Wheeler's {\it observer-participancy} and the related {\it it from bit} credo refer to quantum non-locality and contextuality. The mystery of these concepts slightly starts unveiling if one encodes the (in)compatibilities between qubit observables in the relevant finite geometries. The main objective of this treatise is to outline another conceptual step forward by employing Grothendieck's {\it dessins d'enfants} to reveal the topological and (non)algebraic machinery underlying the measurement acts and their information content.
Quantum nonlocality can be revealed "via local contextuality" in qudit-qudit entangled systems with $d > 2$, that is, through the violation of inequalities containing Alice-Bob correlations that admit a local description, and Alice-Alice correlations (between the results of sequences of measurements on Alice's subsystem) that admit a local (but contextual) description. A fundamental question to understand the respective roles of entanglement and local contextuality is whether nonlocality via local contextuality exists when the parties have only qubit-qubit entanglement. Here we respond affirma…
We describe a scheme of quantum computation with magic states on qubits for which contextuality is a necessary resource possessed by the magic states. More generally, we establish contextuality as a necessary resource for all schemes of quantum computation with magic states on qubits that satisfy three simple postulates. Furthermore, we identify stringent consistency conditions on such computational schemes, revealing the general structure by which negativity of Wigner functions, hardness of classical simulation of the computation, and contextuality are connected.
Contextuality is a fundamental non-classical property of quantum theory, which has recently been proven to be a key resource for achieving quantum speed-ups in some leading models of quantum computation. However, which of the forms of contextuality, and how much thereof, are required to obtain a speed-up in an arbitrary model of quantum computation remains unclear. In this paper, we show that the relation between contextuality and a compuational advantage is more complicated than previously thought. We achieve this by proving that generalized contextuality is present even within the simplest s…
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 non-linear 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 spe…
A central question in quantum computation is to identify the resources that are responsible for quantum speed-up. Quantum contextuality has been recently shown to be a resource for quantum computation with magic states for odd-prime dimensional qudits and two-dimensional systems with real wavefunctions. The phenomenon of state-independent contextuality poses a priori an obstruction to characterizing the case of regular qubits, the fundamental building block of quantum computation. Here, we establish contextuality of magic states as a necessary resource for a large class of quantum computation …
Whereas complementarity manifests itself via two incompatible observables, quantum contextuality can only be revealed via the joint measurements among at least three observables. By incorporating unsharp measurements and joint measurements into a realistic model, we reestablish an inequality due to Liang, Spekkens, and Wiseman rigorously based on the assumption of noncontextuality alone. Its violation therefore unambiguously pinpoints the quantum contextuality of a two-level system. The maximal violation is attained by three triplewise jointly measurable observables that are pairwise jointly m…
Employing five commuting sets of five-qubit observables, we propose specific 160-661 and 160-21 state proofs of the Bell-Kochen-Specker theorem that are also proofs of Bell's theorem. A histogram of the 'Hilbert-Schmidt' distances between the corresponding maximal bases shows in both cases a noise-like behaviour. The five commuting sets are also ascribed a finite-geometrical meaning in terms of the structure of symplectic polar space W(9,2).
We show that three unsharp binary qubit measurements are enough to violate a generalized noncontextuality inequality, the LSW inequality, in a state-dependent manner. For the case of trine spin axes we calculate the optimal quantum violation of this inequality. Besides, we show that unsharp qubit measurements do not allow a state-independent violation of this inequality. We thus provide a minimal state-dependent proof of measurement contextuality requiring one qubit and three unsharp measurements. Our result rules out generalized noncontextual models of these measurements which were previously…
Contextuality is a phenomenon at the heart of the quantum mechanical departure from classical behaviour, and has been recently identified as a resource in quantum computation. Experimental demonstration of contextuality is thus an important goal. The traditional form of contextuality -- as violation of a Kochen-Specker inequality -- requires a quantum system with at least three levels, and the status of the assumption of determinism used in deriving those inequalities has been controversial. By considering `unsharp' observables, Liang, Spekkens and Wiseman (LSW) derived an inequality for gener…
Contextuality has been conjectured to be a super-classical resource for quantum computation, analogous to the role of non-locality as a super-classical resource for communication. We show that the presence of contextuality places a lower bound on the amount of classical memory required to simulate any quantum sub-theory, thereby establishing a quantitative connection between contextuality and classical simulability. We apply our result to the qubit stabilizer sub-theory, where the presence of state-independent contextuality has been an obstacle in establishing contextuality as a quantum comput…
We present a protocol to evaluate the expectation value of the correlations of measurement outcomes for ensembles of quantum systems, and use it to experimentally demonstrate--under an assumption of fair sampling--the violation of an inequality that is satisfied by any non-contextual hidden-variables (NCHV) theory. The experiment is performed on an ensemble of molecular nuclear spins in the solid state, using established Nuclear Magnetic Resonance (NMR) techniques for quantum information processing (QIP).
