On the epistemic view of quantum states
aa r X i v : . [ qu a n t - ph ] J u l On the epistemic view of quantum states
Michael Skotiniotis, Aidan Roy, and Barry C. Sanders
Institute for Quantum Information Science, University of Calgary,2500 University Drive NW,Calgary AB, T2l 1N1, Canada (Dated: December 7, 2018)We investigate the strengths and limitations of the Spekkens toy model, which is a local hiddenvariable model that replicates many important properties of quantum dynamics. First, we presenta set of five axioms that fully encapsulate Spekkens’ toy model. We then test whether these axiomscan be extended to capture more quantum phenomena, by allowing operations on epistemic as wellas ontic states. We discover that the resulting group of operations is isomorphic to the projectiveextended Clifford Group for two qubits. This larger group of operations results in a physicallyunreasonable 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 newoperations do serve as tests for correlation in a two toy bit model, analogous to the well knownHorodecki criterion for the separability of quantum states.
I. INTRODUCTION
Spekkens introduced a toy theory that demonstrates how a local hidden variable model with a classical information-based restriction can capture a great deal of seemingly quantum phenomena, including non-commutativity of mea-surement, remote steering and teleportation [1]. Spekkens’ toy model (STM) builds upon other information-basedmodels with similar aims [2, 3, 4, 5]. Although by no means a proposed axiomatization of quantum theory, STM aimsto strengthen the view that the quantum state is a statistical distribution over a hidden variable space in which thereexists a balance of knowledge and ignorance about the true state of the system.In this paper, we axiomatize STM, and test it by relaxing its axioms. We claim that STM can be formalized intofive axioms describing valid states, allowable transformations, measurement outcomes, and composition of systems.Arguing on empirical grounds, we relax the axiom regarding valid operations on toy bits to obtain larger groups ofoperations for one and two toy bits. We claim that these larger groups are isomorphic to the projective extendedClifford Group for one and two qubits respectively. However, these larger groups of operations contain elements thatdo not necessarily compose under the tensor product. That is to say, there exist operations that do not take validstates to valid states when composed under the tensor product, as one would demand of a physical model. Theseoperations are analogous to positive maps in quantum theory. Just as positive (but not completely positive) mapscan be used to test whether a quantum state is entangled or not [20], validity-preserving (but not completely validity-preserving) maps can be used to test for correlations in the two toy bit STM. Finally, we claim that relaxing thetransformations 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 tothe axioms of quantum theory. We provide a brief review of the original model for an elementary toy system (a toybit) and for two toy bits and provide a number of different ways of representing one and two toy bits. In Section IIIwe propose a relaxation of the criterion for valid operations on elementary systems, identify the resulting groups ofoperations, and analyze both their mathematical and physical properties. We conclude with a discussion of our resultsin 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 anda four-dimensional cube. We show how states, operations, and tensor products stem from the axioms of STM, andwe 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 possessesabout 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 ofquestions that completely determine the actual state of a system. For any given system, at most half of a canonicalset of questions can be answered. The state a system is actually in is called an ontic state, whereas the state ofknowledge 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 onticstates—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 epistemicstates amongst themselves. STM 3: A reproducible measurement is a partition of the ontic states into a set of disjoint epistemic states, withthe outcome of a measurement being a specific epistemic state. The probability of a particular outcome isproportional 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 ofthe measurement. STM 4:
Elementary systems compose under the tensor product giving rise to composite systems; the knowledgebalance 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 Schr¨odinger equationˆ H | ψ i = ı ℏ ∂ | ψ i ∂t (1)whereas ˆ H is a Hermitian operator. QM 3:
Measurement is described by a collection, { M m } , of measurement operators. These are operators acting onthe state space of the system being measured. The index m refers to the measurement outcomes that may occurin the experiment. If the state of the quantum system is | ψ i immediately before the measurement then theprobability that result m occurs is given by p ( m ) = h ψ | ˆ M † m ˆ M m | ψ i , (2)and the state after measurement is given by ˆ M m | ψ i q h ψ | ˆ M † m ˆ M m | ψ i . (3)Measurement operators satisfy P m ˆ M † m ˆ M m = I . QM 4:
The state space of a composite system is the tensor product of the state space of the component systems.The simplest system that can exist is a single toy bit system: there are two yes/no questions in a canonical set,yielding four ontic states, which we label o , o , o , and o . A pair of ontic states forms the answer to one of the twoquestions in a canonical set. The knowledge balance principle restricts us to knowing the answer to at most one oftwo questions, resulting in a pure epistemic state . The six pure states are shown pictorially in Fig. 1. (In Spekkens’original notation, the state e ij was denoted i ∨ j .) e e e e e e o o o o FIG. 1: The six pure epistemic states of the single toy bit model.
