Negative probabilities

线性光学电路输出概率的近似计算

Approximating outcome probabilities of linear optical circuits

本研究提出利用$s$参数化准概率分布近似线性光学电路输出概率的经典算法，可降低电路负性界，实现高效估计，还给出高斯玻色采样经典可模拟的充分条件，提供多种优于现有结果的量子启发算法。

Experimental Demonstration of Negative-Valued Polarization Quasi-Probability Distribution

Polarization quasiprobability distribution defined in the Stokes space shares many important properties with the Wigner function for the position and momentum. Most notably, they both give correct one-dimensional marginal probability distributions and therefore represent the natural choice for the probability distributions in classical hidden-variable models. In this context, negativity of the Wigner function is considered as a proof of non-classicality for a quantum state. On the contrary, the polarization quasiprobability distribution demonstrates negativity for all quantum states. This feat…

Negativity of quasiprobability distributions as a measure of nonclassicality

We demonstrate that the negative volume of any $s$-paramatrized quasiprobability, including the Glauber-Sudashan $P$-function, can be consistently defined and forms a continuous hierarchy of nonclassicality measures that are linear optical monotones. These measures therefore belong to an operational resource theory of nonclassicality based on linear optical operations. The negativity of the Glauber-Sudashan $P$-function in particular can be shown to have an operational interpretation as the robustness of nonclassicality. We then introduce an approximate linear optical monotone, and show that t…

Joint quasiprobability distribution on the measurement outcomes of MUB-driven operators

We propose a method to define quasiprobability distributions for general spin-$j$ systems of dimension $n=2j+1$, where $n$ is a prime or power of prime. The method is based on a complete set of orthonormal commuting operators related to Mutually Unbiased Bases which enable (i) a parameterisation of the density matrix and (ii) construction of measurement operators that can be physically realised. As a result we geometrically characterise the set of states for which the quasiprobability distribution is non-negative, and can be viewed as a joint distribution of classical random variables assuming…

Optical experiment to test negative probability in context of quantum-measurement selection

Negative probability values have been widely employed as an indicator of the nonclassicality of quantum systems. Known as a quasiprobability distribution, they are regarded as a useful tool that provides significant insight into the underlying fundamentals of quantum theory when compared to the classical statistics. However, in this approach, an operational interpretation of these negative values with respect to the definition of probability---the relative frequency of occurred event---is missing. An alternative approach is therefore considered where the quasiprobability operationally reveals …

Verification of joint measurability using phase-space quasiprobability distributions

Measurement incompatibility is a distinguishing property of quantum physics and an essential resource for many quantum information processing tasks. We introduce an approach to verify the joint measurability of measurements based on phase-space quasiprobability distributions. Our results therefore establish a connection between two notions of non-classicality, namely the negativity of quasiprobability distributions and measurement incompatibility. We show how our approach can be applied to the study of incompatibility-breaking channels and derive incompatibility-breaking sufficient conditions …

What is the most general class of quasiprobabilities of work?

How to give a statistical description of thermodynamics in quantum systems is an open fundamental question. Concerning the work, the presence of initial quantum coherence in the energy basis can give rise to a quasiprobability of work, which can take negative values. Our aim is to identify the most general quasiprobability of work satisfying some fundamental conditions. By doing so, we introduce a general notion of quasiprobability in analogy to the Gleason's theorem. Then, we use these quasiprobabilities to define the quasiprobability of work, and finally we discuss the contextuality of the p…

Operational quasiprobabilities for continuous variables

We generalize the operational quasiprobability involving sequential measurements proposed by Ryu {\em et al.} [Phys. Rev. A {\bf 88}, 052123] to a continuous-variable system. The quasiprobabilities in quantum optics are incommensurate, i.e., they represent a given physical observation in different mathematical forms from their classical counterparts, making it difficult to operationally interpret their negative values. Our operational quasiprobability is {\em commensurate}, enabling one to compare quantum and classical statistics on the same footing. We show that the operational quasiprobabili…

弱测量下的准概率分布

Quasiprobability distributions with weak measurements

本文通过光子量子比特弱序列测量实验，结合相称和Margenau-Hill准概率方法，研究了量子相干性在时序测量中的作用，并展示了准分布用于量子监测的潜力。

计算明亮“香蕉”态准概率分布的有效算法

Effective algorithms for calculation of quasiprobability distributions of bright "banana'' states

