorphisms, which we write M(E) ∼ = G/H.
A simple example can be given for a single qubit state under an SU (2) symmetry, where the possible types of group orbit are If one wishes to realise a quantum channel E via a symmetric interaction with a state σ, the most natural way to model this is as If such a relation holds, we say that E can be simulated using the state σ B .
Channel In many cases the quantum operation E has some residual symmetry, which is characterised an isotropy subgroup H of G. The possible isotropy subgroups form an abstract structure called a lattice [30]. The partial ordering of the subgroups is defined by subset inclusion, namely In addition, every pair of elements have a unique supremum and unique infimum, which define binary operations called the meet and join [31]. For two subgroups H 1 and H 2 , their meet is denoted H 1 ∧ H 2 and defined as H 1 ∩ H 2 , while their join is denoted H 1 ∨ H 2 , defined as the subgroup generated by H 1 ∪ H 2 [30]. This structure is illustrated by a Hasse diagram of the subgroups, as shown in Fig. 4.
The lattice of isotropy subgroups immediately gives us a way of comparing the orbits of channels under a group action G, with M(E) ≺ M(F) iff Iso(E) Iso(F). The convention here is to match with resource-theoretic measures of asymmetry [21,23], so that one orbit is 'bigger' than another if it has a smaller isotropy group. i.e. H5 ≺ H4 ≺ H3 etc. Note that for finite-dimensional systems many of the C(H) will be empty. The subsets C(H) can be associated, up to diffeomorphisms, with group orbits (indicated on the right). The left figure shows the action of group-averaging (over H2), going from σ to PH 2 (σ).

B. Simulating Operations Under Symmetry Constraints
Since quantum operations can be combined in a number of ways, we present some basic statements that constrain the residual symmetry of the resultant quantum operation. The proofs for this section are provided in Appendix A.
3. If A = B and A = B , and p some probability These results also apply for quantum states, once we view the state as the state preparation map 1 → ρ. We can now make precise a coarse-grained (necessary, but far from sufficient) requirement that a state σ allow the simulation of a channel E under symmetric dynamics. A similar result can be given in the case of approximate simulation. For any target operation E we can define the noisy version given by where D is the complete depolarization map ρ → 1 d 1 for all ρ. From Lemma 1 we see that Iso(E ) = Iso(E) ∧ G = Iso(E). If σ allows the simulation of E for some < 1 then this corresponds to an approximate, 'isotropic' simulation of the original map with noise parameter . The previous theorem extends to this case in an obvious way.
C. Is 'shape' a resource-theoretic property?
While we normally associate measurable properties with either projective measurements or more generally POVMs, there is an alternative way that is more general again. Specifically, a property is associated with a preorder defined on the set of all quantum states. This recent approach is called the resource-theory method, and has found success in areas such as entanglement, coherence, thermodynamics, and many other scenarios [4,[32][33][34]. Quantum maps that respect the pre-order are called 'free operations' and any real-valued function on quantum states that respects the pre-order is a measure of the property.
Given an ability to order the orbits of quantum states under the group action, we can ask if a meaningful notion of 'shape' can be defined for quantum systems, along such a resource-theoretic line. This would essentially characterise the asymmetry of states and channels without reference to a measure.
The isotropy subgroup gives us a natural equivalence relation ∼ between states that behave in the same way under the group action. We can then say that ρ 1 and ρ 2 are related ρ 1 ∼ ρ 2 if we have that Iso(ρ 1 ) = Iso(ρ 2 ). This equivalence relation partitions the state space into sets C(H) := {ρ : Iso(ρ) = H}, and collects together all states that break the symmetry in the same manner.
Viewed as a mapping, C maps from the subgroup lattice onto a partition of the state space, and so we can simply allow the partition to inherit the lattice structure, and order subsets of states as C( Note however that the sets {C(H)} are not convex in general. For example a qubit system under G = SU (2), both |0 0| and |1 1| have the same U (1) isotropy subgroup, but 1 2 (|0 0| + |1 1|) = 1 2 1, which is symmetric. However, C(G) is always a convex set. This follows because if ρ 1 , ρ 2 ∈ C(G) then Lemma 1 implies that G Iso(p 1 ρ 1 + p 2 ρ 2 ) Iso(ρ 1 ) ∧ Iso(ρ 2 ) = G, and so any mixture has the same isotropy subgroup.
The SU (2) qubit example also highlights that if C(H) = ∅ it does not imply that C(H ) = ∅ for all H H, since there are subgroups of SU (2) containing U (1) which do not appear as the isotropy subgroup of a single qubit state. Not all possible isotropy subgroups will be seen in a given system, and for a finite dimensional system many of the sets C(H) will be empty.
However in order to have a meaningful resource-theory interpretation, the coarse-grained ordering must be compatible with the set of free operations -namely the symmetric operations. It is easy to show that this is in fact the case at the level individual quantum states.

