The NonDisjoint Ontic States of the Grassmann Ontological Model, Transformation Contextuality, and the Single Qubit Stabilizer Subtheory
Abstract
We show that it is possible to construct a preparation noncontextual ontological model that does not exhibit “transformation contextuality” for single qubits in the stabilizer subtheory. In particular, we consider the “blowtorch” map and show that it does not exhibit transformation contextuality under the Grassmann WignerWeylMoyal (WWM) qubit formalism. Furthermore, the transformation in this formalism can be fully expressed at order and so does not qualify as a candidate quantum phenomenon. In particular, we find that the Grassmann WWM formalism at order corresponds to an ontological model governed by an additional set of constraints arising from the relations defining the Grassmann algebra. Due to this additional set of constraints, the allowed probability distributions in this model do not form a single convex set when expressed in terms of disjoint ontic states and so cannot be mapped to models whose states form a single convex set over disjoint ontic states. However, expressing the Grassmann WWM ontological model in terms of nondisjoint ontic states corresponding to the monomials of the Grassmann algebra results in a single convex set. We further show that a recent result by Lillystone et al. that proves a broad class of preparation and measurement noncontextual ontological models must exhibit transformation contextuality lacks the generality to include the ontological model considered here; Lillystone et al.’s result is appropriately limited to ontological models whose states produce a single convex set when expressed in terms of disjoint ontic states. Therefore, we prove that for the qubit stabilizer subtheory to be captured by a preparation, transformation and measurement noncontextual ontological theory, it must be expressed in terms of nondisjoint ontic states, unlike the case for the odddimensional singlequdit stabilizer subtheory.
I Introduction
There has been much interest recently in the study of contextuality by those pursuing the classical simulation of nearterm quantum computation. This is because of its central role in the extension of many efficiently simulatable systems to quantum universality. Contextuality has been shown to be the salient ingredient introduced in the magic state injection of Clifford circuits Howard14; Raussendorf15, measurementbased quantum computation Raussendorf01; Okay17, and the Tgate extension of the Clifford gateset Gross06; Veitch12; Mari12; Kocia16; Kocia17_2.
Contextuality can be present in the operational forms of preparation contextuality, transformation contextuality and measurement contextuality Spekkens05. Measurement contextuality is perhaps the oldest and bestknown form of contextuality, and is the inability to preassign outcomes to a set of observables without prior knowledge of the “context” that they will be taken in Redhead87; Peres90; Mermin90; Spekkens05. In general, contextuality is believed to be a nonclassical property of quantum mechanics and has been shown to require higher than order terms in the WignerWeylMoyal (WWM) representation of the observables Kocia16; Kocia17_2; Kocia17_3. It is most frequently described in the ontological models formalism, wherein measurement contextuality is responsible for multiple possible outcomes in an ontological model (defined in the next section) where a single outcome is expected Spekkens05.
One of the simplest quantum subtheories is the singlequbit stabilizer subtheory, which has long been thought to be completely noncontextual Wallman12; Blasiak13; Kocia17_2. However, recently, Lillystone et al. proved that for a single qubit, a broad class of ontological models that are preparation and measurement noncontextual still exhibit transformation contextuality under the “blowtorch” map Lillystone18. Such a result contradicts the association of the presence of contextuality with the presence of nonclassical properties.
In this paper we relate the Grassmann WWM formalism at order to an ontological model—a preparation and measurement noncontextual epistemic ontological model for a single qubit—and perform Lillystone et al.’s calculations. We find that the Grassmann WWM ontological model does not exhibit transformation contextuality under the “blowtorch” map or any other map consisting of convex combination of onequbit stabilizer states.
This suggests that there must be some aspect of the Grassmann WWM ontological model that is neither captured by Lillystone et al.’s proof nor by many prior ontological models studied in the literature. In particular, we will consider the Grassmann WWM formalism in the framework of ontological models defined over nondisjoint ontic states. We find that they possess unique properties that are not captured by restricting study to ontological models defined only over disjoint ontic states, as in past studies Spekkens05; Wallman12; Karanjai18. The ontological model corresponding to the Grassmann WWM formalism appears to be an example of a novel subclass of ontological models that seem to have been overlooked in the literature.
