Matthew B. Elliott
The central focus of this work is to make progress towards understanding entanglement as a resource for computation by examining the quantum correlations that can be extracted from stabilizer states. As such, we focus on the stabilizer formalism, local realism, and the convergence of the two.
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We show that quantum mechanics predicts a contradiction with local hidden variable theories for photon number measurements which have limited resolving power, to the point of imposing an uncertainty in the photon number result which is macroscopic in absolute terms. We show how this can be interpreted as a failure of a new premise, macroscopic local realism.
We study the relation between local unitary (LU) equivalence and local Clifford (LC) equivalence of stabilizer states. We introduce a large subclass of stabilizer states, such that every two LU equivalent states in this class are necessarily LC equivalent. Together with earlier results, this shows that LC, LU and SLOCC equivalence are the same notions for this class of stabilizer states. Moreover, recognizing whether two given stabilizer states in the present subclass are locally equivalent only requires a polynomial number of operations in the number of qubits.
The classification of stabilizer states under local Clifford (LC) equivalence is of particular importance in quantum error-correction and measurement-based quantum computation. Two stabilizer states are called LC equivalent if there exists a local Clifford operation which maps the first state to the second. We present a finite set of invariants which completely characterizes the LC equivalence class of any stabilizer state. Our invariants have simple descriptions within the binary framework in which stabilizer states are usually described.
The equivalence of stabilizer states under local transformations is of fundamental interest in understanding properties and uses of entanglement. Two stabilizer states are equivalent under the usual stochastic local operations and classical communication criterion if and only if they are equivalent under local unitary (LU) operations. More surprisingly, under certain conditions, two LU equivalent stabilizer states are also equivalent under local Clifford (LC) operations, as was shown by Van den Nest et al. [Phys. Rev. \textbf{A71}, 062323]. Here, we broaden the class of stabilizer states for w…
We suggest to test the premise of ``macroscopic local realism'' which is sufficient to derive Bell inequalities when measurements of photon number are only accurate to an uncertainty of order $n$ photons, where $n$ is macroscopic. Macroscopic local realism is only sufficient to imply, in the context of the original Einstein-Podolsky-Rosen argument, fuzzy ``elements of reality'' which have a macroscopic indeterminacy. We show therefore how the violation of local realism in the presence of macroscopic uncertainties implies the failure of ``macroscopic local realism''. Quantum states violating th…
The conjecture is made that quantum mechanics is compatible with local hidden variables (or local realism). The conjecture seems to be ruled out by the theoretical argument of Bell, but it is supported by the empirical fact that nobody has been able to perform a loophole-free test of local realism in spite of renewed effort during almost 40 years.
We classify, up to local unitary equivalence, local unitary stabilizer Lie algebras for symmetric mixed states into six classes. These include the stabilizer types of the Werner states, the GHZ state and its generalizations, and Dicke states. For all but the zero algebra, we classify entanglement types (local unitary equivalence classes) of symmetric mixed states that have those stabilizers. We make use of the identification of symmetric density matrices with polynomials in three variables with real coefficients and apply the representation theory of SO(3) on this space of polynomials.
A necessary and natural change in our application of quantum mechanics to separated systems is shown to reconcile quantum mechanics and local realism. An analysis of separation and localization justifies the proposed change in application of quantum mechanics. An important EPRB experiment is reconsidered and it is seen that when it is correctly interpreted it supports local realism. This reconciliation of quantum mechanics with local realism allows the axiom sets of quantum mechanics, probability, and special relativity to be joined in a consistent global axiom set for physics.
Quantum mechanics is strictly incompatible with local realism. It has been shown by Bell and others that it is possible, in principle, to experimentally differentiate between local realism and quantum mechanics. Numerous experiments have attempted to falsify local realism; however, they have consistently failed to close the detection loophole under strict locality conditions, thereby allowing local realistic explanations for their observations. In 2015, three experiments took place that tested local realism without the impediments of these significant loopholes. Between these three experiments…
Generalised quantum measurements with two outcomes are fully characterised by two real parameters, dubbed as sharpness parameter and biasedness parameter and they can be linked with different aspects of the experimental setup. It is known that precision of measurements, characterised by the sharpness parameter of the measurements, reduces the possibility of probing quantum features like violation of local-realism (LR) or macro-realism (MR). Here we investigate the effect of biasedness together with sharpness of measurement and find a trade-off between those two parameters in the context of pro…
We show that failure of local realism can be revealed to observers for whom only extremely coarse-grained measurements are available. In our instances, Bell's inequality is violated even up to the maximum limit while both the local measurements and the initial local states under scrutiny approach the classical limit. Furthermore, we can observe failure of local realism when an inequality enforced by non-local realistic theories is satisfied. This suggests that locality alone may be violated while realism cannot be excluded for specific observables and states. Small-scale experimental demonstra…
We investigate the violation of non-local realism using entangled coherent states (ECS) under nonlinear operations and homodyne measurements. We address recently proposed Leggett-type inequalities, including a class of optimized incompatibility ones and thoroughly assess the effects of detection inefficiency.