Lemma 2. Under a symmetric operation E, Iso(E(ρ)) Iso(ρ).
This shows that the coarse-grained ordering of states is consistent with the set of free (symmetric) operations, as expected. However it does not imply that a functional mapping is defined on the sets C(H); it is possible to have ρ 1 and ρ 2 in the same set C(H), but each get sent to different sets C(H 1 ) and C(H 2 ). This aspect means that interpeting the coarse-graining as defining a quantum resource needs care. Another subtlety arises if one tries to naively exploit some metric on quantum states to quantify how far apart the sets {C(H)} are from each other. It is readily seen that the distance been any two such sets is in fact zero.
Lemma 3. Let d(·, ·) be any metric on the space of quantum states. In terms of this metric we define Then This also shows that any set C(H) is arbitrarily close to the set C(G), and raises the question of whether our coarse-graining is meaningful at all. However this is not a problem. Any symmetry properties must always be considered up to some finite resolution scale -arbitrarily small perturbations can always eliminate residual symmetries (like a piece of dust on a sphere), even though this state is practically indistinguishable from the unperturbed one. The uses of Noether's theorem or superselection rules are not ruled out by small perturbations. Therefore any membership of a quantum state ρ to a subset should only be considered up to some smoothing scale based on a distance measure d (such as from the L 1 norm). For each state there is an -ball of nearby states, B (ρ) = {σ : d(ρ, σ) ≤ }. Rather than Iso(ρ), we should consider Iso (ρ) := max σ∈B (ρ) Iso(σ). Intuitively, if a state ρ is within a distance of another state with more residual symmetries, we associate those additional residual symmetries with ρ. The smoothing scale selects the size of symmetry-breaking perturbations that we wish to consider, and often depends on the context.

D. Projecting out residual symmetries via quantum operations
Given the structure of states under the above partition, we can ask how easy it is to move from one set C(H) to another. As discussed one does not in general have quantum operations that map any C(H) neatly into some other subset. Instead it makes sense to consider the setŝ C(H) := W H C(W ). These sets are quite natural to consider because Lemma 2 implies that eachĈ(H) is closed under symmetric operations, and any symmetric operation provides a well-defined mapping ofĈ(H) → C(f (H)).
The quantum operation is the average of the state ρ over a fixed subgroup H weighted by the invariant Haar measure dh. In the case of a finite subgroup of size |H|, the integral H dh is replaced by the sum 1 |H| h∈H .
Lemma 4. The map P H has the following properties: 1. P H is the projector ontoĈ(H).
The mapping P H moves down chains of the subgroup Hasse diagram, however this is not always a symmetric operation. The following tells us when such a transformation can be done for free. [35] of H in G, and therefore if H is a normal subgroup of G (H G) then P H is a symmetric operation.

Lemma 5. Given a group action for
This constrains the kind of resources needed to move from one C(H) to another using P H , since this projects onto a givenĈ(H), although less resource-hungry ways may exist to perform the same transformation.