We begin with a review of the results of Lillystone et al. Lillystone18 in Section II where we also introduce transformation contextuality in ontological models with disjoint ontic states. In Section III we introduce the Grassmann WWM formalism and demonstrate that it does not exhibit transformation contextuality at order . In Section IV we introduce a simple ontological model over nondisjoint states and show how reexpressing it over disjoint ontic states produces more than one convex subset. This motivates why such ontological models cannot be represented by models with disjoint states. We then demonstrate in Section V that the Grassmann WWM formalism is such an ontological model with nondisjoint states and establish more of its properties in Section VI. We prove that it is inequivalent to Lillystone et al.’s representative disjoint eightstate model in Section. VII. We conclude in Section VIII.
Ii Review
We define ontological models according to Leifer14: An ontological model is defined by a measurable space of possible physical states, with an associated algebra , and sets of measures or measurable functions are used to represent preparations, transformations and measurements in the ontological model. is called the ontic space and elements are called ontic states.
An ontological model is a classical probabilty theory and so must satisfy Kolmogorov’s three axioms:

nonnegativity: ,

,

additivitiy: if are disjoint (i.e. correspond to mutually exclusive events).
From these axioms follow Ash08: for any two subsets , ,

probability of an empty set: ,

the sum rule: , and,

monotonicity: if , then and .
and are disjoint if and nondisjoint otherwise. It should be noted that ontological models can be defined over both disjoint and nondisjoint ontic states and past work has been careful to include both cases Leifer14. Nondisjoint ontological models are often treated as a “coarsegraining” of a disjoint ontological model. We will show that in some cases, they must be treated in terms of nondisjoint states in order that their states form a single convex set.
Furthermore, we can distinguish between two different types of ontological models. From Harrigan et al. Harrigan2010:
Definition 1
An ontological model is ontic if for any pair of preparation procedures, and , associated with distinct quantum states and , we have for all .
Definition 2
If an ontological model fails to be ontic, then it is said to be epistemic.
ontic and epistemic ontological models are both also called “hidden variable theories”. Colloquially, ontic ontological models can be thought of as hidden variable theories where the “hidden” variables are not really hidden (because distinct wavefunctions correspond to distinct subsets of ) while epistemic models are models with truly hidden variables.
Lillystone et al. introduce an eightstate ontological model for one qubit Lillystone18, originally developed in Wallman12, which consists of an ontic space that can be indexed by for , , —the eigenvalues of the Pauli matrices , and , respectively. This model is preparation and measurement noncontextual Wallman12. Ontic states evolve under the maps corresponding to , and as
(1) 
(2) 
and
(3) 
respectively. They evolve under the Hadamard gate as
(4) 
Since , , and are each in , these maps are not continuous; they are permutations on defined by Eqs. 14.
Lillystone et al. then consider evolution of an input state under the two operationally equivalent implementations of the following map:
(5) 
and
(6) 
and so this is often called the “blowtorch” map since it is akin to “taking a blowtorch” to the state and heating it up to become the maximally mixed state (a Gibbs distribution at infinite temperature) Brink. Though , the authors point out that under their eightstate model these two transformations are nonequivalent as they produce different outcomes and thus illustrate “transformation contextuality”. Specifically,
Thus maps ontic states with even (odd) sign parity to ontic states with even (odd) sign parity. On the other hand
maps ontic states with even (odd) sign parity to ontic states with odd (even) sign parity. These two sets of four points are different and therefore the two maps can produce different probability distributions, as shown in Fig. 1.
As a result, this model produces different probability distributions over the ontic states depending on whether or is taken. However, since both maps result in the same result—the maximally mixed state—the resultant probability distribution should be the same. Thus, the eightstate model exhibits transformation contextuality. Lillystone et al. then prove that every onequbit nonpreparation contextual ontological model can be mapped to the eightstate model and so all such models exhibit transformation contextuality. This includes both epistemic and ontic ontological models. We examine their proof carefully in Section VII.
Iii The Blowtorch Map in the Grassmann WWM Formalism
The qubit WignerWeylMoyal (WWM) formalism was originally introduced by Berezin Berezin77 and fully developed in Kocia17_2. The Grassmann model at provides a classical Hamiltonian system that yields spin under canonical quantization. It makes use of , and , three real generators of a Grassmann algebra which obey the anticommutation relation:
(9) 
Any element may be represented as a finite sum of homogeneous monomials of the Grassmann elements and is called a Weyl symbol.
In an effort to examine the and maps in this qubit WWM hidden variable theory, we consider the Weyl symbol of a single qubit pure state :
(10) 
where , for . The ’s make the Weyl symbol real, under a generalized conjugation operation Kocia17_2.
