The Non-Disjoint Ontic States of the Grassmann Ontological Model, Transformation Contextuality, and the Single Qubit Stabilizer Subtheory
We show that it is possible to construct a preparation non-contextual 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 Wigner-Weyl-Moyal (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 non-disjoint 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 non-contextual 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 non-contextual ontological theory, it must be expressed in terms of non-disjoint ontic states, unlike the case for the odd-dimensional single-qudit stabilizer subtheory.
There has been much interest recently in the study of contextuality by those pursuing the classical simulation of near-term 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, measurement-based quantum computation Raussendorf01; Okay17, and the T-gate 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 best-known form of contextuality, and is the inability to pre-assign 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 non-classical property of quantum mechanics and has been shown to require higher than order terms in the Wigner-Weyl-Moyal (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 single-qubit stabilizer subtheory, which has long been thought to be completely non-contextual 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 non-contextual still exhibit transformation contextuality under the “blowtorch” map Lillystone18. Such a result contradicts the association of the presence of contextuality with the presence of non-classical properties.
In this paper we relate the Grassmann WWM formalism at order to an ontological model—a preparation and measurement non-contextual -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 one-qubit 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 non-disjoint 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 non-disjoint states and show how re-expressing 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 non-disjoint states and establish more of its properties in Section VI. We prove that it is inequivalent to Lillystone et al.’s representative disjoint eight-state model in Section. VII. We conclude in Section VIII.
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:
-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 non-disjoint otherwise. It should be noted that ontological models can be defined over both disjoint and non-disjoint ontic states and past work has been careful to include both cases Leifer14. Non-disjoint ontological models are often treated as a “coarse-graining” of a disjoint ontological model. We will show that in some cases, they must be treated in terms of non-disjoint 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:
An ontological model is -ontic if for any pair of preparation procedures, and , associated with distinct quantum states and , we have for all .
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 eight-state 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 non-contextual Wallman12. Ontic states evolve under the maps corresponding to , and as
respectively. They evolve under the Hadamard gate as
Lillystone et al. then consider evolution of an input state under the two operationally equivalent implementations of the following map:
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 eight-state model these two transformations are non-equivalent 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 eight-state model exhibits transformation contextuality. Lillystone et al. then prove that every one-qubit non-preparation contextual ontological model can be mapped to the eight-state 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 Wigner-Weyl-Moyal (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:
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 :
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 Wigner-Weyl-Moyal formalism at order by solving the following classical equations of motion:
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 non-Clifford 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 , , :
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 eight-state 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 non-contextual 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 eight-state ontological model, which does exhibit dependence on whether or is taken?
We investigate these questions in the following sections by first defining a simple three-state ontological model example in Section IV, which introduces the key element that the eight-state ontological model does not possess: non-disjoint 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 Non-Disjoint Ontic States
The eight-state model is an example of an ontological model with disjoint ontic states. This means that for any two ontic states and , . By contradistinction, non-disjoint ontic states have non-zero 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 non-disjoint due to an additional set of relations that satisfies Kolmogorov’s axioms. Within this simple model, we show how re-expressing 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:
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:
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 “coarse-grained” non-disjoint 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.
Let the tuple refer to the probability on , , , and , respectively. In cases and we can choose or respectively, and define a one-parameter family of probability distributions :
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. 29-32 (unless ). For instance, consider the convex combination of the probability distributions in Fig. 2a and b corresponding to the following tuples in :
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. 29-32. The only way to obtain a convex combination is to convert the disjoint ontic states , , , and , back into the non-disjoint ontic states , and , perform the convex combination that satisfies the old Eqs. 24-27, 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 non-disjoint states and , produces . From Eqs 26-27, 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 non-disjoint 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. 24-27 produce a single convex set with the “coarse” set of non-disjoint states , , and , the “finer” disjoint set , , , and , cannot satisfy them with a single convex set.
Indeed, additional relations can only non-trivially 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 (non-disjoint) 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 non-disjoint 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 non-disjoint 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 non-negative probability distribution defined over states corresponding to the Grassmann monomials . We now re-present 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 ,
where is the dual (odd) Weyl symbol of Kocia17_2. When is the Weyl symbol of an element of a positive-operator valued measure (POVM) , then is a non-negative measure (a probability distribution) over outcomes of state :
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 one-to-one 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 ,
for all one-qubit states .
We note that
and so is the same statement as . We choose to interpret as proportional to the non-negative measure of ontic state (and vice-versa).
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 non-negative probability of stabilizer state in ontic state to be
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 non-negative convex combination, making use of the Grassmann anticommutation relations.
A stabilizer state has the Weyl symbol
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
This further agrees with
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
Therefore, for a map between and the probabilities to preserve ’s convex combinations under its Weyl algebra, it follows that
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 one-to-one mapping between our probabilities and our measure in such that the set of probability distributions, when considered over the non-disjoint 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 non-disjoint ontic states, we can set , , and then add two additional pairs: and so that:
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:
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. 48-50; 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:
Now there are families of solutions that satisfy Eqs. 52-55. As before in Eqs. 33-34, 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. 33-34 extended to three independent pairs:
where . These cases only contain one common probability distribution: the distribution when .
We have thus established that the Grassmann WWM formalism is equivalent to an ontological model defined by three pairs of non-disjoint 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 eight-state 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 eight-state 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 component-wise convex addition is
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. 48-50 and so the same simple component-wise 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 non-negative (a unique form) Kocia17_2:
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.
The eight ontic states of the eight-state 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