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On the epistemic view of quantum states

We investigate the strengths and limitations of the Spekkens toy model, which is a local hidden variable model that replicates many important properties of quantum dynamics. First, we present a set of five axioms that fully encapsulate Spekkens' toy model. We then test whether these axioms can be extended to capture more quantum phenomena, by allowing operations on epistemic as well as ontic states. We discover that the resulting group of operations is isomorphic to the projective extended Clifford Group for two qubits. This larger group of operations results in a physically unreasonable model; consequently, we claim that a relaxed definition of valid operations in Spekkens' toy model cannot produce an equivalence with the Clifford Group for two qubits. However, the new operations do serve as tests for correlation in a two toy bit model, analogous to the well known Horodecki criterion for the separability of quantum states.

preserving) maps can be used to test for correlations in the two toy bit STM. Finally, we claim that relaxing the transformations of STM to an epistemic perspective gives rise to physically unreasonable alternatives, and as such, no equivalence with the extended Clifford Group for two qubits can be established by relaxing STM's operations.
The outline of the paper is as follows. In Section II, we present STM as a series of axioms and compare them to the axioms of quantum theory. We provide a brief review of the original model for an elementary toy system (a toy bit) and for two toy bits and provide a number of different ways of representing one and two toy bits. In Section III we propose a relaxation of the criterion for valid operations on elementary systems, identify the resulting groups of operations, and analyze both their mathematical and physical properties. We conclude with a discussion of our results in Section IV.

II. THE SPEKKENS TOY MODEL AND QUANTUM THEORY
In this section we present STM in its axiomatic basis and state the axioms of quantum mechanics for comparison.
Using the axioms of STM we develop several ways of representing toy bits including a vector space, a tetrahedron, and a toy analogue of the Bloch sphere. We also develop two ways of representing two toy bits: a product space and a four-dimensional cube. We show how states, operations, and tensor products stem from the axioms of STM, and we draw parallels to the equivalent axioms and concepts in quantum theory.
STM is based on a simple classical principle called the knowledge balance principle: If one has maximal knowledge, then for every system, at every time, the amount of knowledge one possesses about the ontic state of the system at that time must equal the amount of knowledge one lacks.
Spekkens realizes the knowledge balance principle using canonical sets of yes/no questions, which are minimal sets of questions that completely determine the actual state of a system. For any given system, at most half of a canonical set of questions can be answered. The state a system is actually in is called an ontic state, whereas the state of knowledge is called an epistemic state. STM can be succinctly summarized using the following axioms: STM 0: All systems obey the knowledge balance principle. STM 1: A single toy bit is described by a single hidden variable that can be in 1 of 4 possible states, the ontic states.
The knowledge balance principle insists that the hidden variable is known to be in a subset of 2 or 4 of the ontic states-that subset is the epistemic state of the system. STM 2: A valid reversible operation is a permutation of ontic states of the system that also permutes the epistemic states amongst themselves.

STM 3:
A reproducible measurement is a partition of the ontic states into a set of disjoint epistemic states, with the outcome of a measurement being a specific epistemic state. The probability of a particular outcome is proportional to the number of ontic states that outcome has in common with the current epistemic state.
Immediately after the process of measurement, the epistemic state of the system is updated to the outcome of the measurement.

STM 4:
Elementary systems compose under the tensor product giving rise to composite systems; the knowledge balance principle applies to the composite system as well as to the parts.
To help make the comparison with quantum theory, the corresponding axioms of quantum mechanics are given below [6].
QM 1: Any isolated physical system corresponds to a complex vector space with an inner product, a Hilbert space.
A system is completely described by a ray in Hilbert space. QM 2: Evolution of a closed system is described by a unitary transformation through the Schrödinger equation whereasĤ is a Hermitian operator.
QM 3: Measurement is described by a collection, {M m }, of measurement operators. These are operators acting on the state space of the system being measured. The index m refers to the measurement outcomes that may occur in the experiment. If the state of the quantum system is |ψ immediately before the measurement then the probability that result m occurs is given by and the state after measurement is given byM Measurement operators satisfy mM † mM m = I.