By way of example, the questions “Is the ontic state in { o , o } ?” and “Is the ontic state in { o , o } ?” form oneparticular canonical set. The epistemic state e = o + o corresponds to the situation in which the first questioncan be answered, and it is in the affirmative. The model also includes a single mixed epistemic state , namely e = o + o + o + o , corresponding to knowing absolutely nothing about the system.At this point we introduce the linear represention for the toy model which will be convenient for describing operationslater. Let { o , o , o , o } be a basis for a real vector space, and express the epistemic states in that basis. Each pureepistemic state is then a vector with exactly two 1’s and two 0’s; for example, e = . Note that epistemic states that are disjoint (that is, have no ontic states in common) are orthogonal as vectors in R .Now that states in the toy model are defined, we turn our attention to transformations between states. STM 2 statesthat valid operations are permutations of ontic states. The group of permutations of four objects is denoted S ,and permutations are usually summarized using cyclic notation (see [1, p. 7] for details). By way of example, thepermutation (123)(4) maps o to o , o to o , o to o , and o to o . In terms of epistemic states, (123)(4) maps e to e . In the linear representation, each transformation in S is a 4 × . (4)We call this the regular representation of S , 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 ofstates and transformations that may be readily identified with the single toy bit. One such system uses a regulartetrahedron. 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 , then, is o o o o FIG. 2: The regular tetrahedron representation of a toy bit. a symmetry operation on the tetrahedron. For example, the transformation (123)(4) permutes vertices o , o , and o of the tetrahedron by rotating counter-clockwise by 2 π/ o . Since S is the entire group of permutations of { o , o , o , o } , it is also the completegroup of symmetry operations for the regular tetrahedron. Notice that A , the alternating group (or group of evenpermutations), corresponds to the group of rotations, whereas odd permutations correspond to reflections and roto-reflections.As pointed out by Spekkens, another way of viewing the single toy bit is using a toy analogue of the Bloch sphere. Inthe toy Bloch sphere, epistemic states are identified with particular quantum states on the traditional Bloch sphereand are embedded in S accordingly. In particular, e , e , and e are identified with | + i , | ı i , and | i and areembedded on the positive x , y , and z axes respectively (see Fig. 3). States that are orthogonal in the linear model are e = | + i e = |−i e = | i i e = | i e = | i e = |− i i xzy FIG. 3: The Bloch sphere, with both toy and quantum labels. embedded as antipodal points on the toy Bloch sphere, just as orthogonal quantum states are embedded antipodallyon the quantum Bloch sphere. Distance on the toy Bloch sphere corresponds to overlap between states: two epistemicstates have an angle of π/ SO (3), andthey may be characterized using Euler rotations. More precisely, if R x ( θ ) denotes a rotation about the x -axis by θ ,then any T ∈ SO (3) may be written in the form T = R x ( θ ) R z ( φ ) R x ( ψ ) , ≤ θ ≤ π, − π < φ, ψ ≤ π. (5)For example, the rotation by 2 π/ x + y + z axis may be written as R x + y + z ( π ) = R x ( π ) R z (cid:0) − π (cid:1) R x (cid:0) − π (cid:1) (see Fig. 4). ( π ) e e e e e e = ( π ) e e e e e e ( π ) e e e e e e ( π ) e e e e e e FIG. 4: The element R x + y + z (2 π/
3) expressed as a series of Euler rotations.
On the toy Bloch sphere, in contrast, transformations are elements of O (3), not all of which are rotations. For example,the permutation (12)(3)(4) is not a rotation of the toy Bloch sphere but a reflection through the plane perpendicularto the x − y axis (see Fig. 5). Thus, there are operations in the single toy bit model that have no quantum analogue.(We will see shortly that such toy operations correspond to anti-unitary quantum operations.) e e e e e e FIG. 5: The element (12)(3)(4) acts as a reflection on the toy sphere.