针对克尔非线性产生的明亮“香蕉”态在光子数极大时维格纳函数直接计算代价过高的问题，本文提出了可适用于任意实际光子数和非线性值的 Husimi 与 Wigner 准概率函数快速计算算法。

量子伊辛模型中功的准概率分布

Quasiprobability distribution of work in the quantum Ising model

本文研究横场伊辛模型在全局和局域淬火下的功准概率分布，发现局域淬火可通过负四阶矩揭示量子上下文性，并分析了量子相变临界特征及初始相干性的资源作用。

Phase space simulation method for quantum computation with magic states on qubits

We propose a method for classical simulation of finite-dimensional quantum systems, based on sampling from a quasiprobability distribution, i.e., a generalized Wigner function. Our construction applies to all finite dimensions, with the most interesting case being that of qubits. For multiple qubits, we find that quantum computation by Clifford gates and Pauli measurements on magic states can be efficiently classically simulated if the quasiprobability distribution of the magic states is non-negative. This provides the so far missing qubit counterpart of the corresponding result [V. Veitch et …

Quantum Advantage in Postselected Metrology

We show that postselection offers a nonclassical advantage in metrology. In every parameter-estimation experiment, the final measurement or the postprocessing incurs some cost. Postselection can improve the rate of Fisher information (the average information learned about an unknown parameter from an experimental trial) to cost. This improvement, we show, stems from the negativity of a quasiprobability distribution, a quantum extension of a probability distribution. In a classical theory, in which all observables commute, our quasiprobability distribution can be expressed as real and nonnegati…

Measure-theoretic approach to negative probabilities

In this work, we elaborate on a measure-theoretic approach to negative probabilities. We study a natural notion of contextuality measure and characterize its main properties. Then, we apply this measure to relevant examples of quantum physics. In particular, we study the role played by contextuality in quantum computing circuits.

循环系统中的语境性度量与负概率度量 CNT3

Measures of contextuality in cyclic systems and the negative probabilities measure CNT3

本文证明了 Dzhafarov 和 Kujala 关于循环系统中 CNT2 与 CNT3 成比例的猜想，并结合 Cervantes 对 CNT2 与 CNTF 比例关系的证明，完整刻画了循环系统中各语境性度量之间的相互关系。

An Operational Interpretation of Negative Probabilities and No-Signalling Models

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…

Negative probabilities, II: What they are and what they are for

A signed probability distribution may extend a given traditional probability from observable events to all events. We formalize and illustrate this approach. We also illustrate its limitation. We argue that the right question is not what negative probabilities are but what they are for.

The negative probabilities and information gain in weak measurements

We study the outcomes in a general measurement with postselection, and derive upper bounds for the pointer readings in weak measurement. Using the idea of weak measurement, we study Hardy's gedanken experiment and show how the "negative probabilities" emerge in weak measurement. By calculating the information gain of the measuring device about which path the particles pass through, we show that the "negative probabilities" only emerge for cases when the information gain is little due to very weak coupling between the measuring device and the particles. When the coupling strength increases, we …

Interpretations of Negative Probabilities

In this paper, we give a frequency interpretation of negative probability, as well as of extended probability, demonstrating that to a great extent, these new types of probabilities, behave as conventional probabilities. Extended probability comprises both conventional probability and negative probability. The frequency interpretation of negative probabilities gives supportive evidence to the axiomatic system built in (Burgin, 2009; arXiv:0912.4767) for extended probability as it is demonstrated in this paper that frequency probabilities satisfy all axioms of extended probability.

量子光的经典模型中的Wigner函数负值

Wigner function negativity in a classical model of quantum light

本文提出一个经典模型，通过后选择检测事件再现单光子加 coherent 状态的非经典行为，并展示Wigner函数的负值特征。

Representation of entanglement by negative quasi-probabilities

Any bipartite quantum state has quasi-probability representations in terms of separable states. For entangled states these quasi-probabilities necessarily exhibit negativities. Based on the general structure of composite quantum states, one may reconstruct such quasi-propabilities from experimental data. Because of ambiguity, the quasi-probabilities obtained by the bare reconstruction are insufficient to identify entanglement. An optimization procedure is introduced to derive quasi-probabilities with a minimal amount of negativity. Negativities of optimized quasi-probabilities unambiguously pr…

Indistinguishability and Negative Probabilities

In this paper, we examined the connection between quantum systems' indistinguishability and signed (or negative) probabilities. We do so by first introducing a measure-theoretic definition of signed probabilities inspired by research in quantum contextuality. We then argue that ontological indistinguishability leads to the no-signaling condition and negative probabilities.