Contextuality, a generalization of non-locality, has been proposed as the resource that provides the computational speed-up for quantum computation. For universal quantum computation using qudits, of odd-prime dimension, contextuality has been shown to be a necessary and possibly sufficient resource. However, the role of contextuality in quantum computation with qubits remains open. The $n$-qubit stabilizer formalism, which by itself cannot provide a quantum computer super-polynomial computational advantage over a classical counterpart, is contextual. Therefore contextuality cannot be identifi…
We report on the realization and application of non-destructive three-qubit parity measurements on nuclear spin qubits in diamond. We use high-fidelity quantum logic to map the parity of the joint state of three nuclear spin qubits onto an electronic spin qubit that acts as an ancilla, followed by single-shot non-destructive readout of the ancilla combined with an electron spin echo to ensure outcome-independent evolution of the nuclear spins. Through the sequential application of three such parity measurements, we demonstrate the generation of genuine multipartite entangled states out of the …
It was presented by Cabello and Nakamura [A. Cabello, Phys. Rev. Lett. 90, 190401 (2003)], that the Kochen-Specker theorem applies to two dimensions if one uses Positive Operator-Valued Measures. We show that contextuality in their models is not of the Kochen-Specker type. It is rather the result of not keeping track of the whole system on which the measurement is performed. This is connected to the fact that there is no one-to-one correspondence between POVM elements and projectors on the extended Hilbert space and the same POVM element has to originate from two different projectors when used…
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 an…
We show that qubit stabilizer states can be represented by non-negative quasi-probability distributions associated with a Wigner-Weyl-Moyal formalism where Clifford gates are positive state-independent maps. This is accomplished by generalizing the Wigner-Weyl-Moyal formalism to three generators instead of two---producing an exterior, or Grassmann, algebra---which results in Clifford group gates for qubits that act as a permutation on the finite Weyl phase space points naturally associated with stabilizer states. As a result, a non-negative probability distribution can be associated with each …
Probe of Generic Quantum Contextuality and Nonlocal Resources for Qubits
Hebei Normal University（河北师范大学）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；South China Normal University（华南师范大学）；The University of Hong Kong（香港大学）；Beijing Academy of Quantum Information Sciences（北京量子信息科学研究院）
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 $\hbar^0$ and so does not qualify as a candidate quantum phenomenon. In particular, we find that the Grassmann WWM formalism at order $\hbar^0$ corresponds to …
Randomness is a potential resource for cryptography, simulations and algorithms. Non-local correlations violating Bell's inequality certify the generation of bit strings whose randomness is guaranteed in a device-independent manner. We provide interesting semi-device independent randomness certification protocols by Kochen-Specker (KS) contextuality. For this, we first cast the Mermin's magic-square proof of KS contextuality for two-qubit system as a semi-device independent communication game in prepare-measure scenario. This provides a semi-device independent certification of two-bit of rando…
The most well-known tool for studying contextuality in quantum computation is the n-qubit stabilizer state tableau representation. We provide an extension that describes not only the quantum state, but is also outcome deterministic. The extension enables a value assignment to exponentially many Pauli observables, yet remains quadratic in both memory and computational complexity. Furthermore, we show that the mechanisms employed for contextuality and measurement disturbance are wholly separate. The model will be useful for investigating the role of contextuality in $n$-qubit quantum computation…
One of the defining differences between classical and quantum systems is how measurements affect them. Here, we compare the approaches of contextuality and quantum discord in capturing quantum correlations in special classes of two-qubit states, demonstrating that although non-discordant states are non-contextual, discordant states are not always contextual.