The toy operations that do correspond to rotations on the Bloch sphere are precisely the operations in A , the groupof even permutations. In terms of the linear model, these are the transformations of S with determinant 1. Toyoperations not in A may be expressed as a rotation composed with a single reflection. When T is a rotation on thetoy Bloch sphere, its Euler rotations R x ( θ ) R z ( φ ) R x ( − ψ ) satisfy θ ∈ { , π/ , π } and φ, ψ ∈ {− π/ , , π/ , π } . Forexample, the permutation (123)(4) corresponds to the rotation R x + y + z (2 π/
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 questionfrom a canonical set; thus there are a total of six measurements that may be performed. After a measurement isperformed and a result is obtained, the observer has acquired new information about the system and updates his stateof 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 ofgetting “yes” as the outcome is then p mn = r Tmn e ij , (6)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 inQM 3.STM 4 concerns the composition of one or more toy bits. For the case of two toy bits there are four questions ina canonical set, two per bit, giving rise to 16 ontic states, which we denote o ij , i, j = 1 . . .
4. In the linear modelthis is simply the tensor product of the 4-dimensional vector space with itself, and the ontic state o ij is understoodto be o i ⊗ o j . The types of epistemic states arising in this case are of three types; maximal, non maximal, andzero knowledge, corresponding to knowing the answers to two, one, or zero questions respectively. It suffices for ourpurposes to consider only states of maximal knowledge (pure states). These, in Spekkens’ representation, are of twotypes (see Fig. 6), called uncorrelated and correlated states respectively. An uncorrelated state is the tensor product o o o o o o o o o o o o o o o o FIG. 6: (a) Uncorrelated and (b) correlated states in the toy model. of two pure single toy bit states. If each of the single toy bits satisfy the knowledge balance principle, then theircomposition will also satisfy the knowledge balance principle for the composite system. A correlated state is one inwhich nothing is known about the ontic state of each elementary system, but everything is known about the classicalcorrelations between the ontic states of the two elementary toy systems. If the two single bit systems in Fig. 6(b) arelabelled A and B, then nothing is known about the true state of either A or B, but we know that if A is in the state o i , then B is also in the state o i .According to STM 2, operations on two toy bits are permutations of ontic states that map epistemic states toepistemic 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 givensystem, then it should still be valid when an ancilla is added to that system. That is, if T is a valid operation on asingle toy bit, then T ⊗ I ought to be valid on two toy bits. It follows that valid operations should compose underthe tensor product.Finally, STM 3 implies that a measurement of the two toy bit space is a partition of ontic states into disjoint epistemic o o o o o o o o o o o o o o o o FIG. 7: Operations on two toy bits: (a) a tensor product operation and (b) an indecomposable permutation. states: each epistemic state consists of 4 or 8 ontic states. There are in total 105 partitions of the two toy bit spaceinto epistemic states of size 4.In the linear model, epistemic states, operations, and measurements extrapolate in the manner anticipated. A pureepistemic state is a { , } -vector of length 16 containing exactly 4 ones, whereas an operation is a 16 ×
16 permutationmatrix. The group of operations can be computationally verified to be of order 11520. Measurement is a row vector r o ijkl ∈ { , } with the state after measurement updated according to the outcome obtained. In the linear modelSTM 4 is understood as the composition of valid states and operations under the tensor product.Finally, a two toy bit system can be geometrically realized by the four-dimensional cube (see Fig. 8). This is a newrepresentation for the two toy bit system that in some ways generalizes Spekkens’ tetrahedral description of the singletoy bit. By mapping the ontic states o . . . o of an elementary system to the vertices ( x, y ) , x, y ∈ {− , } of a square,the four-dimensional cube is the result of the tensor product of two elementary systems. Every epistemic state is anaffine plane containing four vertices, and the group of permutations of two toy bits is a subgroup of B [3 , , o o o o o o o o o o o o o o o o Y X WZ
FIG. 8: The four-dimensional hypercube representation for the space of two toy bits.
In this section we reviewed STM, identifying its axioms and drawing a correspondence with the axioms of quantumtheory. 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 operationswhich contains a subgroup isomorphic to the projective Clifford Group for two qubits, a characteristic of quantumtheory 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 composeunder 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 operationsact on the ontic states in a reversible manner (ontic determinism). Now consider an empiricist living in a universegoverned by the axioms of STM—a toy universe. Such an empiricist has access only to epistemic states. As a resultan empiricist sees determinism only at the epistemic scale (epistemic determinism); the knowledge balance principleforbids 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.