Paths, negative "probabilities", and the Leggett-Garg inequalities

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

Negative Probabilities, Fine's Theorem and Linear Positivity

Many situations in quantum theory and other areas of physics lead to quasi-probabilities which seem to be physically useful but can be negative. The interpretation of such objects is not at all clear. In this paper, we show that quasi-probabilities naturally fall into two qualitatively different types, according to whether their non-negative marginals can or cannot be matched to a non-negative probability. The former type, which we call viable, are qualitatively similar to true probabilities, but the latter type, which we call non-viable, may not have a sensible interpretation. Determining the…

Negative probabilities and counter-factual reasoning in quantum cognition

In this paper we discuss quantum-like decision-making experiments using negative probabilities. We do so by showing how the two-slit experiment, in the simplified version of the Mach-Zehnder interferometer, can be described by this formalism. We show that negative probabilities impose constraints to what types of counter-factual reasoning we can make with respect to (quantum) internal representations of the decision maker.

How to simulate a quantum computer using negative probabilities

The concept of negative probabilities can be used to decompose the interaction of two qubits mediated by a quantum controlled-NOT into three operations that require only classical interactions (that is, local operations and classical communication) between the qubits. For a single gate, the probabilities of the three operations are 1, 1, and -1. This decomposition can be applied in a probabilistic simulation of quantum computation by randomly choosing one of the three operations for each gate and assigning a negative statistical weight to the outcomes of sequences with an odd number of negativ…

Negative Probabilities and Contextuality

There has been a growing interest, both in physics and psychology, in understanding contextuality in experimentally observed quantities. Different approaches have been proposed to deal with contextual systems, and a promising one is contextuality-by-default, put forth by Dzhafarov and Kujala. The goal of this paper is to present a tutorial on a different approach: negative probabilities. We do so by presenting the overall theory of negative probabilities in a way that is consistent with contextuality-by-default and by examining with this theory some simple examples where contextuality appears,…

极值Wigner负性自旋态的恒星表示

Stellar representation of extremal Wigner-negative spin states

本文利用马约拉纳恒星表示研究最大Wigner负性自旋态，发现其对称性部分且无简单几何模式，并证明所有自旋相干态及纯自旋态均具有非零Wigner负性，结果适用于置换不变量子比特系综。

Simulating all non-signalling correlations via classical or quantum theory with negative probabilities

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.

Bounds on integrals of the Wigner function: the hyperbolic case

Wigner functions play a central role in the phase space formulation of quantum mechanics. Although closely related to classical Liouville densities, Wigner functions are not positive definite and may take negative values on subregions of phase space. We investigate the accumulation of these negative values by studying bounds on the integral of an arbitrary Wigner function over noncompact subregions of the phase plane with hyperbolic boundaries. We show using symmetry techniques that this problem reduces to computing the bounds on the spectrum associated with an exactly-solvable eigenvalue prob…

Decision Making for Inconsistent Expert Judgments Using Negative Probabilities

In this paper we provide a simple random-variable example of inconsistent information, and analyze it using three different approaches: Bayesian, quantum-like, and negative probabilities. We then show that, at least for this particular example, both the Bayesian and the quantum-like approaches have less normative power than the negative probabilities one.

Negative probabilities and Counterfactual Reasoning on the double-slit Experiment

In this paper we attempt to establish a theory of negative (quasi) probability distributions from fundamental principles and apply it to the study of the double-slit experiment in quantum mechanics. We do so in a way that preserves the main conceptual issues intact but allow for a clearer analysis, by representing the double-slit experiment in terms of the Mach-Zehnder interferometer, and show that the main features of quantum systems relevant to the double-slit are present also in the Mach-Zehnder. This converts the problem from a continuous to a discrete random variable representation. We th…

Quantum eigenstates from classical Gibbs distributions

We discuss how the language of wave functions (state vectors) and associated non-commuting Hermitian operators naturally emerges from classical mechanics by applying the inverse Wigner-Weyl transform to the phase space probability distribution and observables. In this language, the Schrödinger equation follows from the Liouville equation, with $\hbar$ now a free parameter. Classical stationary distributions can be represented as sums over stationary states with discrete (quantized) energies, where these states directly correspond to quantum eigenstates. Interestingly, it is now classical mecha…

