Many-party correlations between measurement outcomes in general probabilistic theories are given by conditional probability distributions obeying the non-signalling condition. We show that any such distribution can be obtained from classical or quantum theory, by relaxing positivity constraints on either the mixed state shared by the parties, or the local functions which generate measurement outcomes. Our results apply to generic non-signalling correlations, but in particular they yield two distinct quasi-classical models for quantum correlations.
Negative probabilities have long been discussed in connection with the foundations of quantum mechanics. We have recently shown that, if signed measures are allowed on the hidden variables, the class of probability models which can be captured by local hidden-variable models are exactly the no-signalling models. However, the question remains of how negative probabilities are to be interpreted. In this paper, we present an operational interpretation of negative probabilities as arising from standard probabilities on signed events. This leads, by virtue of our previous result, to a systematic sc…
Zurek has derived the quantum probabilities for Schmidt basis states of bipartite quantum systems in pure joint states, from the assumption that they should be not be affected by one party's action if the action can be undone by the other party (``envariance of probability'') and an auxiliary assumption. We argue that a natural generalization of the auxiliary assumption is actually strong enough to yield the Born rule itself, but that Zurek's argument and protocol can be adapted to do without this assumption, at the cost of using envariance of probability in both directions. We consider altern…
Non-signalling quantum channels -- relevant in, e.g., the study of Bell and Einstein-Podolsky-Rosen scenarios -- may be simulated via affine combinations of local operations in bipartite scenarios. Moreover, when these channels correspond to stochastic maps between classical variables, such simulation is possible even in multipartite scenarios. These two results have proven useful when studying the properties of these channels, such as their communication and information processing power, and even when defining measures of the non-classicality of physical phenomena (such as Bell non-classicali…
We present a path analysis of the condition under which the outcomes of previous observation affect the results of the measurements yet to be made. It is shown that this effect, also known as "signalling in time", occurs whenever the earlier measurements are set to destroy interference between two or more virtual paths. We also demonstrate that Feynman's negative "probabilities" provide for a more reliable witness of "signalling in time", than the Leggett-Garg inequalities, while both methods are frequently subject to failure
We present an approach to computing probabilities in quantum field theory for a wide class of source-detector models. The approach works directly with probabilities and not with squared matrix elements, and the resulting probabilities can be written in terms of expectation values of nested commutators and anti-commutators. We present results that help in the evaluation of these, including an expression for the vacuum expectation values of general nestings of commutators and anti-commutators in scalar field theory. This approach allows one to see clearly how faster-than-light signalling is prev…
No-Signalling is a fundamental constraint on the probabilistic predictions made by physical theories. It is usually justified in terms of the constraints imposed by special relativity. However, this justification is not as clear-cut as is usually supposed. We shall give a different perspective on this condition by showing an equivalence between No-Signalling and Lambda Independence, or "free choice of measurements", a condition on hidden-variable theories which is needed to make no-go theorems such as Bell's theorem non-trivial. More precisely, we shall show that a probability table describing…
Bipartite and tripartite EPR-Bell type systems are examined via joint quasi-probability distributions where elementary probabilities are permitted to be negative. It is shown that such distributions exist only when the no-signalling condition is satisfied. A characteristic measure, the probability mass, is introduced and, via its minimization, limits the number of quasi-distributions describing a given marginal probability distribution. The minimized probability mass is shown to be an alternative way to characterize non-local systems. Non-signalling polytopes for two, three, and four settings …
Wigner's form of the local realist inequality is used to derive its temporal version for an oscillating two-level system involving two-time joint probabilities. Such an inequality may be regarded as a novel form of the Leggett-Garg inequality (LGI) constituting a necessary condition for macrorealism. The robustness of its quantum mechanical (QM) violation against unsharpness of measurement is investigated by using a suitable model of unsharp measurements. It is found that there exists a range of values of the sharpness parameter (characterizing precision of the relevant measurements) for which…
The Leggett-Garg (LG) inequalities were proposed in order to assess whether sets of pairs of sequential measurements on a single quantum system can be consistent with an underlying notion of macrorealism. Here, the LG inequalities are explored using a simple quasi-probability linear in the projection operators to describe the properties of the system at two times. We show that this quasi-probability is measurable, has the same correlation function as the usual two-time measurement probability (for the bivalent variables considered here) and has the key property that the probabilities for the l…
A phenomenological model for the calculation of reduction probabilities of a superposition of several states is presented. The approach bases only the idea that quantum state reduction has its origin in a mutual physical interaction between the states. The model is explicitly worked out for the gravitational reduction hypothesis of Diosi and Penrose. It agrees for typical quantum mechanical experiments with the projection postulate and predicts regimes in which other behavior could be observed. An outlook is given, how the new effects can possibly become of interest for biology. For verificati…
We introduce a novel generalization of the Clauser-Horne-Shimony-Holt (CHSH) game to a multiplayer setting, i.e., Hypercube game, where all $m$ players are required to assign values to vertices on corresponding facets of an $m$-dimensional hypercube. The players win if and only if their answers satisfy both parity and consistency conditions. We completely characterize the maximum winning probabilities (game value) under classical, quantum and no-signalling strategies, respectively. In contrast to the original CHSH game designed to demonstrate the superiority of quantumness, we find that the qu…
Two results are proved at the quantal level in Sorkin's hierarchy of measure theories. One is a strengthening of an existing bound on the correlations in the EPR-Bohm setup under the assumption that the probabilities admit a strongly positive joint quantal measure. It is also proved that any set of no-signalling probabilities, for two distant experimenters with a choice of two alternative experiments each and two possible outcomes per experiment, admits a joint quantal measure, though one that is not necessarily strongly positive.