Algorithm to Verify Local Equivalence of Stabilizer States
提出将图态与稳定子态局域幺正等价性验证简化为模运算线性方程组求解的算法，证明图态间LU变换的特定形式并推广LC变换类，验证n≤11时稳定子态LU与LC轨道数量一致。
Local realism is in conflict with special quantum Bell-type models. Recently, several experiments have demonstrated violation of local realism if we trust their setup assuming special relativity valid. In this paper we question the assumption of relativity, point out not commented anomalies and show that the experiments have not closed objectivity loophole because clonability of the result has not been demonstrated. We propose several improvements in further experimental tests of local realism make the violation more convincing.
k-Uniform complete hypergraph states stabilizers in terms of local operators
本研究提出了一种将k-均匀完全超图状态稳定子表示为局部算符线性组合的方法，为超图状态在稳定子形式化中的研究提供了新视角。
It is shown that a careful analysis of the `wrong' events (those not present in the usual formulations of the GHZ argument), which are a necessary feature of the tests of local realism involving independent sources, permits one to show that there is no local realistic model, which is capable to describe recent GHZ experiments performed in Innsbruck
We exhibit a local-hidden-variable model in agreement with the results of the two-photon coincidence experiment made by Torgerson et al. [Phys. Lett. A 204 (1995) 323]. The existence of any such model shows that the experiment does not exclude local realism.
It is argued that local realism is a fundamental principle, which might be rejected only if experiments clearly show that it is untenable. A critical review is presented of the derivations of Bell's inequalities and the performed experiments, with the conclusion that no valid, loophole-free, test exists of local realism vs. quantum mechanics. It is pointed out that, without any essential modification, quantum mechanics might be compatible with local realism. This suggests that the principle may be respected by nature.
Local unitary stabilizer subgroups constitute powerful invariants for distinguishing various types of multipartite entanglement. In this paper, we show how stabilizers can be used as a basis for entanglement verification protocols on distributed quantum networks using minimal resources. As an example, we develop and analyze the performance of a protocol to verify membership in the space of Werner states, that is, multi-qubit states that are invariant under the action of any 1-qubit unitary applied to all the qubits.
Quantum codewords are highly entangled combinations of two-state systems. The standard assumptions of local realism lead to logical contradictions similar to those found by Bell, Kochen and Specker, Greenberger, Horne and Zeilinger, and Mermin. The new contradictions have some noteworthy features that did not appear in the older ones.
Following the theoretical suggestion of Ref. [1,2], we present experimental results addressed to test restricted families of local realistic models, but without relying on the fair sampling assumption.
I make some critical comments on the article "An experimental test of non-local realism", by S. Groblacher, T. Paterek, R. Kaltenbaek, C. Brukner, M. Zukowski, M. Aspelmeyer, & A. Zeilinger, Nature 446, p. 871-875 (2007). This article makes incorrect claims concerning Bell's theorem. Moreover, I point to the fact that a hypothesis referred to as "realism" in the article (Groblacher et al.) is not used in the deduction of Leggett's inequality.
Stabilizer states constitute a set of pure states which plays a dominant role in quantum error correction, measurement--based quantum computation, and quantum communication. Central in these applications are the local symmetries of these states. We characterize all local symmetries of arbitrary stabilizer states and provide an algorithm which determines them. We demonstrate the usefulness of these results by showing that the additional local symmetries find applications in entanglement theory and quantum error correction.
We present a linear program that is capable of determining whether a set of correlations can be captured by a local realistic model. If the correlations can be described by such a model, the linear program outputs a joint probability distribution that produces the given correlations. If the correlations cannot be described under the assumption of local realism, the program outputs a Bell inequality violated by the correlations.