Certifying Non-Classical Behavior for Negative Keldysh Quasi-Probabilities

We introduce an experimental test for ruling out classical explanations for the statistics obtained when measuring arbitrary observables at arbitrary times using individual detectors. This test requires some trust in the measurements, represented by a few natural assumptions on the detectors. In quantum theory, the considered scenarios are well captured by von Neumann measurements. These can be described naturally in terms of the Keldysh quasi-probability distribution (KQPD), and the imprecision and backaction exerted by the measurement apparatus. We find that classical descriptions can be rul…

Using weak values to experimentally determine "negative probabilities" in a two-photon state with Bell correlations

Bipartite quantum entangled systems can exhibit measurement correlations that violate Bell inequalities, revealing the profoundly counter-intuitive nature of the physical universe. These correlations reflect the impossibility of constructing a joint probability distribution for all values of all the different properties observed in Bell inequality tests. Physically, the impossibility of measuring such a distribution experimentally, as a set of relative frequencies, is due to the quantum back-action of projective measurements. Weakly coupling to a quantum probe, however, produces minimal back-a…

The intensity of the random variable intercept in the sector of negative probabilities

We consider properties of the measurement intensity $ρ$ of a random variable for which the probability density function represented by the corresponding Wigner function attains negative values on a part of the domain. We consider a simple economic interpretation of this problem. This model is used to present the applicability of the method to the analysis of the negative probability on markets where there are anomalies in the law of supply and demand (e.g. Giffen's goods). It turns out that the new conditions to optimize the intensity $ρ$ require a new strategy. We propose a strategy (so-calle…

Weak measurements, non-classicality and negative probability

This paper establishes a direct, robust and intimate connection between (i) non classicality tests for various quantum features, e.g., non-Boolean logic, quantum coherence, nonlocality, quantum entanglement, quantum discord; (ii) negative probability, and (iii) anomalous weak values. It has been shown [Adhikary et al. Eur. Phys. J. D, 74(68):68, 2020] that nonexistence of a classical joint probability scheme gives rise to sufficiency conditions for nonlocality, a nonclassical feature not restricted to quantum mechanics. The conditions for nonclassical features of quantum mechanics are obtained…

Negative probabilities: What they are and what they are for

An observation space $\mathcal S$ is a family of probability distributions $\langle P_i: i\in I \rangle$ sharing a common sample space $Ω$ in a consistent way. A \emph{grounding} for $\mathcal S$ is a signed probability distribution $\mathcal P$ on $Ω$ yielding the correct marginal distribution $P_i$ for every $i$. A wide variety of quantum scenarios can be formalized as observation spaces. We describe all groundings for a number of quantum observation spaces. Our main technical result is a rigorous proof that Wigner's distribution is the unique signed probability distribution yielding the cor…

Logical Entropy and Negative Probabilities in Quantum Mechanics

The concept of Logical Entropy, $S_L = 1- \sum_{i=1}^n p_i^2$, where the $p_i$ are normalized probabilities, was introduced by David Ellerman in a series of recent papers. Although the mathematical formula itself is not new, Ellerman provided a sound probabilistic interpretation of $S_L$ as a measure of the distinctions of a partition on a given set. The same formula comes across as a useful definition of entropy in quantum mechanics, where it is linked to the notion of purity of a quantum state. The quadratic form of the logical entropy lends itself to a generalization of the probabilities th…

负概率

Negative Probability

本文旨在建立物理学中负概率与贝叶斯建模中负混合分布之间的联系，基于Bartlett定义给出负权重贝叶斯规则，并讨论半硬币分布、Polya-Gamma混合及对偶密度等例子。

量子相干与负准概率在一种上下文三路径干涉仪中

Quantum coherence and negative quasi probabilities in a contextual three-path interferometer

本文提出利用五阶段干涉仪中的正交关系和负Kirkwood-Dirac分布，对三维希尔伯特空间中的纯态进行非经典关联分类。

p-adic probability prediction of correlations between particles in the two-slit and neutron interferometry experiments

We start from Feynman`s idea to use negative probabilities to describe the two slit experiment and other quantum interfernce experiments. Formally by using negative probability distributions we can explain the results of the two slit experiment on the basis of the pure corpuscular picture of quantum mechnanics. However, negative probabilities are absurd objects in the framework of the standard Kolmogorov theory of probability. We present a large class of non-Kolmogorovean probability models where negative probabilities are well defined on the frequency basis. These are models with probabilitie…