The ontological model framework for an operational theory has generated much interest in recent years. The debate concerning reality of quantum states has been made more precise in this framework. With the introduction of generalized notion of contextuality in this framework, it has been shown that completely mixed state of a qubit is \emph{preparation contextual}. Interestingly, this new idea of preparation contextuality has been used to demonstrate nonlocality of some $ψ$-epistemic models without any use of Bell's inequality. In particular, nonlocality of a non maximally $ψ$-epistemic model …
Exploring quantum phenomena beyond predictions of any classical model has fundamental importance to understand the boundary of classical and quantum descriptions of nature. As a typical property that a quantum system behaves distinctively from a classical counterpart, contextuality has been studied extensively and verified experimentally in systems composed of at least three levels (qutrit). Here we extend the scope of experimental test of contextuality to a minimal quantum system of only two states (qubit) by implementing the minimum error state discrimination on a single $^{171}$Yb$^+$ ion. …
"All-Possible-Worlds" is a novel interpretation of quantum physics, which results from a unified reformulation of Many-Worlds and Copenhagen ("collapse") in the light of quantum contextuality, and proposes "nonlocality at detection" as a principle ruling the whole quantum realm, including single qubit and qutrit experiments.
It is known that there are two non-equivalent embeddings of the split Cayley hexagon of order two into $\mathcal{W}(5,2)$, the binary symplectic polar space of rank three, called classical and skew. Labelling the 63 points of $\mathcal{W}(5,2)$ by the 63 canonical observables of the three-qubit Pauli group subject to the symplectic polarity induced by the (commutation relations between the elements of the) group, the two types of embedding are found to be quantum contextuality sensitive. In particular, we show that the complement of a classically-embedded hexagon is not contextual, whereas tha…
In this article we present a new prepare and measure quantum key distribution protocol that uses an experimentally accessible measure of single qubit contextuality to warranty the security of the quantum channel. The definition of contextuality used is that due to Spekkens in which any underlying hidden variable model of the physical system is noncontextual if the probability distribution of the model's hidden variables is independent of the preparation and measurement context. Under this more general definition of noncontextuality the measurement outcomes of a single qubit can be shown to req…
We present a new and feasible test proving quantum contextuality in four-dimensional Hiltbert space. In our scheme, a contradiction between quantum mechanics and noncontextual hidden variables is revealed through the measurement statistics of two joint measurements of an ensemble of non-entangled qubits. The origin of nonclassicality is fully attributed to the measurement process. Quantum optical realization using photon pairs and linear optics are presented.
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, $\hbar$, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of $\hbar$---all orders higher than $\hbar^0$ can be interpreted as quantum corrections to the order $\hbar^0$ term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner-Weyl-Moyal (WWM) formalism that require higher than order …
We show, under natural assumptions for qubit systems, that measurement-based quantum computations (MBQCs) which compute a non-linear Boolean function with high probability are contextual. The class of contextual MBQCs includes an example which is of practical interest and has a super-polynomial speedup over the best known classical algorithm, namely the quantum algorithm that solves the Discrete Log problem.
I examine Pan and Home's reply to my Comment on their proposal for testing noncontextual models. I show that the Kochen-Specker model for a qubit does explain all outcomes of a test based on such a proposal, so that it would be inconclusive about the untenability of realistic, noncontextual models.
We show that, for any system with a number of levels which can be identified with n qubits, there is an inequality for the correlations between three compatible dichotomic measurements which must be satisfied by any noncontextual theory, but is violated by any quantum state. Remarkably, the violation grows exponentially with n, and the tolerated error per correlation also increases with n, showing that state-independent quantum contextuality is experimentally observable in complex systems.
We uncover a form of quantum contextuality that connects maximal contextuality to boson indistinguihability in a similar way maximal nonlocality with respect to the Clauser-Horne-Shimony-Holt Bell inequality is connected to maximal entanglement. Unlike previous forms of photonic contextuality, this form cannot be simulated with classical light, as it relies on indistinguishability and higher-order interference. Ideal measurements on the bosonic system can be performed by means of dispersive coupling with an ancillary qubit. This allows us delaying at will the ending of each measurement and tar…
Quantum contextuality takes an important place amongst the concepts of quantum computing that bring an advantage over its classical counterpart. For a large class of contextuality proofs, aka. observable-based proofs of the Kochen-Specker Theorem, we formulate the contextuality property as the absence of solutions to a linear system and define for a contextual configuration its degree of contextuality. Then we explain why subgeometries of binary symplectic polar spaces are candidates for contextuality proofs. We report the results of a software that generates these subgeometries, decides their…
An experiment has recently been performed to demonstrate quantum nonlocality by establishing contextuality in one of a pair of photons encoding four qubits; however, low detection efficiencies and use of the fair-sampling hypothesis leave these results open to possible criticism due to the detection loophole. In this Letter, a physically motivated local hidden-variable model is considered as a possible mechanism for explaining the experimentally observed results. The model, though not intrinsically contextual, acquires this quality upon post-selection of coincident detections.