The structure of a local hidden variable model for experiments involving sequences of measurements rigorously is analyzed. Constraints imposed by local realism on the conditional probabilities of the outcomes of such measurement schemes are explicitly derived. The violation of local realism in the case of ``hidden nonlocality'' is illustrated by an operational example.
We show that it is, in principle, possible to perform local realism violating experiments of the Hardy type in which only position and momentum measurements are made on two particles emanating from a common source. In the optical domain, homodyne detection of the in-phase and out-of-phase amplitude components of an electromagnetic field is analogous to position and momentum measurement. Hence, local realism violations of the Hardy type are possible in optical systems employing only homodyne detection.
A standard approach in the foundations of quantum mechanics studies local realism and hidden variables models exclusively in terms of violations of Bell-like inequalities. Thus quantum nonlocality is tied to the celebrated no-go theorems, and these comprise a long list that includes the Kochen-Specker and Bell theorems, as well as elegant refinements by Mermin, Peres, Hardy, GHZ, and many others. Typically entanglement or carefully prepared multipartite systems have been considered essential for violations of local realism and for understanding quantum nonlocality. Here we show, to the contrar…
It is shown that the model introduced by Cabello et al. to criticize the Franson experiment suffers from the same weakness as the previous model introduced by Aerts et al. with the same purpose. It is also shown why we can assume with confidence that the Franson experiment does violate local realism.
We develop and exploit a source of two-photon four-dimensional entanglement to report the first two-particle all-versus-nothing test of local realism with a linear optics setup, but without resorting to a non-contextuality ssumption. Our experimental results are in well agreement with quantum mechanics while in extreme contradiction with local realism. Potential applications of our experiment are briefly discussed.
Nonproduct n-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized n-qubit Greenberger-Horne-Zeilinger states and their local unitary equivalents, for n greater than or equal to 3 but not equal to 4. We characterize the Lie algebra of the stabilizer subgroup for these states. For n=4, there is an additional maximal stabilizer subalgebra, not local unitary equivalent to the former. We give a canonical form for states with this stabilizer as well.
In Kaszlikowski [Phys. Rev. Lett. {\bf 85}, 4418 (2000)], it has been shown numerically that the violation of local realism for two maximally entangled $N$-dimensional ($3 \leq N$) quantum objects is stronger than for two maximally entangled qubits and grows with $N$. In this paper we present the analytical proof of this fact for N=3.
We put bounds on the minimum detection efficiency necessary to violate local realism in Bell experiments. These bounds depends of simple parameters like the number of measurement settings or the dimensionality of the entangled quantum state. We derive them by constructing explicit local-hidden variable models which reproduce the quantum correlations for sufficiently small detectors efficiency.
Recently various papers have proposed to test local realism (LR) by considering electroweak CP-violation parameters values in neutral pseudoscalar meson systems. Considering the large interest for a conclusive test of LR and the experimental accessibility to these tests, in this paper we critically consider these results showing how they, albeit very interesting, require anyway additional assumptions and therefore cannot be considered conclusive tests of LR.
We show all-versus-nothing proofs of Bell's theorem in the one-dimensional transverse-field Ising model, which is one of the most important exactly solvable models in the field of condensed matter physics. Since this model can be simulated with nuclear magnetic resonance, our work might lead to a fresh approach to experimental test of the Greenberger-Horne-Zeilinger contradiction between local realism and quantum mechanics.
Tests of local realism vs quantum mechanics based on Bell's inequality employ two entangled qubits. We investigate the general case of two entangled quNits, i.e. quantum systems defined in an N-dimensional Hilbert space. Via a numerical linear optimization method we show that violations of local realism are stronger for two maximally entangled quNits (N=3,4,...,9), than for two qubits and that they increase with N. The two quNit measurements can be experimentally realized using entangled photons and unbiased multiport beamsplitters.
We show using a numerical approach that gives necessary and sufficient conditions for the existence of local realism, that the bound entangled state presented in Bennett et. al. Phys. Rev. Lett. 82, 5385 (1999) admits a local and realistic description. We also find the lowest possible amount of some appropriate entangled state that must be ad-mixed to the bound entangled state so that the resulting density operator has no local and realistic description and as such can be useful in quantum communication and quantum computation.
Are Bell's conditions for local realism general enough?