Some forms of classical simulations of quantum type probabilities and correlations are capable of violating Boole's conditions of possible experience, such as the Clauser-Horne-Shimony-Holt inequality, even beyond the Tsirelson bound. This can be achieved by communicating a single bit encoding the measurement context.
Advances in photonics require photon-number resolved simulations of quantum optical experiments with Gaussian states. We demonstrate a simple and versatile method to simulate the photon statistics of general multimode Gaussian states. The derived generating functions enable simulations of the photon number distribution, cumulative probabilities, moments, and factorial moments of the photon statistics of Gaussian states as well as of multimode photon-added and photon-subtracted Gaussian states. Numerical results are obtained by automatic differentiation of these generating functions by employin…
Shannon's theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly, and in the presence of various resources that are all classes of non-signalling correlations: Shared randomness, shared entanglement and arbitrary non-signalling correlations. Specifically, when the channel being simulated is noiseless, this reduces to the zero-error capacity of the channel, assisted by the various classes of non-signalling correlations. When the resource channel is noiseless, it results in the "reverse" problem of simulating a noisy ch…
In this paper we introduce the three main notions of probability used by physicists and discuss how these are to be used when invoking spacelike separated observers in a relativistic format. We discuss a standard EPRB experiment and concentrate upon problems of the interpretation of probabilities. We promote a particularly conservative interpretation of this experiment (which need not invoke an objective notion of collapse) where probabilities are, tentatively, passively Lorentz invariant. We also argue that the Heisenberg picture is preferable in relativistic situations due to a conflict betw…
We present a novel approach to computing transition probabilities in quantum field theory, which allows them to be written directly in terms of expectation values of nested commutators and anti-commutators of field operators, rather than squared matrix elements. We show that this leads to a diagrammatic expansion in which the retarded propagator plays a dominant role. As a result, one is able to see clearly how faster-than-light signalling is prevented between sources and detectors. Finally, we comment on potential implications of this approach for dealing with infra-red divergences.