E. Discussion of the section results
In this section we have given an analysis of the what happens when one classifies quantum states or channels in terms of their residual symmetries. We discussed how composition and mixing affect the ordering, and how one can relate these features to the issue of simulation (exact or approximate) of a target quantum operation with some resource state. We also saw that the ordering has subtleties if one wishes to interpret it in a resource theoretic sense. The statements one makes are also quite blunt, and a good example of this is the fact that the sets {C(H)} are all arbitrarily close to one another. However the relations still carry non-trivial content, and for example Theorem 1 can be viewed as a form of superselection rule on quantum operations that tells us which states are ruled out and which are not.
While this is conceptually neat and provides high-level insight into the complexities of the full classification of states (as described in [7]), it is less clear how computationally useful or simple these structures are in practice. To address this point, in the next section we consider the important case of the two-qubit system under an SU (2) action. We find that already in just this simple scenario the hierarchy of states is quite complex, however the example does provide insight into what to expect in the more general case.
The goal of this section is to illustrate the classification of quantum states in a simple quantum system that has sufficient structure, yet is tractable to the point of being fully solvable. The state space of a two-qubit system is 15 dimensional and is sufficiently non-trivial, moreover there is a very natural group action to consider, namely the tensor product representation of SU (2). The orbits of 2-qubit states have been studied in relation to thermodynamics and correlations within these states [36,37].
Any two qubit state ρ AB can be written with local Bloch vectors a and b and correlation terms determined by the correlation matrix T ij [38]. In general |a| ≤ 1 and |b| ≤ 1. Moreover, we can put all 2-qubit states into diagonal form, where {c i } and {d i } are orthonormal bases of R 3 . The coefficients τ i specify a position (τ 1 , τ 2 , τ 3 ) in a tetrahedron with vertices (−1, −1, −1), (1, 1, −1), (1, −1, 1) and (−1, 1, 1). When a = b = 0, then ρ AB is a quantum state with maximally mixed marginals represented by a point in the tetrahedron. In the case a or b being nonzero, every quantum state ρ AB corresponds to a point in the tetrahedron, however the converse is not true: not all triples (a, b, t) correspond to a valid quantum states. Via local unitaries U 1 ⊗ U 2 , these can be transformed into the canonical form The states in this canonical form with maximally mixed marginals are called T-states [38], The set of T-states is the convex hull of the four Bell states. The corners of the tetrahedron defined by the coefficients {τ i } correspond to the Bell states. We now consider the symmetry properties of 2-qubit states under SU (2) symmetry constraints, assuming an SU (2) tensor product representation for the group ac- Our description of the group orbit as the quotient space G/Iso(ρ) starts with the group manifold of G and eliminates the redundancy caused by the symmetries of ρ. The group transformations in Iso(ρ) have trivial group action on ρ, so we quotient out the equivalence classes g Iso(ρ) := {gh i : h i ∈ Iso(ρ)}.
Certain groups have structure which permits us to further simplify our description of the group orbit manifold. Here we consider SU (2) symmetry constraints, however this SU (2) action has a subgroup, Z 2 = {±1} that is always a symmetry of a quantum state, and which is associated with SU (2) being the double cover of SO (3). Moreover this sub-group has a key property which allows us to simplify our description of the SU (2) isotropy groups to those of SO(3). We can therefore think in terms of SO(3) subgroups, which are the familiar chiral point groups one encounters in for example crystallography [1].  (3), as detailed in [1]. All quantum states have isotropy subgroups under SU (2) that can be related to particular chiral point group.
In the next section we look at the isotropy subgroups of the Bell states, and then go on to classify the symmetry properties of the T-states. Then we continue onto the wider class of 2-qubit states with maximally mixed marginals and finally we complete the classification for general 2-qubit states by considering non-zero local Bloch vectors.

A. Bell States
The Bell states correspond to the extremal points of the tetrahedron of T-states. Their isotropy subgroups under the SU (2) group action can be calculated directly, as described in Appendix B.
The singlet Bell state ψ − is symmetric under SU (2), therefore Iso(ψ − ) = SU(2) and so M(ψ − ) = {e} as expected. The triplet Bell states do break the SU (2) symmetry, with We will call this subgroup K ∞ ∼ = U (1) Z 4 , and it provides an example of a semi-direct product group 1 . Similarly, and Given the triplet Bell states have isomorphic isotropy subgroups, their group orbits will have the same shape, In the previous section we described how normal subgroups simplify our description of the group orbit, and we can use this here to put the description in terms of SO(3) subgroups. There is a 2-to-1 homomorphism from the quotient G/{1, −1}. Elimination of the Z 2 normal subgroup from the group orbit description gives SU (2)/K ∞ ∼ = SO(3)/D ∞ , where D ∞ is the isotropy subgroup of the cylinder.