Necessary and sufficient condition for a realistic theory of quantum systems

We study the possibility to describe pure quantum states and evens with classical probability distributions and conditional probabilities and show that the distributions and/or conditional probabilities have to assume negative values, except for a simple model whose realistic space dimension is not smaller than the Hilbert space dimension of the quantum system. This gives a negative answer to a question proposed by Montina [Phys.Rev.Lett.{\bf 97}, 180401 (2006)] whether or not does there exist a classical theory whose phase-space dimension is much smaller than the Hilbert space dimension for a…

How quantum paradoxes originate from the non-classical statistics of physical properties related to each other by half-periodic transformations

Quantum paradoxes show that quantum statistics can exceed the limits of positive joint probabilities for physical properties that cannot be measured jointly. It is therefore impossible to describe the relations between the different physical properties of a quantum system by assigning joint realities to their observable values. Instead, recent experimental results obtained by weak measurements suggest that non-classical correlations could be expressed by complex valued quasi-probabilities, where the phases of the complex probabilities express the action of transformations between the non-commu…

从含噪玻色采样中进行经典采样与负概率

Classical sampling from noisy Boson Sampling and the negative probabilities

本文研究含噪玻色采样经典采样中由有限阶多玻色子干涉截断引起的负概率问题，指出该问题仅在接近完全可区分玻色子的消失区域内可解，并提出基于对称群特征渐近行为的新研究方向。

Transition probabilities and transition rates in discrete phase space

The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the disc…

量子态的复值Wigner熵

Complex-valued Wigner entropy of a quantum state

本文提出复值Wigner熵，用于分析量子态的相空间熵性质，其实部和虚部在Gaussian操作下保持不变，并与噪声下的熵变化相关联。

SU(1,1) 态的维格纳负性与星秩

Wigner negativity and stellar rank for SU(1,1) states

研究SU(1,1)态的量子特性，通过构建全协变准概率分布族填补相关研究空白，得出维格纳函数对特定相干态严格为正等结果，还研究了星秩及引入广义多极层次结构，提供了量化该系统量子资源的工具包。

模拟量子计算：需要多少“比特”？

Simulating quantum computation: how many "bits" for "it"?

本文分析了一种基于魔态采样且无准概率负值的经典量子模拟方法所需跟踪的经典数据量，证明对任意 n 个魔态，描述系统所需比特数仅为 2n²+O(n)，表明该模拟所需经典资源很小。

Wigner and Bell Inequalities relationships and Kolmogorov's Axioms

In this work, we show that Bell's inequality violation of arise from the fact that the condition imposed upon the development of inequality is not respected when it is applied in the idealized experiment. Such a condition is that the quantities taken by probability must be non negative, and such a condition is represented by $|\cal{P(w)}|=\cal{P(w)}$. We will also show that, when trying to define the values of the joint probabilities of $ (Z_1, Z_2, Z_3) $, through the values obtained from the $ (Z_j, Z_k) $ pairs, we find that these values are negative, so not Kolmogorov's axiom is respected:…

Diffraction in time in terms of Wigner distributions and tomographic probabilities

Long ago appeared a discussion in quantum mechanics of the problem of opening a completely absorbing shutter on which were impinging a stream of particles of definite velocity. The solution of the problem was obtained in a form entirely analogous to the optical one of diffraction by a straight edge. The argument of the Fresnel integrals was though time dependent and thus the first part in the title of this article. In section 1 we briefly review the original formulation of the problem of diffraction in time. In section 2 and 3 we reformulate respectively this problem in Wigner distributions an…

Inversion of moments to retrieve joint probabilities in quantum sequential measurements

A sequence of moments encode the corresponding probability distribution. Probing if quantum joint probability distribution can be retrieved from the associated set of moments -- realized in the sequential measurement of a dichotomic observable at different time intervals -- reveals a negative answer i.e., the joint probabilities of sequential measurements do not agree with the ones obtained by inverting the moments. This is indeed a reflection of the non-existence of a bonafide grand joint probability distribution, consistent with all the physical marginal probability distributions. Here we ex…

Negative Full Counting Statistics Arise From Interference Effects

The Keldysh-ordered full counting statistics is a quasi-probability distribution describing the fluctuations of a time-integrated quantum observable. While it is well known that this distribution can fail to be positive, the interpretation and origin of this negativity has been somewhat unclear. Here, we show how the full counting statistics can be tied to trajectories through Hilbert space, and how this directly connects negative quasi-probabilities to an unusual interference effect. Our findings are illustrated with the example of energy fluctuations in a driven bosonic resonator; we discuss…

Exploring non-signalling polytopes with negative probability

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 …