No physical measurement can be performed with infinite precision. This leaves a loophole in the standard no-go arguments against non-contextual hidden variables. All such arguments rely on choosing special sets of quantum-mechanical observables with measurement outcomes that cannot be simulated non-contextually. As a consequence, these arguments do not exclude the hypothesis that the class of physical measurements in fact corresponds to a dense subset of all theoretically possible measurements with outcomes and quantum probabilities that \emph{can} be recovered from a non-contextual hidden var…
Modal quantum theory (MQT) is a "toy model" of quantum theory in which amplitudes are elements of a general field. The theory predicts, not the probabilities of a measurement result, but only whether or not a result is possible. In this paper we review MQT and extend it to include mixed states, generalized measurements and open system dynamics. Even though MQT does not have density operators, superoperators or any concept of "positivity", we can nevertheless establish a precise analogue to the usual representation theorem for CP maps. We also embed MQT in a larger class of modal theories. We s…
University of Chicago（芝加哥大学）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）；University of Michigan（密歇根大学）；IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM托马斯·J·沃森研究中心）
We present here a classical optics device based on an imaging architecture as analogy of a quantum system where the violation of the Bell inequality can be evidenced. In our case, the two qbits entangled state needed to obtain non classical correlations is encoded using an electromagnetic wave modulated in amplitude and phase. Computational states are represented in a way where each one of the two qbits is associated with two orthogonal directions in the input plane. In addition, unitary operations involved in the measurement of the observables are simulated with the use of a coherent optical …
We propose a general tensor network method for simulating quantum circuits. The method is massively more efficient in computing a large number of correlated bitstring amplitudes and probabilities than existing methods. As an application, we study the sampling problem of Google's Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. Using our method, employing a small computational cluster containing 60 graphical processing units (GPUs), we have generated one million correlated bitstrings with some entries f…
Efficiently simulating the work distribution of multiple identical bosons with boson sampling
Center for Quantum Technology Research and Key Laboratory of Advanced Optoelectronic Quantum Architecture and Measurements (MOE), School of Physics, Beijing Institute of Technology（北京理工大学物理学院量子技术研究中心及先进光电子量子架构与测量教育部重点实验室）
We use the mathematical language of sheaf theory to give a unified treatment of non-locality and contextuality, in a setting which generalizes the familiar probability tables used in non-locality theory to arbitrary measurement covers; this includes Kochen-Specker configurations and more. We show that contextuality, and non-locality as a special case, correspond exactly to obstructions to the existence of global sections. We describe a linear algebraic approach to computing these obstructions, which allows a systematic treatment of arguments for non-locality and contextuality. We distinguish a…
We investigate the boundary between classical and quantum computational power. This work consists of two parts. First we develop new classical simulation algorithms that are centered on sampling methods. Using these techniques we generate new classes of classically simulatable quantum circuits where standard techniques relying on the exact computation of measurement probabilities fail to provide efficient simulations. For example, we show how various concatenations of matchgate, Toffoli, Clifford, bounded-depth, Fourier transform and other circuits are classically simulatable. We also prove th…
Gell-Mann and Hartle (GMH) have recently considered time-neutral cosmological models in which the initial and final conditions are independently specified, and several authors have investigated experimental tests of such models. We point out here that GMH time-neutral models can allow superluminal signalling, in the sense that it can be possible for observers in those cosmologies, by detecting and exploiting regularities in the final state, to construct devices which send and receive signals between space-like separated points. In suitable cosmologies, any single superluminal message can be tr…
Predictions for measurement outcomes in physical theories are usually computed by combining two distinct notions: a state, describing the physical system, and an observable, describing the measurement which is performed. In quantum theory, however, both notions are in some sense identical: outcome probabilities are given by the overlap between two state vectors - quantum theory is self-dual. In this paper, we show that this notion of self-duality can be understood from a dynamical point of view. We prove that self-duality follows from a computational primitive called bit symmetry: every logica…
This paper presents a novel quantum walk approach to simulating parton showers on a quantum computer. We demonstrate that the quantum walk paradigm offers a natural and more efficient approach to simulating parton showers on quantum devices, with the emission probabilities implemented as the coin flip for the walker, and the particle emissions to either gluons or quark pairs corresponding to the movement of the walker in two dimensions. A quantum algorithm is proposed for a simplified, toy model of a 31-step, collinear parton shower, hence significantly increasing the number of steps of the pa…
A classical computer simulating Schrodinger dynamics of a quantum system requires resources which scale exponentially with the size of the system, and is regarded as inefficient for such purposes. However, a quantum computer made up of a controllable set of quantum particles has the potential to efficiently simulate other quantum systems of matching dimensions. In this work we studied quantum simulations of single particle Schrodinger equation for certain one-dimensional potentials. In particular, we report the following cases: (i) spreading of wave-function of a free-particle, (ii) evolution …