QM 4:
The state space of a composite system is the tensor product of the state space of the component systems.
Note that epistemic states that are disjoint (that is, have no ontic states in common) are orthogonal as vectors in R 4 . Now that states in the toy model are defined, we turn our attention to transformations between states. STM 2 states that valid operations are permutations of ontic states. The group of permutations of four objects is denoted S 4 , and permutations are usually summarized using cyclic notation (see [1, p. 7] for details). By way of example, the We call this the regular representation of S 4 , and we will call this description of STM the linear model.
Since the group of operations on a single toy bit is such a well-studied group, there are other classical systems of states and transformations that may be readily identified with the single toy bit. One such system uses a regular tetrahedron. In this geometric representation, the vertices of the tetrahedron represent the ontic states of the system, whereas pure epistemic states are represented by edges (see Fig. 2). The action of a transformation in S 4 , then, is For example, the rotation by 2π/3 about the x + y + z axis may be written as R x+y+z ( 2π Fig. 4).
, π/2, π} and φ, ψ ∈ {−π/2, 0, π/2, π}. For example, the permutation (123)(4) corresponds to the rotation R x+y+z (2π/3) seen in Fig. 4. STM 3 addresses the problem of measurement in the toy theory. For a single toy bit, a measurement is any one question from a canonical set; thus there are a total of six measurements that may be performed. After a measurement is performed and a result is obtained, the observer has acquired new information about the system and updates his state of knowledge to the result of the measurement. This ensures that a repeat of the question produces the same outcome.
Note that the outcome of a measurement is governed by the ontic state of the system and not the measurement itself.
The question "Is the ontic state in {o m , o n }?" can be represented by a vector r mn = o m + o n . The probability of getting "yes" as the outcome is then where e ij is the current epistemic state of the system. After this outcome, the epistemic state is updated to be e mn .
The vectors r mn and probabilities p mn are analogous to the measurement operators and outcome probabilities in  of two pure single toy bit states. If each of the single toy bits satisfy the knowledge balance principle, then their composition will also satisfy the knowledge balance principle for the composite system. A correlated state is one in which nothing is known about the ontic state of each elementary system, but everything is known about the classical correlations between the ontic states of the two elementary toy systems. If the two single bit systems in Fig. 6 According to STM 2, operations on two toy bits are permutations of ontic states that map epistemic states to epistemic states. These permutations are of two types: tensor products of permutations on the individual systems, and indecomposable permutations (see Fig. 7). Moreover, STM 4 suggests that if an operation is valid on a given system, then it should still be valid when an ancilla is added to that system. That is, if T is a valid operation on a single toy bit, then T ⊗ I ought to be valid on two toy bits. It follows that valid operations should compose under the tensor product.
Finally, STM 3 implies that a measurement of the two toy bit space is a partition of ontic states into disjoint epistemic Finally, a two toy bit system can be geometrically realized by the four-dimensional cube (see Fig. 8). This is a new representation for the two toy bit system that in some ways generalizes Spekkens' tetrahedral description of the single In this section we reviewed STM, identifying its axioms and drawing a correspondence with the axioms of quantum theory. In the next section, we investigate a relaxation of STM 2.
III. RELAXING THE SPEKKENS TOY MODEL.
In this section we relax STM 2, the axiom describing valid reversible operations. We obtain a new group of operations which contains a subgroup isomorphic to the projective Clifford Group for two qubits, a characteristic of quantum theory not captured by STM. However, the operations in these new group fail to compose under the tensor product, rendering the relaxation of STM 2 physically unreasonable. Nevertheless, we claim that operations that fail to compose under the tensor product can be used as tests for correlations in STM.
Recall that STM 2 describes valid operations on toy states. In particular, STM 2 requires that valid operations act on the ontic states in a reversible manner (ontic determinism). Now consider an empiricist living in a universe governed by the axioms of STM-a toy universe. Such an empiricist has access only to epistemic states. As a result an empiricist sees determinism only at the epistemic scale (epistemic determinism); the knowledge balance principle forbids exact knowledge of the ontic state of the system. For an empiricist, ontic determinism is too strict a condition.
We thus propose the following amendment. states is preserved. This linearity condition is essential if the toy theory is to emulate significant aspects of quantum theory. Investigations into a non-linear theory of quantum mechanics [9,10,11] have been experimentally tested and found to be "measurably not different from the linear formalism" [12]. Furthermore it was shown by Peres that a non-linear quantum mechanical theory would violate the second law of thermodynamics [13].
We let T G(1) denote the group of operations obtained by replacing STM 2 with STM 2 ′ . In terms of the linear model, an operation is in T G(1) if it can be represented as a 4 × 4 orthogonal matrix that maps epistemic states to epistemic states. This includes all the operations in S 4 , but it also includes operations such as On the toy Bloch sphere, T G(1) is the subgroup of operations in O(3) that preserve the set of six pure epistemic states. On the toy Bloch sphere, Eq. (7), are the Euler rotations respectively. We have called these operations √ Z and H because their action on the toy Bloch sphere resembles the quantum operations √ Z and H respectively.
The order of T G(1) is 48, as the next lemma shows. Once a pair is chosen, there are two ways to permute the states within a pair. Therefore, there are a total of 3!·2 3 = 48 distinct permutations that map pairs of antipodal states to pairs of antipodal states.
T G(1) exhibits a relationship with the operations in quantum mechanics acting on a single qubit restricted to the six states shown in Fig. 3. To describe the connection, we must first describe the extended Clifford Group.