Transformations , , , , and are all Clifford transformations and so can be captured in the WignerWeylMoyal formalism at order by solving the following classical equations of motion:
(11) 
where the right derivative is as defined in Kocia17_2, , , , , for and for . For , these equations of motion are deformed to the Weyl algebra Kocia17_2 for nonClifford unitaries. They are then described by a Weyl bracket instead of a Poisson bracket and the Grassmann elements become the usual Pauli matrices in quantum mechanics. However, since this is unnecessary for Clifford transformations, we will not need to explore this regime.
Clifford transformations take stabilizer states to stabilizer states. Solving the equations of motion for transformations , , , , and , can be written in the same way as the transformations in Lillystone18 by using tuples for , , :
(12) 
(13) 
(14) 
(15) 
and
(16) 
Substituting in the maps given by Eqs. 1216, we find that , , , , and are
(17) 
(18) 
(19) 
(20) 
and
(21) 
respectively.
Thus, we see that under the transformation,
This is the Weyl symbol for . The simplification of the convex combination above is accomplished by the Weyl algebra of . Such a simplification is not possible under , which lacks such algebraic operations.
On the other hand, acting on this evolution with the Hadamard gate to effect transformation produces:
Again, this is the Weyl symbol for . Both of these results are obtained without quantizing the Weyl symbols and so this result is possible all while working at order .
This result raises an interesting question when compared to the result obtained using Lillystone et al.’s eightstate ontological model: since the WWM formalism is able to obtain the maximally mixed state at order regardless of whether map or is taken, does this suggest that there exists an analogous classical probability theory (a preparation noncontextual ontological model) that similarly does not depend on whether transformation or is taken? If so, how can this be reconciled with Lillystone et al.’s proof that every such ontological model can be mapped to their eightstate ontological model, which does exhibit dependence on whether or is taken?
We investigate these questions in the following sections by first defining a simple threestate ontological model example in Section IV, which introduces the key element that the eightstate ontological model does not possess: nondisjoint ontic states. This then leads us to develop a larger ontological model equivalent to the Grassmann WWM formalism in Section V and VI.
Iv Example of a Simple Ontological Model with NonDisjoint Ontic States
The eightstate model is an example of an ontological model with disjoint ontic states. This means that for any two ontic states and , . By contradistinction, nondisjoint ontic states have nonzero overlaps () and so satisfy the classical relation: . This can be derived directly from Kolmogorov’s three axioms as we noted in Section II.
Here we introduce a simple example of a classical probability theory that is defined over only three ontic states, two of which are nondisjoint due to an additional set of relations that satisfies Kolmogorov’s axioms. Within this simple model, we show how reexpressing the ontic states in terms of only disjoint states does not produce a convex set of probability distributions due to this additional set of constraints.
Consider a probability space with three elements, . We wish to deal with a proper probability space, and so must satisfy all of Kolmogorov’s axioms given in Section II.
We specify additional constraints on our probability space that we will show are compatible with these axioms:
(24) 
(25) 
(26) 
and
(27) 
These additional constraints impose that our ontic states and are disjoint.
Axiom is satisfied since and and axiom can be imposed.
Axiom is satisfied since and are not disjoint and so satisfy the sum rule:
(28) 
Since , all probability distributions only cover a part of our ontic space. We show three example probability distributions in Fig. 2 that satisfy the additional constraints given by Eqs. 2427.
It is of course perfectly acceptable to split up our ontic space into four “finer” disjoint ontic states Wallman12; Lillystone18; Karanjai18, which we label , , and as in Fig. 3.
However, while with the “coarsegrained” nondisjoint states (, and ) the probability distributions form a single convex set, with the “atomic” or “finer” disjoint states (, , and ), the additional set of constraints splits this convex set into more than one subset.
To see this, note that for the probability space labelled by the disjoint ontic states , , , and , incorporating the additional system of equations given by Eqs 2427 produces:
(29) 
(30) 
(31) 
(32) 
respectively.
Allowed probability distributions are points in the threesimplex defined by Eq. 31 that also satisfy Eqs 29, 30, and 32. There are only two cases of solutions:

,

.
Let the tuple refer to the probability on , , , and , respectively. In cases and we can choose or respectively, and define a oneparameter family of probability distributions :
(33)  
(34) 
respectively, for . corresponds to the only probability distribution that lies in both cases and is the one indicated by Fig. 2c. These two cases correspond to two convex subsets of the original single convex set.