STM 2 ′ : A valid reversible operation is a linear transformation that permutes the epistemic states of the system.. The requirement that transformations be linear implies that as e = e + e , then T ( e ) = T ( e ) + T ( e )for any valid T : in other words, mixtures of epistemic states are transformed into other mixtures. It follows that pairsof disjoint epistemic states are mapped to other pairs of disjoint states, and the amount of overlap between epistemicstates is preserved. This linearity condition is essential if the toy theory is to emulate significant aspects of quantumtheory. Investigations into a non-linear theory of quantum mechanics [9, 10, 11] have been experimentally tested andfound to be “measurably not different from the linear formalism” [12]. Furthermore it was shown by Peres that anon-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 × S , but it also includes operations such as g √ Z = 12 − − − − , e H = 12 − − − − . (7)On the toy Bloch sphere, T G (1) is the subgroup of operations in O (3) that preserve the set of six pure epistemicstates. On the toy Bloch sphere, Eq. (7), are the Euler rotations g √ Z = R z (cid:16) − π (cid:17) , e H = R x (cid:16) π (cid:17) R z (cid:16) π (cid:17) R x (cid:16) π (cid:17) , (8)0respectively. We have called these operations g √ Z and e H because their action on the toy Bloch sphere resembles thequantum operations √ Z and H respectively.The order of T G (1) is 48, as the next lemma shows.
Lemma 1.
T G (1) is the set of all permutations of { e , e , e , e , e , e } such that pairs of antipodal states aremapped to pairs of antipodal states.Proof. Since
T G (1) contains S as a proper subgroup, T G (1) has order at least 48. Moreover, every element of
T G (1)is a permutation of epistemic states mapping pairs of antipodal points to pairs of antipodal points. We prove thelemma by counting those permutations; as only 48 such operations exist, they must all be in
T G (1).There are three pairs of antipodal states on the toy sphere, namely { e , e } , { e , e } , and { e , e } . Thereforea map that preserves pairs of antipodal points must permute these three pairs: there are 3! = 6 such permutations.Once a pair is chosen, there are two ways to permute the states within a pair. Therefore, there are a total of 3! · = 48distinct permutations that map pairs of antipodal states to pairs of antipodal states.By the argument in Lemma 1, T G (1) may be formally identified with the semidirect product ( Z ) ⋊ S , where g ∈ S acts on Z by g : ( x , x , x ) ( x g (1) , x g (2) , x g (3) ) , ( x , x , x ) ∈ Z . (9)An element of S permutes the three pairs of antipodal states, whereas an element of Z determines whether ornot to permute the states within each antipodal pair. The following result explains how Spekkens’ original group ofoperations fits into T G (1).
Lemma 2. S is the subgroup of Z ⋊ S consisting of elements (( x, y, z ) , g ) such that ( x, y, z ) ∈ Z has Hammingweight of zero or two.Proof. Label the antipodal pairs { e , e } , { e , e } , and { e , e } with their Bloch sphere axes of x , y , and z . Now S is generated by the elements (12)(3)(4), (23)(1)(4), and (34)(1)(2), and by considering the action on the Blochsphere, we see that these elements correspond to ((0 , , , ( z )( xy )), ((0 , , , ( zx )( y )) and ((1 , , , ( z )( xy )) in Z ⋊ S respectively. Note that ((0 , , , ( z )( xy )) and ((0 , , , ( zx )( y )) generate all elements of the form ((0 , , , g ) with g ∈ S , so adding ((1 , , , ( z )( xy )) generates all elements of the form (( x, y, z ) , g ) whereas ( x, y, z ) has Hammingweight zero or two. T G (1) exhibits a relationship with the operations in quantum mechanics acting on a single qubit restricted to the sixstates shown in Fig. 3. To describe the connection, we must first describe the extended Clifford Group.1Recall that the
