The goal of the paper is to check whether the real eigenstates of the observables in the single qudit Pauli group may lead to quantum contextuality, the property that mutually compatible and independent experiments depend on each other. We find that quantum contextuality crops up at dimension twelve in various configurations with a few rays. We use the Shannon capacity for characterizing the corresponding orthogonality graphs. Some arithmetical properties underlying the qudit contextuality are outlined.
The goal of the paper is to check whether the real eigenstates of the observables in the single qudit Pauli group may lead to quantum contextuality, the property that mutually compatible and independent experiments depend on each other. We find that quantum contextuality crops up at dimension twelve in various configurations with a few rays. We use the Shannon capacity for characterizing the corresponding orthogonality graphs. Some arithmetical properties underlying the qudit contextuality are outlined.
The Pauli groups are ubiquitous in quantum information theory because of their usefulness in describing quantum states and operations and their readily understood symmetry properties. In addition, the most well-understood quantum error correcting codes—stabilizer codes—are built using Pauli operators. The eigenstates of these operators—stabilizer states—display a structure (e.g., mutual orthogonality relationships) that has made them useful in examples of multi-qubit non-locality and contextuality. Here, we apply the graph-theoretical contextuality formalism of Cabello, Severini and Winter to sets of stabilizer states, with particular attention to the effect of generalizing two-level qubit systems to odd prime d-level qudit systems. While state-independent contextuality using two-qubit states does not generalize to qudits, we show explicitly how state-dependent contextuality associated with a Bell inequality does generalize. Along the way we note various structural properties of stabilizer states, with respect to their orthogonality relationships, which may be of independent interest.
Contextuality—the obstruction to describing quantum mechanics in a classical statistical way—has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality—the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits , which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for state-independent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality , which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.
Contextuality is a necessary resource for universal quantum computation and non-contextual quantum mechanics can be simulated efficiently by classical computers in many cases. Orders of Planck's constant, ℏ, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of ℏ—all orders higher than ℏ 0 can be interpreted as quantum corrections to the order ℏ 0 term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner–Weyl–Moyal (WWM) formalism that require higher than order ℏ 0 terms to be included in order to violate the classical bounds on their expectation values. As a result, we show that contextuality as a resource is closely related to orders of ℏ as a resource within the WWM formalism. This offers an explanation for why qubits can only exhibit state-independent contextuality under Pauli observables as in the Peres–Mermin square while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, qubit states exhibit contextuality when measured by qubit Pauli observables regardless of the state being measured and so the Weyl symbol of these observables lack an order ℏ 0 contribution altogether. On the other hand, odd-dimensional qudit states exhibit contextuality when measured by qudit observables depending on the state measured and so odd-dimensional qudit observables generally possess non-zero order ℏ 0 terms, and higher, in their WWM formulation: odd-dimensional qudit states that exhibit measurement contextuality have an order ℏ 1 contribution in their expectation values with the observable that allows for the violation of classical bounds while states that have insufficiently large order ℏ 1 contributions do not exhibit measurement contextuality.
It has been recently conjectured that the state-independency of quantum contextuality may be lost when the indistinguishability of identical particles is taken into account. Here, we show that quantum state-independent contextuality exists for any system of more than one identical bosonic qudit, and for most systems of fermionic qudits. The only exception is the case of d fermionic qudits, since there the dimension of the antisymmetric subspace is 1, which is insufficient for contextuality. For all the other cases, either the symmetry precludes the existence of physical states or we provide an explicit method to produce quantum state-independent contextual correlations.
Quantum theory shows a striking nonclassical feature of contextuality, which is also a crucial resource for quantum information processing. In this paper, we study the dependence of violating noncontextuality inequalities against the dimension of quantum systems. To this end, we consider the restrictions imposed on relations among measurements by limitation of dimensions. Based on the graph-theoretic approach to quantum contextuality, we construct noncontextuality inequalities using multiple mutually orthogonal projectors which are unique to high-dimensional quantum systems. We report two typical classes of dimension-dependent noncontextuality inequalities for one of which a violation implies the dimension of the quantum system is higher than a threshold, and for the other, different amounts of violations correspond to different thresholds of dimension. Our paper is expected to inspire more dimension-dependent noncontextuality inequalities.
In the intriguing realm of quantum physics, the concept of contextuality raises interesting questions about the nature of quantum systems. Since the commutation relation in the N-qubits Pauli group corresponds to the colinearity relation in the symplectic polar space W(2N − 1, 2), we study subgeometries of this space in order to discover new contextual quantum configurations. We have developed algorithms and a corresponding software in C language, that can effectively determine quantum contextuality and quantify it, by charting these symplectic polar spaces of various ranks. This tool allowed us to uncover new insights that go beyond the achievements of earlier work, such as that of de Boutray et al (J. Phys. A: Math. Theor. 55 475301, 2022). Our study spans across these geometric structures of ranks from two to seven, offering significant findings.In my talk I will explicitly illustrate the concept of quantum contextuality on certain wellknown finite-geometric configurations living in W(5,2), like the doily (and its two-spread), an elliptic quadric (and its spread of lines) and a hyperbolic quadric (and its associated Heawoodgraph); I will also consider configurations whose contexts are all the subspaces with a given dimension k < N.