We consider a topological Hamiltonian and establish a correspondence between its eigenstates and the resource for a causal order game introduced in Ref. [1] known as process matrix. We show that quantum correlations generated in the quantum many-body energy eigenstates of the model can mimic the statistics that can be obtained by exploiting different quantum measurements on the process matrix of the game. This provides an interpretation of the expectation values of the observables computed for the quantum many-body states in terms of the success probabilities of the game. As a result, we show …
We define a family of 'no signaling' bipartite boxes with arbitrary inputs and binary outputs, and with a range of marginal probabilities. The defining correlations are motivated by the Klyachko version of the Kochen-Specker theorem, so we call these boxes Kochen-Specker-Klyachko boxes or, briefly, KS-boxes. The marginals cover a variety of cases, from those that can be simulated classically to the superquantum correlations that saturate the Clauser-Horne-Shimony-Holt inequality, when the KS-box is a generalized PR-box (hence a vertex of the `no signaling' polytope). We show that for certain m…
We show that several quantum circuit families can be simulated efficiently classically if it is promised that their output distribution is approximately sparse i.e. the distribution is close to one where only a polynomially small, a priori unknown subset of the measurement probabilities are nonzero. Classical simulations are thereby obtained for quantum circuits which---without the additional sparsity promise---are considered hard to simulate. Our results apply in particular to a family of Fourier sampling circuits (which have structural similarities to Shor's factoring algorithm) but also to …
A possible notion of nonclassicality for single systems can be defined on the basis of the notion of memory cost of classically simulating probabilities observed in a temporal sequence of measurements. We further explore this idea in a theory-independent framework, namely, from the perspective of general probability theories (GPTs), which includes classical and quantum theory as special examples. Under the assumption that each system has a finite memory capacity, identified with the maximal number of states perfectly distinguishable with a single measurement, we investigate what are the tempor…
We introduce a distributed classical simulation algorithm for general quantum circuits, and present numerical results for calculating the output probabilities of universal random circuits. We find that we can simulate more qubits to greater depth than previously reported using the cluster supported by the Data Infrastructure and Search Technology Division of the Alibaba Group. For example, computing a single amplitude of an $8\times 8$ qubit circuit with depth $40$ was previously beyond the reach of supercomputers. Our algorithm can compute this within $2$ minutes using a small portion ($\appr…
Due to the unreliability and limited capacity of existing quantum computer prototypes, quantum circuit simulation continues to be a vital tool for validating next generation quantum computers and for studying variational quantum algorithms, which are among the leading candidates for useful quantum computation. Existing quantum circuit simulators do not address the common traits of variational algorithms, namely: 1) their ability to work with noisy qubits and operations, 2) their repeated execution of the same circuits but with different parameters, and 3) the fact that they sample from circuit…
In one-way quantum computation (1WQC) model, universal quantum computations are performed using measurements to designated qubits in a highly entangled state. The choices of bases for these measurements as well as the structure of the entanglements specify a quantum algorithm. As scalable and reliable quantum computers have not been implemented yet, quantum computation simulators are the only widely available tools to design and test quantum algorithms. However, simulating the quantum computations on a standard classical computer in most cases requires exponential memory and time. In this pape…
Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information Invariance & Continuity, at the foundation of those axiomatic reconstructions maps to "no preferred reference frame" (NPRF, aka "the relativity principle") as it pertains to the invariant measurement of Planck's constant h for Stern-Gerlach (SG) spin measure…
Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al, utilizes a randomized sparsification technique to approximate the output state of a quantum circuit by a stabilizer sum with a reduced number of terms. In this paper, we describe an improved Monte Carlo algorithm for performing randomized sparsification. This algorithm reduces the runtime of computing the approximate state by the factor $\ell/m$, where $\ell…
In the minimal scenario of quantum correlations, two parties can choose from two observables with two possible outcomes each. Probabilities are specified by four marginals and four correlations. The resulting four-dimensional convex body of correlations, denoted $\mathcal{Q}$, is fundamental for quantum information theory. We review and systematize what is known about $\Qm$, and add many details, visualizations, and complete proofs. In particular, we provide a detailed description of the boundary, which consists of three-dimensional faces isomorphic to elliptopes and sextic algebraic manifolds…
This paper is a comment on the paper "Quantum Mechanics and Algorithmic Randomness" was written by Ulvi Yurtsever \cite{Yurtsever} and the briefly explanation of the algorithmic randomness of quantum measurements results. There are differences between the computability of probability sources, ( which means there is an algorithm that can define the way that random process or probability source generates the numbers ) and the algorithmic randomness of the sequences or strings which are produced by a source. We may have the source without a computable algorithm for that but it can produce compres…
Advances with trapped ultracold atoms intensified interest in simulating complex physical phenomena, including quantum magnetism and transitions from itinerant to non-itinerant behavior. Here we show formation of antiferromagnetic ground states of few ultracold fermionic atoms in single and double well (DW) traps, through microscopic Hamiltonian exact diagonalization for two DW arrangements: (i) two linearly oriented one-dimensional, 1D, wells, and (ii) two coupled parallel wells, forming a trap of two-dimensional, 2D, nature. The spectra and spin-resolved conditional probabilities reveal for …