B. T-States
From the Bell states, we can begin to map out the possible shapes of group orbit on the tetrahedron of Tstates. Consider the convex hull of triplet Bell states to be the base of the tetrahedron, and ψ − to be the peak. The vertical height above the base of this tetrahedron indicates the proportion of the singlet state in the convex 1 The semi-direct product [1] is constructed from two groups, in this case Z 4 and U (1). Each element of the semidirect product group can be thought of as a pairing (h, n) where h ∈ H = Z 4 and n ∈ N = U (1). However the multiplication law of the semidirect product group does not treat the elements of this pairing independently, with (h 1 , n 1 )(h 2 , n 2 ) = (h 1 h 2 , f h 2 (n 1 )n 2 ) where f h : N → N is a automorphism on the second group specified by an element of the first group. The semi-direct product is defined by the choice of automorphism f h . Note that N (N H). If a trivial (identity) automorphism is chosen, where f h (n) = n for any h and n, we have a direct product group [1]. In the U (1) Z 4 example, the group multiplication law is  (2). The remaining states retain their K2 isotropy subgroup. c) Hasse diagram of the observed isotropy subgroups (up to isomorphism) for two qubits. Arrows indicate subset inclusion. This is a sublattice of the full SU (2) subgroup lattice.
Since ψ − is symmetric under the SU (2) group action, therefore the possible types of group orbit for T-states will all be exhibited in the convex hull of the triplet states. These can be visualized as a triangle of states, as shown in Fig. 6. The vertices of this triangle are the triplet Bell states, and each have a K ∞ isotropy subgroup. Appendix B also shows that the midpoints of the edges of this triangle, corresponding to states 1 4 (1 ⊗ 1 + σ i ⊗ σ i ), also have K ∞ isotropy subgroups. Therefore the diagonals of this triangle, corresponding to states for 0 ≤ p < 1, have K ∞ ≺ Iso(ρ). The states not on these diagonals remain invariant under the subgroup K 2 = {±1, ±iX, ±iY, ±iZ}.
The possible isotropy subgroups exhibited by the Tstates can be seen with the projectors P H , as detailed in Appendix C. These are: Werner states along the central axis of the T-state tetrahedron, where τ 1 = τ 2 = τ 3 . These are the states for 0 ≤ p ≤ 1.
T-states with τ i = τ j = τ k . These states lie on or above the diagonals of the triangle of triplet Bell states, as seen in Fig.6, excluding the Werner states on the central axis of the tetrahedron. The smoothed isotropy subgroups are illustrated in the lower part of Fig. 6, with a smoothing scale of = 0.04 under the trace distance. Under this smoothing, the horizontal slices through the tetrahedron are no longer equivalent; near ψ − all states are assigned SU (2) as their isotropy subgroup, while at the base there remain many states with Iso (ρ) = K 2 .

C. Maximally Mixed Marginals
The set of 2-qubit states with maximally mixed marginals is a wider class of states reached from the T-states through local SU (2) unitaries U (g 1 ) ⊗ U (g 2 ). Appendix C details how the isotropy subgroups are enumerated.
If these local unitaries are the same for each qubit, we reach states of the form where the vectors {c i } form an orthonormal basis of R 3 . These states behave similarly to the T-states, but with isomorphic isotropy subgroups.
The possible types of group orbit for these states are the same as for the T-states: The same symmetric states as the T-states, unchanged by the unitary transformation U (g)⊗U (g). These are statesρ T with τ 1 = τ 2 = τ 3 . Further states with maximally mixed marginals are reached from the T-states by different local unitaries on each qubit. Suppose that these two local unitaries have the same generator r · σ, such that U (g 1 ) ⊗ U (g 2 ) = e iθ1r·σ ⊗ e iθ2r·σ . This gives states of the form where {r, c 2 , c 3 } and {r, d 2 , d 3 } are two sets of orthonormal bases for R 3 . If U (g 1 ) = U (g 2 ) in the local unitaries, these sets of bases will be the same and we retrieveρ T . However, in the case that θ 1 = θ 2 in the local unitaries, new possibilities for the isotropy subgroup are introduced: States ρ M with τ 2 = τ 3 . These are reached from T-states with K ∞ isotropy subgroups via local unitaries e iθ1r·σ ⊗ e iθ2r·σ where θ 1 = θ 2 .
The statesρ T are only invariant under a K 2 subgroup because the correlation terms are put into diagonal form by the same orthonormal bases {c i } and {d i } on each qubit. For example, the correlation terms of the T-states are put into the canonical form by the standard cartesian orthonormal basis {x,ŷ,ẑ} on each qubit. When the local unitaries are applied, this can break the K 2 symmetry, and the local bases become 'misaligned'. However, when c 1 = d 1 = r there is still a subgroup Z 4 = {±1, ±i(r · σ)} that leaves these states invariant.
The remaining 2-qubit states with maximally mixed marginals are reached from the T-states through local SU (2) unitaries that do not share a generator, such that U (g 1 ) ⊗ U (g 2 ) = e iθ1r1·σ ⊗ e iθ2r2·σ where |r 1 | = |r 2 | = 1 and r 1 = r 2 . These states can be expressed in the form where {c i } and {d i } are orthonormal bases of R 3 that do not share any elements, i.e. c i = d j for all i and j. These states exhibit another type of group orbit: Statesρ M . The orthonormal bases {c i } and {d i } do not share any elements, therefore the only subgroup that leaves these states invariant is Z 2 = {±1}. These are in a sense maximally asymmetric states, as they have no non-trivial residual symmetries.