本文重新审视贝尔局域现实条件，提出更现实的物理条件以实现违反贝尔不等式的光学实验模型。
Fundamental principle of classical physics -- local realism, means that freely chosen observations can be explained by a local (slower than light) real process. It is apparently violated in quantum mechanics as shown by Bell theorem. Despite extreme efforts experiments have not conclusively confirmed this violation due to loopholes. We propose a new postulate that the description of quantum processes must be consistent with local realism, It also assumes existence of many worlds/copies of the same system, interacting weakly microscopically but strongly macroscopically, whose number can be esti…
We examine the prevalent use of the phrase ``local realism'' in the context of Bell's Theorem and associated experiments, with a focus on the question: what exactly is the `realism' in `local realism' supposed to mean? Carefully surveying several possible meanings, we argue that all of them are flawed in one way or another as attempts to point out a second premise (in addition to locality) on which the Bell inequalities rest, and (hence) which might be rejected in the face of empirical data violating the inequalities. We thus suggest that the phrase `local realism' should be banned from future…
Employing a procedure called monitoring---via a completely positive trace-preserving map that is able to interpolate between weak and projective measurements---we investigate the resilience of the recently proposed realism-based nonlocality to local and bilocal weak measurements. This analysis indicates realism-based nonlocality as the most ubiquitous and persistent form of quantumness within a wide class of quantum-correlation quantifiers. In particular, we show that the set of states possessing this type of quantumness forms a strict superset of symmetrically discordant states and, therefore…
Mermin's observation [Phys. Rev. Lett. {\bf 65}, 1838 (1990)] that the magnitude of the violation of local realism, defined as the ratio between the quantum prediction and the classical bound, can grow exponentially with the size of the system is demonstrated using two-photon hyper-entangled states entangled in polarization and path degrees of freedom, and local measurements of polarization and path simultaneously.
Let $|Ψ>$ be an arbitrary stabilizer state distributed between three remote parties, such that each party holds several qubits. Let $S$ be a stabilizer group of $|Ψ>$. We show that $|Ψ>$ can be converted by local unitaries into a collection of singlets, GHZ states, and local one-qubit states. The numbers of singlets and GHZs are determined by dimensions of certain subgroups of $S$. For an arbitrary number of parties $m$ we find a formula for the maximal number of $m$-partite GHZ states that can be extracted from $|Ψ>$ by local unitaries. A connection with earlier introduced measures of multipa…
We show that quantum mechanics predicts an Einstein-Podolsky-Rosen paradox (EPR), and also a contradiction with local hidden variable theories, for photon number measurements which have limited resolving power, to the point of imposing an uncertainty in the photon number result which is macroscopic in absolute terms. We show how this can be interpreted as a failure of a new, very strong premise, called macroscopic local realism. We link this premise to the Schrodinger-cat paradox. Our proposed experiments ensure all fields incident on each measurement apparatus are macroscopic. We show that an…
In [Phys. Rev. A 58, 1833 (1998)] a family of polynomial invariants which separate the orbits of multi-qubit density operators $ρ$ under the action of the local unitary group was presented. We consider this family of invariants for the class of those $ρ$ which are the projection operators describing stabilizer codes and give a complete translation of these invariants into the binary framework in which stabilizer codes are usually described. Such an investigation of local invariants of quantum codes is of natural importance in quantum coding theory, since locally equivalent codes have the same …
The usual entanglement swapping protocol can entangle two particles that never interact. Here we demonstrate an all-versus-nothing violation of local realism for a partial entanglement swapping process. A Clauser-Horne-Shimony-Holt-type inequality for two particles entangled by entanglement swapping is proposed for a practical experiment and can be violated quantum mechanically up to 4 beyond Cirel'son's bound 2sqr(2). Our result uncovers the intriguing nonlocal character of entanglement swapping and may lead to a conclusive (loophole-free) test of local realism versus quantum mechanics. The e…
An ensemble consisting on systems of two entangled spin 1/2 particles, all of them in the same global quantum state, are considered. The two spins are measured, each of them, on a fixed direction, at two randomly selected measurement times. Realism, plus locality and freedom of choice referred to these chosen times, are assumed. Then, from the sole correlation function related to the two measurements, without considering any Bell inequalities, a contradiction between these assumptions and quantum mechanics is found.