Recall that the Pauli Group for a single qubit, denoted P(1), is the group of matrices generated by X = ( 0 1 1 0 ) and Z = ( 1 0 0 −1 ). The Clifford Group, denoted C(1), is the normalizer of the Pauli Group in U (2), and is generated by the matrices (see [14]) Since U and e ıθ U are equivalent as quantum operations, we focus on the projective group of Clifford operations, namely C(1)/U (1) ∼ = C(1)/ e ıθ I . This is a finite group of 24 elements. For our purposes, the significance of the Clifford Group is that it is the largest group in U (2) that acts invariantly on the set of the six quantum states {|0 , |1 , |+ , |− , |ı , |−ı } ⊂ C 2 (with |ψ and e ıθ |ψ considered equivalent).
An anti-linear map on C 2 is a transformation T that satisfies the following condition for all u, v ∈ C 2 and α ∈ C: Every anti-linear map may be written as a linear map composed with the complex conjugation operation, namely We now look at the composition of two elementary systems. In the linear model of two toy bits, every valid operation is an orthogonal matrix. As STM 2 ′ requires that valid operations map epistemic states to epistemic states reversibly, We discover that T G(2) is very simply related to the extended Clifford Group for two qubits, EC(2). Let P(2) be the Pauli Group for two qubits; then the extended Clifford Group for two qubits, EC (2), is the group of all unitary and anti-unitary operators U such that It is generated by the conjugation operation, and unitary multiples of the identity matrix. Working projectively, it can be shown that EC(2)/U (1) is a group of order 23040 (see [15]). The two-qubit Clifford Group C(2) is a subgroup of EC(2), and C (2) is the largest group in U (4) that acts invariantly on a set of sixty states; this is the same size as the set of epistemic states for two toy bits. The following isomorphism was verified using the computation program GAP [21].
We give one such isomorphism explicitly. Let SW AP denote the toy operation that swaps rows and columns of ontic states, and let P 1 and P 2 be as shown in Fig. 9. For convenience, we use the generating set {conj, CNOT, H ⊗ I, H ⊗ H, √ Z ⊗ √ Z} for EC (2). Then the following map, extended to the entire group, is an isomorphism from EC(2)/U (1) to T G (2): A similar GAP computation shows that Spekkens' group of operations for two toy bits is not isomorphic to C(2)/U (1), despite the fact that both groups have 11520 elements. One way to verify that the two groups are not isomorphic is the following: while the projective Clifford group contains no maximal subgroups of order 720, Spekkens' group does.
As T G(2) is isomorphic to the extended Clifford group-which contains the Clifford Group as a proper subgroup-the relaxation of STM 2 to STM 2 ′ results in a group of operations that is isomorphic to the Clifford Group of two qubits.
We emphasize that this equivalence is a direct consequence of applying empiricism to STM.
Unfortunately, the relaxation of STM 2 to STM 2 ′ gives rise to a physically unreasonable state of affairs. For a physical model, we expect that if an operation is valid for a given system, then it should also be valid when we attach an ancilla to that system; the operations of T G(2) violate this condition. Consider the operation H ⊗ I: under STM 2 ′ , both H and I are valid operations on an elementary system, yet H ⊗ I is not a valid operation on the composite system, as it fails to map the correlated state shown in Fig. 6(b) to a valid epistemic state. In fact, the subgroups of T G(1) and T G(2) that preserve valid epistemic states when an ancilla is added are simply Spekkens' original groups of operations for one and two toy bits respectively.
However, just as positive maps serve as tests for entanglement in quantum theory, validity-preserving maps serve as tests of correlation in the toy theory, as we now explain.
is positive for every identity map I : A 3 → A 3 . In other words, a completely positive map takes valid density operators to valid density operators even if an ancilla is attached to the system. Also recall that an operator ρ ∈ A 1 ⊗ A 2 is separable if it can be written in the form Theorem 1 says is that maps that are positive but not not completely positive serve as tests for detecting whether or not a density matrix is separable. An analogous statement can be made for validity preserving maps and correlated states in a two toy bit system.
Define a transformation ∆ in STM to be validity-preserving if it maps all valid epistemic states to valid epistemic states in a toy system; all operations in T G(1) and T G(2) are validity-preserving. Define ∆ to be completely validitypreserving if ∆⊗I is validity-preserving for every I, where I is the identity transformation on some ancilla toy system.
For example, H ∈ T G(1) is validity-preserving but not completely validity-preserving. Finally, a two toy bit state is perfectly correlated if for any acquisition of knowledge about one of the systems, the description of the other system is refined. The perfectly correlated two toy bit states are precisely the correlated pure states: no mixed states are perfectly correlated.
Theorem 2. Let σ be a two toy bit epistemic state (pure or mixed). Then σ is perfectly correlated if and only if there exists a one toy bit validity-preserving operation ∆ such that (∆ ⊗ I)σ is an invalid two toy bit state.
Proof. First suppose σ is a pure state. If σ is uncorrelated, then it has the form e ab ⊗ e cd , and for any ∆ ∈ T G(1), the state (∆ ⊗ I)(e ab ⊗ e cd ) = (∆e ab ) ⊗ e cd is a valid two toy bit state. On the other hand, if σ is correlated, then it has the form (I ⊗ P )σ 0 , where σ 0 is the correlated state shown in Fig. 6(b) and P ∈ S 4 is some permutation of the second toy bit system. In this case, the state ( H ⊗ I)(I ⊗ P )σ 0 = (I ⊗ P )( H ⊗ I)σ 0 is an invalid state, as we have already seen that ( H ⊗ I)σ 0 is invalid.
Next suppose σ is a mixed state. Then either σ is uncorrelated, in which case it has the form e ab ⊗ e 1234 , e 1234 ⊗ e ab , or e 1234 ⊗ e 1234 , or it is correlated, in which case it has the form (e ab ⊗ e cd + e mn ⊗ e pq ), with {a, b} disjoint from {m, n} and {c, d} disjoint from {p, q}. Any of these mixed states may be written as a sum of pure uncorrelated states.
Since pure uncorrelated states remain valid under ∆ ⊗ I for any validity preserving ∆, it follows that (∆ ⊗ I)σ is also a valid state whenever σ is a mixed state. Thus, invalidity of a state under a local validity-preserving map is a necessary and sufficient condition for a bipartite epistemic state (pure or mixed) to have perfect correlation.
In this section we introduced a possible relaxation of STM. Motivated by empiricism, we argued for the relaxation of STM 2, from ontic to epistemic determinism. We showed that this relaxation gives rise to a group of operations that is equivalent to the projective extended Clifford Group for one and two qubits. However, the operations of T G(1) and T G(2) are physically unreasonable as they do not represent completely validity-preserving maps. They do, however, serve as tests for correlations in the toy model. In the next section we discuss these results further.