D. Local Bloch Vectors
Finally, the introduction of local Bloch vectors completes the set of 2-qubit states. In the states with maximally mixed marginals, the alignment of the orthonormal bases describing the correlation terms played a key role in determining the symmetry properties. The effect of the local Bloch vectors depends on the how they relate to each other, as well as how they relate to the bases for the correlation terms.
There are three possible isotropy subgroups for these states: This will occur for any states whose local Bloch vectors a · σ and b · σ are not parallel or anti-parallel, such that a = γb for any γ ∈ R.
This isotropy subgroup will also occur in states where the basis in which the correlation terms are diagonalised does not align with the local Bloch vectors. These are states of the form

States of the form
where {r, c 2 , c 3 } and {r, d 2 , d 3 } are two sets of orthonormal bases for R 3 , a, b ∈ R and τ 2 = τ 3 . The local Bloch vectors are invariant under the same Z 4 subgroup that leaves the correlation terms invariant. This also includes the case where c i = d i for i = 2, 3. The correlation terms are invariant under a K 2 subgroup, but the local Bloch vectors break this to a Z 4 subgroup.
Similarly, states of the form where {c 1 , c 2 , c 3 } is an orthonormal basis for R 3 and τ 1 = τ 2 = τ 3 . The correlation terms are invariant under a K ∞ subgroup, but the local Bloch vectors break the residual symmetry down to a Z 4 subgroup as the local Bloch vectors are orthogonal to c 3 · σ, the generator of the relevant U (1) subgroup.
These are states This completes the classification of the isotropy subgroups of 2-qubit states.

E. Some simple consequences of the classification
This classification allows us to infer constraints on the resources required to simulate asymmetric channels under symmetry constraints. If we wish to simulate an axial channel (i.e. M(E) = S 2 ) under SU (2) symmetry constraints using the T-states, the resource state must have M(σ) S 2 . T-states on the diagonals of the tetrahedron, with M(ρ) ≺ SO(3)/D ∞ , will not be able to perform such simulations; only the T-states with τ 1 = τ 2 = τ 3 will achieve this task up to some approximation as discussed earlier.
However, convex combinations of different T-states can create more states that break the symmetry to a larger degree. For example, Group orbits also constrain the dynamics of the Tstates. For instance, Lemma 2 implies that symmetric operations move states situated on a diagonal of the tetrahedron only along that diagonal. The T-states are closed under symmetric dynamics -leaving would entail further symmetries being broken.
Also note that H 1 ⊆ H 2 does not imply there exists a symmetric channel from C(H 1 ) into C(H 2 ). T-states on the diagonals of the base of the tetrahedron cannot be reached from those that lie off these diagonals under symmetric operations [25], despite K 2 ≺ K ∞ . Group orbits impose the highest level constraints; further techniques such as those of [25] then impose more detailed constraints.

V. OUTLOOK
Our analysis has shown that the basic problem of state interconversion under a symmetry constraint has a rich and non-trivial structure. We have established a high-level description of the problem that is consistent with the resource-theoretic picture, however the question whether the framework can find practical use in the same way that superselection rules simplify computation remains to be explored.
We have classified the set of all two-qubit states under the SU (2) tensor product representation, and which may be of use in its own right for the study of quantum correlations. However, beyond two-qubits, the general problem can be extremely difficult -for example even the finite subgroups of SU (5) remain unclassified [17]. Despite this complexity, we believe that the tools we have developed here provide useful insight into the structure of information processing under symmetry constraints.
Other techniques for studying the effect of symme-try constraints on quantum operations connect with the present analysis. As already mentioned, recent work on the harmonic analysis of quantum channels suggests that the geometry of group orbits provides an insight into the use of finite quantum resources [25]. The present paper has focused purely on the 'shape' of this orbit, but the role of the induced geometry on the orbits is only partially understood. To extend such an analysis would require adapting techniques in metrology [15] and differential geometry [39]. It would also be of interest to draw connections with recent work [40] that extends classical coarse-graining into the quantum regime with symmetries present. Finally, these ideas may inform us about other resource theories, using the symmetry constraints directly (e.g. in thermodynamics). Along similar lines, the techniques we have described may find application in the context of realising universal quantum computation via the combination of resource states with simple gate-sets, for example Clifford operations and magic states.