Multiple observers who independently harvest nonclassical correlations from a single physical system share the system's ability to enable quantum correlations. We show that any number of independent observers can share the preparation contextual outcome statistics enabled by state ensembles in quantum theory. Furthermore, we show that even in the presence of any amount of white noise, there exists quantum ensembles that enable such shared preparation contextuality. The findings are experimentally realised by applying sequential unsharp measurements to an optical qubit ensemble which reveals th…
The simulation of quantum effects requires certain classical resources, and quantifying them is an important step in order to characterize the difference between quantum and classical physics. For a simulation of the phenomenon of state-independent quantum contextuality, we show that the minimal amount of memory used by the simulation is the critical resource. We derive optimal simulation strategies for important cases and prove that reproducing the results of sequential measurements on a two-qubit system requires more memory than the information carrying capacity of the system.
We invoke some ideas from finite geometry to map bijectively 135 heptads of mutually commuting three-qubit observables into 135 symmetric four-qubit ones. After labeling the elements of the former set in terms of a seven-dimensional Clifford algebra, we present the bijective map and most pronounced actions of the associated symplectic group on both sets in explicit forms. This formalism is then employed to shed novel light on recently-discovered structural and cardinality properties of an aggregate of three-qubit Mermin's 'magic' pentagrams. Moreover, some intriguing connections with the so-ca…
Dimension witness provides a device-independent certification of the minimal dimension required to reproduce the observed data without imposing assumptions on the functioning of the devices used to generate the experimental statistics. In this paper, we provide a family of Bell expressions where Alice and Bob perform $2^{n-1}$ and $n$ number of dichotomic measurements respectively which serve as the device-independent dimension witnesses of Hilbert space of $2^{m}$ dimensions with $m=1,2..2^{\lfloor n/2\rfloor}$. The family of Bell expressions considered here determines the success probability…
Simulating quantum contextuality with classical systems requires memory. A fundamental yet open question is what is the minimum memory needed and, therefore, the precise sense in which quantum systems outperform classical ones. Here, we make rigorous the notion of classically simulating quantum state-independent contextuality (QSIC) in the case of a single quantum system submitted to an infinite sequence of measurements randomly chosen from a finite QSIC set. We obtain the minimum memory needed to simulate arbitrary QSIC sets via classical systems under the assumption that the simulation shoul…
In the standard formulation of quantum mechanics, there exists an inherent feedback of the measurement setting on the elementary object under scrutiny. Thus one cannot assume that an 'element of reality' prexists to the measurement and, it is even more intriguing that unperformed/counterfactual observables enter the game. This is called quantum contextuality. Simple finite projective geometries are a good way to picture the commutation relations of quantum observables entering the context, at least for systems with two or three parties. In the essay, it is further discovered a mathematical mec…
A theory is universal contextual if its prediction cannot be reproduced by an ontological model satisfying both preparation and measurement noncontextuality assumptions. In this report, we first generalize the logical proofs of quantum preparation and measurement contextuality for qubit system for any odd number of preparations and measurements. Based on the logical proof, we derive testable universally non-contextual inequalities violated by quantum theory. We then propose a class of two-party communication games and show that the average success probability of winning such games is solely li…
We experimentally observed the violation of Kujula-Dzhafarov noncontextuality inequalities by non-separable spin-orbit laser modes. Qubits are encoded on polarization and transverse modes of an intense laser beam. A spin-orbit non-separable mode was produced by means of a radial polarization converter (S-plate). The results show that contextuality can be investigated by using spin-orbit modes in a simple way, corroborating recent works stating that such system can emulate single-photon experiments. Additionally, an improvement in the non-separable mode preparation allowed us to observe a great…
Postselection following weak measurements has long been investigated for its peculiar manifestation of quantum signatures. In particular, the postselected events can give rise to anomalous values lying outside the spectrum of the measured quantity, and may provide enhanced Fisher information. Furthermore, the Pusey inequality highlights that, for extremely weak measurements, non-contextual models can account for the outcome probabilities. It is then interesting to investigate whether these are linked in a unified framework. Here we discuss on the existence of a possible connection in the case …