Pauli Group for a single qubit, denoted P (1), is the group of matrices generated by X = ( ) and Z = ( − ). The Clifford Group , denoted C (1), is the normalizer of the Pauli Group in U (2), and is generated by thematrices (see [14]) H = 1 √ − , √ Z = i , (cid:8) e ıθ I | ≤ θ < π (cid:9) . (10)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) / h e ıθ I i . This is a finite group of 24 elements. For our purposes, the significance of theClifford Group is that it is the largest group in U (2) that acts invariantly on the set of the six quantum states {| i , | i , | + i , |−i , | ı i , |− ı i} ⊂ C (with | ψ i and e ıθ | ψ i considered equivalent).An anti-linear map on C is a transformation T that satisfies the following condition for all u, v ∈ C and α ∈ C : T ( αu + v ) = ¯ αT ( u ) + T ( v ) . (11)Every anti-linear map may be written as a linear map composed with the complex conjugation operation, namelyconj : α | i + β | i 7→ ¯ α | i + ¯ β | i . (12)An anti-unitary map is an anti-linear map that may be written as a unitary map composed with conjugation. Theunitary maps U (2) and their anti-unitary counterparts together form a group, which we denote EU (2). Finally, the extended Clifford Group EC (1) is the normalizer of the Pauli Group in EU (2). Working projectively, EC (1) /U (1) isa finite group of 48 elements, generated by √ Z h e ıθ I i , H h e ıθ I i , and conj h e ıθ I i . For more details about the extendedClifford Group, see for example [15].The following proposition demonstrates the relationship between T G (1) and EC (1) /U (1). Proposition 1.
The toy group
T G (1) is isomorphic to the projective extended Clifford Group EC (1) /U (1) .Proof. By Lemma 1,
T G (1) consists of all possible ways of permuting { e , e , e , e , e , e } such that an-tipodal points are mapped to antipodal points. Now consider the quantum analogues of these states, namely | + i , |−i , | ı i , |− ı i , | i , and | i respectively. For each T h e ıθ I i in EC (1) /U (1), T is a normalizer of the Pauli Group,so T h e ıθ I i acts invariantly on the six quantum states as a set. Since T is also unitary or anti-unitary, it preservesdistance on the Bloch sphere and therefore maps antipodal points to antipodal points. By the argument in Lemma1, there are only 48 such operations, and it is easy to verify that no two elements of EC (1) /U (1) act identically. Itfollows that EC (1) /U (1) and T G (1) are isomorphic, as both are the the group of operations on six points of the Blochsphere that map pairs of antipodal points to pairs of antipodal points.We now look at the composition of two elementary systems. In the linear model of two toy bits, every valid operationis an orthogonal matrix. As STM 2 ′ requires that valid operations map epistemic states to epistemic states reversibly,2 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o FIG. 9: (a) ^ SW AP , (b) f P , (c) f P and (d) f P : four operations on two toy bits. it can be shown that operations such as I ⊗ e HP , with P ∈ S , fail to map correlated states to valid epistemic statesand therefore are not valid operations. On the other hand, operations such as P e H ⊗ Q e H , with P, Q ∈ S , are validunder STM 2 ′ . Let T G (2) denote the group of valid operations for two toy bits. The order of
T G (2) can be verifiedcomputationally to be 23040, and Spekkens’ group of operations is a subgroup of
T G (2).We discover that
T G (2) is very simply related to the extended Clifford Group for two qubits, EC (2). Let P (2) be thePauli Group for two qubits; then the extended Clifford Group for two qubits, EC (2), is the group of all unitary andanti-unitary operators U such that U P (2) U † = P (2) . (13)It is generated by √ Z ⊗ I, I ⊗ √
Z, H ⊗ I, I ⊗ H, CNOT = , (14)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 epistemicstates for two toy bits. The following isomorphism was verified using the computation program GAP [21]. Proposition 2.