VI. ACKNOWLEDGEMENTS
We would like to thank Cristina Cirstoiu, Erick Hinds Mingo, Zoe Holmes and Markus Frembs for many useful discussions. TH is funded by the EPSRC Centre for Doctoral Training in Controlled Quantum Dynamics. DJ is supported by the Royal Society. Proof. Viewing the state σ B as a CPTP map, we can view the simulation as a composite channel Alternatively, from the Covariant Stinespring theorem [41,42], we can write Lemma 2. Under a symmetric operation E, Iso(E(ρ)) Iso(ρ).
Lemma 3. Let d(·, ·) be any metric on the space of quantum states. In terms of this metric we define Then d(C(H 1 ), C(H 2 )) = 0 for all H 1 , H 2 ≺ G.
Proof. For a state ρ 1 ∈ C(H 1 ), Lemma 1 tells us that  Proof. For states ρ ∈ C(W ) for W H, the unitary U (h) for h ∈ H will leave the state unchanged. Therefore We follow the proof in [24]. By Klein's Inequality, we have S(P H (ρ)||σ) ≥ 0, with equality iff σ = P H (ρ), therefore min σ∈Ĉ(H) S(P H (ρ)||σ) = 0.  Proof. For some g ∈ G to satisfy U g •P H •U g −1 = P H , we need gHg −1 = H. This is the definition of the normalizer, so we have that Iso(P H ) = N G (H). Averaging over a normal subgroup H G is therefore symmetric, because N G (H) = G by the definition of a normal subgroup.
In the φ = π case, the conditions become If b = 0, then cos ϕ = sin ϕ = 0, which is not possible. Therefore the only solutions for these conditions is b = 0 and cos ϕ = 0, which gives ϕ = π/2 or 3π/2. In this case a and c are only constrained by a 2 +c 2 = 1. The ϕ = π/2 solution corresponds to the SU (2) transformations since a 2 + c 2 = 1, so we parametrise in terms of cos θ and sin θ. Similarly U (g) = (−iZ)e iθY for the ϕ = 3π/2 case. This shows that This gives the conditions solved in the same way as the previous example. The isotropy subgroup is 3. Iso(ψ + ) = K∞ |ψ + ∝ |vec(X) gives the condition e iϕ(aX+bY +cZ) = e iφ e iϕ(−aX−bY +cZ) .
This is satisfied when which gives the isotropy subgroup as The singlet Bell state |ψ − ∝ |vec(Y) behaves differently to the other Bell states. The isotropy subgroup condition is e iϕ(aX+bY +cZ) = e iφ e iϕ(aX+bY +cZ) , which is satisfied for all elements of SU (2), therefore Iso(ψ − ) = SU(2).
For states, ρ = 1 4 (1 ⊗ 1 + τ i σ i ⊗ σ i ), the 1 ⊗ 1 term can be neglected because Iso(1 ⊗ 1 + ρ ) = Iso(ρ ). For ρ = 1 4 (1⊗1+τ 1 X ⊗X), an SU (2) group element belongs to Iso(ρ) when satisfied when or equivalently This is the same condition as for ψ + , therefore Likewise, elements of Iso(Y ⊗ Y) satisfy e iϕ(aX+bY +cZ) = ±e iϕ(−aX+bY −cZ) , the same as for φ + , therefore while the condition for membership of Iso(Z ⊗ Z) is showing that The isotropy subgroups of 2-qubit states under a tensor product SU (2) group action can be found with projection operators P H . These do not indicate which subgroups of SU (2) will appear as isotropy subgroups, however the following lemma restricts the possibilities.
Lemma 6. For a 2-qubit state ρ transforming under an SU (2) tensor product representation, any cyclic subgroup of the isotropy subgroup Iso(ρ) must be either Z 2 , Z 4 or U (1).