Convex combination of probability distributions from these two convex sets do not satisfy the constraints given by Eqs. 2932 (unless ). For instance, consider the convex combination of the probability distributions in Fig. 2a and b corresponding to the following tuples in :
(35) 
For the probability space labelled by disjoint ontic states , , , and , since the two terms in the convex combination given by Eq. 35 correspond to two different cases ( and ) with , it follows that their result cannot satisfy Eqs. 2932. The only way to obtain a convex combination is to convert the disjoint ontic states , , , and , back into the nondisjoint ontic states , and , perform the convex combination that satisfies the old Eqs. 2427, and then convert back to the disjoint ontic states.
Doing so, we can find that the convex combination given by Eq. 35, when converted to be in terms of nondisjoint states and , produces . From Eqs 2627, this means that . Moreover, by Eq. 24, the resultant probability distribution must have zero support on and . Converting back to the disjoint states , , , , this means that and . This is represented by the tuple , which is the probability distribution in Fig. 2c.
In other words, given the information that there is a probability of being found in and a probability of being found in , this model enforces that and are nondisjoint and so produces the physically intuitive result that the probability of being found in or is (and so the probability of being found in neither or is too). This is a very different outcome from the one obtained if and are assumed to be disjoint, which given the information that there is a probability of being found in and a probability instead implies that the probability of being found in or is .
Though the constraints given by Eqs. 2427 produce a single convex set with the “coarse” set of nondisjoint states , , and , the “finer” disjoint set , , , and , cannot satisfy them with a single convex set.
Indeed, additional relations can only nontrivially supplement Kolmogorov’s axioms if they produce two or more convex subsets when the ontic states are expressed disjointly (with no overlaps). This is because additional relations that satisfy additivity for all ontic states (i.e. all ontic states are disjoint) add nothing new to the probability theory unless they produce more than one convex subset. However, for the theory to still describe the subtheory of interest, i.e. for the additional relations not to be too constraining, there must exist some other set of (nondisjoint) ontic states with respect to which all the probability distributions fall into the same convex set. This example demonstrates that such a middle ground between “unconstrained” ontological models, which produce one convex set regardless of which set of disjoint or nondisjoint ontic states they are expressed with, and “overconstrained” ontological models, which produce more than one convex set regardless of which set of ontic states they are expressed with, exists. This middle ground consists of constrained ontological models, which produce one convex set with respect to a particular set of nondisjoint ontic states and more than one for all other sets. This possibility appears to have been overlooked in the literature.
V Grassmann WWM as an Ontological Model
In our prior work Kocia17_2 we showed that it is possible to construct a local hidden variable theory (an ontological model) from the Grassmann WWM formalism to describe qubit stabilizer propagation using a nonnegative probability distribution defined over states corresponding to the Grassmann monomials . We now represent these results with respect to the nomenclature used to examine the simple ontological model in Section IV.
A measure on the algebra can be defined for any state ,
(36) 
where is the dual (odd) Weyl symbol of Kocia17_2. When is the Weyl symbol of an element of a positiveoperator valued measure (POVM) , then is a nonnegative measure (a probability distribution) over outcomes of state :
(37) 
However, we cannot rely on the measure as a probability measure over ontic states since it can be negative for as they are not elements of POVMs. Nevertheless, we can define a onetoone map between and a bone fide probability measure that also preserves convex combination if we consider the Weyl algebra that the satisfy.
For the ontic state ,
(38) 
for all onequbit states .
We note that
(39) 
and so is the same statement as . We choose to interpret as proportional to the nonnegative measure of ontic state (and viceversa).
Given a probability , we further choose the probability of the other ontic state, , to be zero under the heuristic motivation that does not need to track something if it is zero. Thus, given an ontic state , we define the nonnegative probability of stabilizer state in ontic state to be
(40) 
where the factor of allows the probability to saturate an upper bound of . We note that this is perhaps an arbitrary definition, we shall see that it is an acceptable one as it produces a theory consistent with Kolmogorov’s axioms once unions and intersections are included, and reproduces the Grassmann WWM formalism for the stabilizer subtheory.
Any single qubit state’s Weyl symbol is represented by a linear combination of Grassmann monomials as in Eq. 10. Thus, our choice of definition for equates the “addition” operator in the Weyl algebra to a “convex addition” operator since it treats any linear combination involving negative coefficients in front of Grassmann monomials as a unique nonnegative convex combination, making use of the Grassmann anticommutation relations.
A stabilizer state has the Weyl symbol
(41) 
We now consider a convex combination of the two distinct stabilizer states and under the Weyl algebra:
for , such that . Note that .