Hypergraph states of many quantum bits share the rich interplay between simple combinatorial description and nontrivial entanglement properties enjoyed by the graph states that they generalize. In this paper, we consider hypergraph states that are also permutationally invariant. We characterize the states in this class that have nontrivial local Pauli stabilizers and give applications to nonlocality and error correction.
One hurdle to performing reliable quantum computations is overcoming noise. One possibility is to reduce the number of particles needing to be protected from noise and instead use systems with more states, so called qudit quantum computers. In this paper we show that codes for these systems can be derived from already known codes, and in particular that degenerate stabilizer codes can have their distance also promised upon sufficiently large local-dimension, as well as a new bound on the local-dimension required to preserve the distance of local-dimension-invariant codes, which is a result whi…
The concept of realism in quantum mechanics means that results of measurement are caused by physical variables, hidden or observable. Local hidden variables were proved unable to explain results of measurements on entangled particles tested far away from one another. Then, some physicists embraced the idea of nonlocal hidden variables. The present article proves that this idea is problematic, that it runs into an impasse vis-à-vis the special relativity.
We report a local hidden-variable model which reproduces quantum predictions for the two-photon interferometric experiment proposed by Franson [Phys. Rev. Lett. 62, 2205 (1989)]. The model works for the ideal case of full visibility and perfect detection efficiency. This result changes the interpretation of a series of experiments performed in the current decade.
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describ…
A sequence of Bell inequalities for N-particle systems, which involve three settings of each of the local measuring apparatuses, is derived. For Greenberger-Horne-Zeilinger states, quantum mechanics violates these inequalities by factors exponentially growing with N. The threshold visibilities of the multiparticle sinusoidal interference fringes, for which local realistic theories are ruled out, decrease as (2/3)^N.
The entanglement properties of a multiparty pure state are invariant under local unitary transformations. The stabilizer dimension of a multiparty pure state characterizes how many types of such local unitary transformations existing for the state. We find that the stabilizer dimension of an $n$-qubit ($n\ge 2$) graph state is associated with three specific configurations in its graph. We further show that the stabilizer dimension of an $n$-qubit ($n\ge 3$) graph state is equal to the degree of irreducible two-qubit correlations in the state.
We show that for macroscopic measurements which cannot reveal full information about microscopic states of the system, the monogamy of Bell inequality violations present in quantum mechanics implies that practically all correlations between macroscopic measurements can be described by local realistic models. Our results hold for sharp measurement and arbitrary closed quantum systems.
Objections to the use of inequalities to address foundational issues are considered and shown to be invalid. The Clauser-Horne (CH) inequality is considered and interpreted in this light. It is shown that, applied correctly, the CH inequality can validly test aspects of locality. This paper establishes a firm methodological ground for a following paper analyzing a recent CH experiment.
We investigate the influence of environmental noise on polarization entangled light generated by parametric emission in a cavity. By adopting a recently developed separability criterion, we show that: i) self-stimulation may suppress the detrimental influence of noise on entanglement; ii) when self-stimulation becomes effective, a classical model of parametric emission incorporating noise provides the same results of quantum theory for the expectation values involved in the separability criterion. Moreover we show that, in the macroscopic limit, it is impossible to observe violations of local …
Certifying bipartite entangled states with few local measurements: from separable stabilizers to applications
通过少量本地测量认证双粒子纠缠态，提供了一种简单有效的方法来确定量子态的保真度和纠缠特性。
We investigate the nonlocal property of the fractional statistics in Kitaev's toric code model. To this end, we construct the Greenberger-Horne-Zeilinger paradox which builds a direct conflict between the statistics and local realism. It turns out that the fractional statistics in the model is purely a quantum effect and independent of any classical theory.We also discuss a feasible experimental scheme using anyonic interferometry to test this contradiction.
Most working scientists hold fast to the concept of 'realism' - a viewpoint according to which an external reality exists independent of observation. But quantum physics has shattered some of our cornerstone beliefs. According to Bell's theorem, any theory that is based on the joint assumption of realism and locality (meaning that local events cannot be affected by actions in space-like separated regions) is at variance with certain quantum predictions. Experiments with entangled pairs of particles have amply confirmed these quantum predictions, thus rendering local realistic theories untenabl…
Loophole-free Bell inequality violations cannot disprove local realism
该研究指出贝尔不等式推导中误将局域实在论的充分条件当作必要条件，通过构建三个违反贝尔不等式的局域实在论模型，证明无漏洞的贝尔不等式违反也无法证伪局域实在论。
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