T G (2) is isomorphic to EC (2) /U (1) , the two qubit extended Clifford Group modulo phases. We give one such isomorphism explicitly. Let ^ SW AP denote the toy operation that swaps rows and columns of onticstates, and let f P and f P 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)3to T G (2): conj h e ıθ I i 7→ − − − − ⊗ , CNOT h e ıθ I i 7→ ^ SW AP · − − − − ⊗ , ( H ⊗ I ) h e ıθ I i 7→ − − − − ⊗ −
11 1 − − − , ( H ⊗ H ) h e ıθ I i 7→ f P , ( √ Z ⊗ √ Z ) h e ıθ I i 7→ f P . 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 isthe following: while the projective Clifford group contains no maximal subgroups of order 720, Spekkens’ group does.One such maximal subgroup is generated by the operations (12) ⊗ (23), I ⊗ (12), and f P (also shown in Fig. 9).As T G (2) is isomorphic to the extended Clifford group— which contains the Clifford Group as a proper subgroup—therelaxation 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 physicalmodel, we expect that if an operation is valid for a given system, then it should also be valid when we attach anancilla to that system; the operations of T G (2) violate this condition. Consider the operation e H ⊗ I : under STM 2 ′ ,both e H and I are valid operations on an elementary system, yet e H ⊗ I is not a valid operation on the compositesystem, 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 groupsof 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 astests of correlation in the toy theory, as we now explain.4Formally, let A i denote the set of operators acting on the Hilbert space H i . Then a linear map ∆ : A → A is positive if it maps positive operators in A to positive operators in A : in other words, ρ ≥ ρ ≥
0. Onthe other hand ∆ is completely positive if the map∆ ⊗ I : A ⊗ A → A ⊗ A is positive for every identity map I : A → A . In other words, a completely positive map takes valid density operatorsto valid density operators even if an ancilla is attached to the system. Also recall that an operator ρ ∈ A ⊗ A is separable if it can be written in the form ̺ = n X i =1 p i ̺ i ⊗ ˜ ̺ i , (15)for ̺ i ∈ A , ˜ ̺ i ∈ A , and some probability distribution { p i } . A well known result in quantum information is thatpositive maps can distinguish whether or not a state is separable (Theorem 2 [20, p. 5]): Theorem 1.
Let ̺ act on H ⊗ H . Then ̺ is separable if and only if for any positive map ∆ : A → A , the operator (∆ ⊗ I ) ̺ is positive. Theorem 1 says is that maps that are positive but not not completely positive serve as tests for detecting whether ornot a density matrix is separable. An analogous statement can be made for validity preserving maps and correlatedstates in a two toy bit system.Define a transformation ∆ in STM to be validity-preserving if it maps all valid epistemic states to valid epistemicstates in a toy system; all operations in
T G (1) and
T G (2) are validity-preserving. Define ∆ to be completely validity-preserving if ∆ ⊗ I is validity-preserving for every I , where I is the identity transformation on some ancilla toy system.For example, e 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 systemis refined. The perfectly correlated two toy bit states are precisely the correlated pure states: no mixed states areperfectly correlated.
Theorem 2.
Let σ be a two toy bit epistemic state (pure or mixed). Then σ is perfectly correlated if and only if thereexists 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 , where σ is thecorrelated state shown in Fig. 6(b) and P ∈ S is some permutation of the second toy bit system. In this case, the5state ( e H ⊗ I )( I ⊗ P ) σ = ( I ⊗ P )( e H ⊗ I ) σ is an invalid state, as we have already seen that ( e H ⊗ I ) σ is invalid.Next suppose σ is a mixed state. Then either σ is uncorrelated, in which case it has the form e ab ⊗ e , e ⊗ e ab ,or e ⊗ e , 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 isalso a valid state whenever σ is a mixed state. Thus, invalidity of a state under a local validity-preserving map is anecessary 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 ofSTM 2, from ontic to epistemic determinism. We showed that this relaxation gives rise to a group of operations thatis 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 ′ —inits assumptions. The motivation for proposing STM 2 ′ is the empirical fact that in a toy universe, an observer isrestricted to knowledge of epistemic states. We discovered that replacing STM 2 with STM 2 ′ gave rise to a group ofoperations that exhibit an isomorphism with the projective extended Clifford Group of operations (and consequentlythe projective Clifford group of operations) in quantum mechanics. This characteristic is not present in STM; while S 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 arenot 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 Horodeckicriterion for separability in the toy model. The same operations that render the toy model physically unreasonableserve as tools for detecting correlations in the toy model. We believe that the investigation into possible relaxations ofthe axioms of STM gives rise demonstrates the power as well as the limitations of STM. Most significantly, we discoverthat no physically reasonable toy model can arise from relaxing STM 2 to an epistemic perspective; this robustnessis an indication of the model’s power. On the other hand, we conclude that there is at least one characteristic ofquantum theory that the STM cannot capture, an equivalence with the Clifford Group of operations.6 Acknowledgments
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