WLOG, let us assume that . Eq. 40 for means that the probability of being in ontic state after this convex combination is two times the coefficient in front of the resultant Weyl symbol’s term, , and the probability of being in ontic state is . Before we simplified the convex combination, and so
(43) 
This further agrees with
(44) 
since . In other words, the convex combination takes a probability density of from ontic state to the intersection between the two ontic states. This means that
(45) 
Therefore, for a map between and the probabilities to preserve ’s convex combinations under its Weyl algebra, it follows that
(46) 
and
(47) 
As a result, we have the same probability space as that considered in the simple example of Section IV, except that instead of one independent pair and , we have three independent pairs. Moreover, we accomplished this via a onetoone mapping between our probabilities and our measure in such that the set of probability distributions, when considered over the nondisjoint ontic states, is a convex set. Most importantly, as we showed in the previous section, these additional constraints satisfy Kolmogorov’s axioms and so form a valid classical probability theory or ontological model.
Using nondisjoint ontic states, we can set , , and then add two additional pairs: and so that:
(48) 
(49) 
and
(50) 
now, and is enforced by Kolmogorov’s first axiom.
The allowed probability distributions all belong in the same family and for , where is the probability to be in and so on, they take the form:
(51) 
where .
Since there are no relations that govern the probabilities between the different pairs, these three sets of ontic states (, , and ) are independent of each other. Convex combinations of any probability distribution defined on these disjoint ontic states produce another probability distribution on the disjoint ontic states that satisfies the constraints given by Eqs. 4850; there is only one convex set of probability distributions.
On the other hand, using disjoint ontic states, we can set , , , and , and then add two additional pairs: and , where we define , , , and in a similar manner.
, , , and satisfy all the constraints that , , , and did:
(52) 
(53) 
(54) 
(55) 
Now there are families of solutions that satisfy Eqs. 5255. As before in Eqs. 3334, we can find that and so we discard when listing these cases , where is the probability of being in and so on. The set of solutions corresponds to all possible permutations of the two solutions given in Eqs. 3334 extended to three independent pairs:
(56)  
(57)  
(58)  
(59)  
(60)  
(61)  
(62)  
(63) 
where . These cases only contain one common probability distribution: the distribution when .
Again, these are eight convex subsets of the simplex of all distributions over ontic states; convex combinations of the probability distributions above do not satisfy Eqs. 5255 (unless ).
We have thus established that the Grassmann WWM formalism is equivalent to an ontological model defined by three pairs of nondisjoint ontic states for the stabilizer subtheory and produces eight convex subsets when expressed in terms of disjoint ontic states. In the subsequent Section VI, we develop more of its properties.
Vi Properties of the Grassmann WWM Ontological Model
In the eightstate model, the ontic space is partitioned into eight disjoint states that are indexed by the eight tuples in :
Convex combinations of these eight tuples defines any valid probability distribution in the eightstate model.
These tuples can be converted into equivalent tuples by defining the tuples to be , where and if the first entry of the corresponding tuple is ‘’ and and if it the first entry is ‘’ and so on. This produces a partition of the ontic space into eight tuples
Using tuples () instead of tuples () simplifies the resultant probability distribution of convex combinations because they can now be represented by a single tuple. For instance, the probability distribution cannot be simplified any further but . For general probability distributions, the equation for componentwise convex addition is
(66) 
The tuple notation is still useful in simplifying convex combinations of ontic states into a single tuple when applied to the Grassmann WWM ontological model’s probability distributions, defined to be . However, now convex combinations of probability distributions must additionally satisfy Eqs. 4850 and so the same simple componentwise addition rule of Eq. 66 does not hold.
Nevertheless, the tuple is useful in another way for the Grassmann WWM ontological model because for probability distributions that correspond to quantum states , its entries correspond to the coefficients in front of the ontic states in the Weyl symbol of the state when it is written with the minimal number of terms such that all coefficients are nonnegative (a unique form) Kocia17_2:
(67) 
where , , etc. Since a stabilizer state is given by Eq. 41, and the entries in a tuple in correspond to , the six stabilizer states correspond to the probability distributions,
Therefore, for stabilizer states the entries in the tuple are five s and a single . This leads to a generalized discrete notion of conserved area or symplecticity for Clifford gates on stabilizer states Kocia17_2. For all these reasons, we will proceed to use this tuple notation from this point onwards.
Note that uniquely identifies any probability distribution in the Grassmann WWM ontological model since and can be determined from and () as we showed in the last Section and Eqs. 5663.
The eight ontic states of the eightstate model given by Eq. VI in the tuple notation, also serve as a valid basis for the convex combination (vector space) operation in the Grassmann WWM ontological model with the