Proof.
A group element belongs to Iso(ρ) when U g (ρ) = ρ, or in vectorised form [43], For ρ to be invariant under the group transformation U g , it requires |vec(ρ) to be in ker [U(g) All groups contain cyclic subgroups from repeated application of a single generator, because g ∈ G implies that g n ∈ G. For any cyclic subgroup of Iso(ρ), there is some basis in which U (g) = diag[e iθ , e −iθ ], denoting a diagonal matrix with those entries. Therefore Assuming for simplicity that U (g) is diagonal in the computational basis, then vectors in for θ = 0, π, and are therefore invariant under a Z 2 subgroup. Finally, span(|0011 , |1100 ) are in ker[U(g) ⊗ U(g) ⊗ U * (g) ⊗ U * (g) − 1] when θ = 0, π/2, π, 3π/2, and hence invariant under a Z 4 subgroup.
This holds for any cyclic subgroup of SU (2), therefore any cyclic subgroup of Iso(ρ) must be Z 2 , Z 2 or U (1).
This plays a similar role to the crystallographic restriction theorem in crystallography [1], and reduces the number of subgroups to check. For example, any 2-qubit state invariant under a Z 6 must be invariant under the U (1) subgroup with the same generator. This allows us to enumerate the possible isotropy subgroups for 2-qubit states. It also suggests that an N -qubit state will have only cyclic subgroups Z 2 , Z 4 , . . . Z 2N , U (1) within Iso(ρ).
The channels P H provide a way to find the sets C(H). Consider two subgroups H 1 and H 2 , with H 1 ≺ H 2 and there are no subgroups H that could be the isotropy subgroup of some state such that H 1 ≺ H ≺ H 2 . If a state ρ satisfies P H1 (ρ) = ρ but P H2 (ρ) = ρ, then Iso(ρ) = H 1 , and ρ ∈ C(H 1 ). We now use this technique to identify the isotropy subgroups of all 2-qubit states.

Z4 ≺ Iso(ρ)
From Lemma 6, if a 2-qubit state has Z 3 ≺ Iso(ρ), then it also has U (1) ≺ Iso(ρ). The next subgroup to check is Z 4 = {±1, ±ir · σ}. For illustrative purposes, let us consider the particular Z 4 subgroup {±1, ±iZ}. For a general 2-qubit state ρ, Now consider the more general Z 4 = {±1, ±ir · σ} subgroup. The useful identities allow us to calculate P Z4 (ρ). The local Bloch vectors give and similarly P Z4 (1 ⊗ b · σ) = (r · b)(1 ⊗ r · σ). The correlation terms may be written where {c i } is an orthonormal basis for R 3 and we choose allows us to show that when j = 1, because r · c j = c 1 · c j = 0. The same holds for the c j · σ ⊗ c 1 · σ terms with j = 1, which also vanish when P Z4 is applied. Finally, when i, j = 1, where {r, c i } and {r, d i } are orthonormal bases of R 3 .

Iso(ρ) = Z2
The image of P Z4 forms a proper subset of the image of P Z2 , therefore there exist 2-qubit states with Iso(ρ) = Z 2 , and these arise in several ways: • States of the form where {c i } and {d i } are orthonormal bases of R 3 and c i = d i for i = 1, 2, 3.
• States of the form where {r, c i } and {r, d i } are orthonormal bases of R 3 but a and/or b are not parallel or anti-parallel to r, i.e. there are no γ 1 , γ 2 ∈ R such that both a = γ 1 r and b = γ 2 r.
• States of the form where a = γb for any γ ∈ R, so the local Bloch vectors are neither parallel nor anti-parallel to each other.

Iso(ρ) = Z4
We have now calculated P H (ρ) for the two subgroups directly above Z 4 on the subgroup lattice that are possible isotropy subgroups. This identifies the 2-qubit states in C(Z 4 ). They are: • States of the form where τ 2 = τ 3 and c i = d i for i = 2, 3.
• States of the form where τ 2 = τ 3 and c i = d i for i = 2, 3, and a, b ∈ R.

K∞ ≺ Iso(ρ)
The next subgroup to consider is K ∞ , which contains the K 2 subgroup. Therefore we need only apply P K∞ to states P K2 (ρ). We can also say the same about states P U (1) (ρ), however the K 2 subgroup projects onto a simpler subset of states.

Iso(ρ) = K∞
These are states in the image of P K∞ but not in that of P SU (2) . They have the form where τ 1 = τ 2 = τ 3 i.e. two of the τ i are the same, but not equal to the third.