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arXiv:2504.02944 · Quantum

Quantifiers and witnesses for the nonclassicality of measurements and of states

Yujie Zhang, Yìlè Yīng, David Schmid

Published Jul 30, 2026• 2 citations
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Abstract

In recent work [Phys. Rev. X 16, 021050], we proposed a unified notion of nonclassicality that applies to arbitrary processes in quantum theory, including individual quantum states, measurements, and sets thereof. This notion is derived from the principle of generalized noncontextuality, but in a novel manner that applies to individual processes rather than full experiments or theories. In the present work, we develop semidefinite-programming-based certificates and witnesses for the nonclassicality of states, sources, measurements, and sets thereof. These theory-dependent methods complement theory-independent approaches based on noncontextuality inequalities. We demonstrate the framework through a variety of explicit examples.

DOI
10.22331/q-2026-07-30-2180

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In recent work [Phys. Rev. X 16, 021050], we proposed a unified notion of nonclassicality that applies to arbitrary processes in quantum theory, including individual quantum states, measurements, and sets thereof. This notion is derived from the principle of generalized noncontextuality, but in a novel manner that applies to individual processes rather than full experiments or theories. In the present work, we develop semidefinite-programming-based certificates and witnesses for the nonclassicality of states, sources, measurements, and sets thereof. These theory-dependent methods complement theory-independent approaches based on noncontextuality inequalities. We demonstrate the framework through a variety of explicit examples.
We provide the first systematic technique for deriving witnesses of contextuality in prepare-transform-measure scenarios. More specifically, we show how linear quantifier elimination can be used to compute a polytope of correlations consistent with generalized noncontextuality in such scenarios. This polytope is specified as a set of noncontextuality inequalities that are necessary and sufficient conditions for observed data in the scenario to admit of a classical explanation relative to any linear operational identities, if one ignores some constraints from diagram preservation. While including these latter constraints generally leads to tighter inequalities, it seems that nonlinear quantifier elimination would be required to systematically include them. We also provide a linear program which can certify the nonclassicality of a set of numerical data arising in a prepare-transform-measure experiment. We apply our results to get a robust noncontextuality inequality for transformations that can be violated within the stabilizer subtheory. Finally, we give a simple algorithm for computing all the linear operational identities holding among a given set of states, of transformations, or of measurements.
The predictions of quantum theory resist generalised noncontextual explanations. In addition to the foundational relevance of this fact, the particular extent to which quantum theory violates noncontextuality limits available quantum advantage in communication and information processing. In the first part of this work, we formally define contextuality scenarios via prepare-and-measure experiments, along with the polytope of general contextual behaviours containing the set of quantum contextual behaviours. This framework allows us to recover several properties of set of quantum behaviours in these scenarios, including contextuality scenarios and associated noncontextuality inequalities that require for their violation the individual quantum preparation and measurement procedures to be mixed states and unsharp measurements. With the framework in place, we formulate novel semidefinite programming relaxations for bounding these sets of quantum contextual behaviours. Most significantly, to circumvent the inadequacy of pure states and projective measurements in contextuality scenarios, we present a novel unitary operator based semidefinite relaxation technique. We demonstrate the efficacy of these relaxations by obtaining tight upper bounds on the quantum violation of several noncontextuality inequalities and identifying novel maximally contextual quantum strategies. To further illustrate the versatility of these relaxations, we demonstrate monogamy of preparation contextuality in a tripartite setting, and present a secure semi-device independent quantum key distribution scheme powered by quantum advantage in parity oblivious random access codes.
A key aim of quantum foundations is to characterize the sense in which nature goes beyond classical physics. Understanding nonclassicality is one of our best avenues towards finding a satisfactory interpretation of quantum theory. By determining which classical principles cannot be satisfied in any empirically adequate physical theory, we begin to see which principles can be preserved, which in turn gives us insight into the ontology of the world. These insights then guide us in determining which questions to ask and which experiments to perform next. Furthermore, it is these nonclassical aspects of nature that give rise to new technologies such as the speed-ups of quantum computation or the security of quantum key distribution. \n \nThe gold standard for establishing that a phenomenon is truly nonclassical is to prove that it violates the principle of local causality or the principle of noncontextuality. Much of this thesis reports on my research relating to these two principles. This research primarily involves (i) finding new justifications for our notions of nonclassicality; (ii) refining their fundamental definitions; (iii) quantifying and characterizing their various manifestations; and (iv) finding applications where nonclassical phenomena act as resources for information processing. \n \nUltimately, all of this work on nonclassicality is woven together into a novel framework for physical theories introduced by myself, John Selby, and Rob Spekkens. Its main advantage over preexisting frameworks is that it maintains a clear distinction between which elements of a given physical theory directly describe causal processes, and which refer only to one’s inferences about causal processes. This clarifies a number of confusions in the literature which arose precisely because previous frameworks scrambled causal and inferential concepts. Furthermore, local causality and noncontextuality emerge in this framework as the assumptions that the causal and inferential structures (respectively) that are operationally observed must be respected in the underlying ontology. This work constitutes a first step in developing a new interpretation of quantum theory—the first interpretation designed to satisfy the spirit of both local causality and noncontextuality.
Identifying when observed statistics cannot be explained by any reasonable classical model is a central problem in quantum foundations. A principled and universally applicable approach to defining and identifying nonclassicality is given by the notion of generalized noncontextuality. Here, we study the typicality of contextuality, namely, the likelihood that randomly chosen quantum preparations and measurements produce nonclassical statistics. Using numerical linear programs to test for the existence of a generalized-noncontextual model, we find that contextuality is fairly common: even in experiments with only a modest number of random preparations and measurements, contextuality arises with probability over 99 % . We also show that while typicality of contextuality decreases as the purity (sharpness) of the preparations (measurements) decreases, this dependence is not especially pronounced, so contextuality is fairly typical even in settings with realistic noise. Finally, we show that although nonzero contextuality is quite typical, quantitatively high degrees of contextuality are not as typical, and so large quantum advantages (like for parity-oblivious multiplexing, which we take as a case study) are not as typical. We provide an open-source toolbox that outputs the typicality of contextuality as a function of tunable parameters (such as lower and upper bounds on purity and other constraints on states and measurements). This toolbox can inform the design of experiments that achieve the desired typicality of contextuality for specified experimental constraints.
We show that the phenomenon of quantum contextuality can be used to certify lower bounds on the dimension accessed by the measurement devices. To prove this, we derive bounds for different dimensions and scenarios of the simplest noncontextuality inequalities. Some of the resulting dimension witnesses work independently of the prepared quantum state. Our constructions are robust against noise and imperfections, and we show that a recent experiment can be viewed as an implementation of a state-independent quantum dimension witness.
We introduce an experimentally accessible method to measure a unique degree of nonclassicality, based on the quantum superposition principle, for arbitrary quantum states. We formulate witnesses and test a given state for any particular value of this measure. The construction of optimal tests is presented as well as the general numerical implementation. We apply this approach on examples such as squeezed states, and we show how to formulate conditions to certify a particular degree of nonclassicality for single- and multimode radiation fields.