IV. DISCUSSION
In this paper we formulated STM in an axiomatic framework and considered a possible relaxation-STM 2 ′ -in its assumptions. The motivation for proposing STM 2 ′ is the empirical fact that in a toy universe, an observer is restricted to knowledge of epistemic states. We discovered that replacing STM 2 with STM 2 ′ gave rise to a group of operations that exhibit an isomorphism with the projective extended Clifford Group of operations (and consequently the projective Clifford group of operations) in quantum mechanics. This characteristic is not present in STM; while S 4 is isomorphic to C(1)/U (1), the group of operations for two toy bits in STM is not isomorphic to C(2)/U (1).
However, due to the fact that operations arising from STM 2 ′ do not compose under the tensor product-they are not completely validity-preserving-the proposed relaxation does not give rise to a physically reasonable model.
Despite this failure, the group of operations generated by STM 2 ′ gives rise to a very useful tool; namely, the Horodecki criterion for separability in the toy model. The same operations that render the toy model physically unreasonable serve as tools for detecting correlations in the toy model. We believe that the investigation into possible relaxations of the axioms of STM gives rise demonstrates the power as well as the limitations of STM. Most significantly, we discover that no physically reasonable toy model can arise from relaxing STM 2 to an epistemic perspective; this robustness is an indication of the model's power. On the other hand, we conclude that there is at least one characteristic of quantum theory that the STM cannot capture, an equivalence with the Clifford Group of operations.