aa r X i v : . [ qu a n t - ph ] N ov Negative probabilities
Yuri Gurevich ∗ Computer Science & EngineeringUniversity of MichiganAnn Arbor, Michigan, U.S.A.
Vladimir Vovk † Dept. of Computer ScienceRoyal Holloway, University of LondonEgham, Surrey, UK
Abstract
We explain, on the example of Wigner’s quasiprobability distribution,how negative probabilities may be used in the foundations of probability.
Negative probabilities may appear devoid of any empirical meaning. The fre-quentist semantics does not apply to them. “It is absurd to talk about an urncontaining −
17 red balls” [7, page 148]. Yet negative probabilities are usefullyemployed in quantum physics. The situation reminds us of the introduction ofimaginary numbers in the theory of algebraic equations, even though the stan-dard semantics, according to which numbers are quantities, does not apply toimaginary numbers.In this note we illustrate that negative probabilities may be usefully em-ployed in the game-theoretic approach to probabilities [8, 9].
We work in the framework of the most standard formalization of (non-relativistic)quantum mechanics. It was originally proposed by John von Neumann [5, 6].A modern, mathematically rigorous exposition is found in the book [4].Like any physical system, a quantum system Q has a state space. The statespace of Q is a Hilbert space H . The states of Q are represented by unit vectorsin H .For simplicity of exposition, we consider a quantum system Q of one particleΠ moving in one dimension, though all our results generalize to more (butfinitely many) particles moving in more (but finitely many) dimensions. Thestate space of our quantum system Q is the Hilbert space L ( R ) of squareintegrable functions f : R → C with the inner product of f, g ∈ L ( R ) given by ∗ Partially supported by the US Army Research Office under W911NF-20-1-0297. † Partially supported by Amazon, Astra Zeneca, and Stena Line. Z f ∗ ( t ) g ( t ) dt . In this note, by default, the integrals arefrom −∞ to + ∞ .Consider physical properties of Q , like the position of particle Π, which takereal values and can be measured. Such a physical property is represented bya self-adjoint operator A over L ( R ). It is common to speak about measuring A itself and to call A an observable. The result of the measurement of (thephysical property represented by) A in a given state ψ of Q is determinedprobabilistically. The probability that the result lies in real interval ( p, q ] isgiven by formula Prob A ( p, q ] = k ( E q − E p ) ψ k ( ∗ )where { E r : r ∈ R } is the spectral resolution of the identity for A , whoseexistence follows from the spectral theorem for linear operators in Hilbert spacesproved originally by von Neumann [5]; a modern treatment of the issue is foundin [4, § Protocol 1 (Quantum measurement) . K := 1.FOR n = 1 , , . . . ,1. Experimenter (prepares and) announces a state ψ n of system Q and an observable A n to be measured in state ψ n .2. Quantum Mechanics announces a probability distribution µ n on R .3. Skeptic announces a measurable function f n ∈ [0 , ∞ ] R such that Z f n dµ n = 1.4. Reality announces the result r n ∈ R of the measurement of observable A n in state ψ n .5. K n := K n − f n ( r n ).Here Experimenter and Reality are free agents, who do not have to followany strategy, deterministic or probabilistic. The strategy of Quantum Mechanicsis given by formula ( ∗ ) where, at stage n , the observable is A n so that µ n =Prob A n . The goal of Skeptic is to test Quantum Mechanics, and we will beinterested in testing strategies for Skeptic.Protocol 1 is a simplified version of the protocols given in [9, § § ψ governed bythe Schr¨odinger equation. Here is one corollary of game-theoretic limit theorems(cf. [9, Corollary 10.14]). 2 orollary 1. Let F : R → R be a bounded measurable function. Skeptic canforce the event lim N →∞ N N X n =1 (cid:18) F ( r n ) − Z F dµ n (cid:19) = 0 , in the sense of having a strategy ensuring K n → ∞ whenever the equality fails. This corollary (a law of large numbers) is unusual in that it lies outsideKolmogorov’s framework for probability. The reason for that phenomenon isthat Experimenter does not have to follow any strategy. Of course, real-worldexperimenters usually follow deterministic or probabilistic testing strategies,which brings us into Kolmogorov’s framework.
Recall that our quantum system Q is a particle Π moving in one dimension.For technical reasons, we assume that Experimenter only prepares Q in states ψ which are smooth and compactly supported on R ; such states ψ will be called nice . Nice states are everywhere dense in L ( R ). Below, by default, states of Q are nice.Two important physical properties of Q are the position x and momentum p of particle Π. According to quantum mechanics, they are represented byself-adjoint operators( Xψ )( x ) := xψ ( x ) and ( P ψ )( x ) := − i ~ dψdx ( x )respectively where ~ is a real constant, the so-called reduced Planck constant.The uncertainty principle of quantum mechanics asserts a limit to the preci-sion with which position x and momentum p can be determined simultaneouslyin a given state ψ , even if ψ is nice. You can know the distribution of x andthat of p , but there is no joint probability distribution of x, p with the correctmarginal distributions of x and p .The situation changes if one allows negative probabilities. To address theissue, we need the following definitions. A quasiprobability distribution µ ona measurable space (Ω , Σ) is a real-valued, countably additive function on themeasurable sets such that µ (Ω) = 1. A quasiprobability density function for agiven quasiprobability distribution is the obvious generalization of a probabilitydensity function for a given probability distribution. Remark 1.
Qasiprobability distributions are special signed probability mea-sures and are also known as signed probability distributions . One may worrywhether countable additivity makes sense in signed probability spaces, but itdoes [3, § W ψ ( x, p ) := 12 π Z ψ ∗ (cid:18) x + β ~ (cid:19) ψ (cid:18) x − β ~ (cid:19) e iβp dβ ψ is an arbitrary unit vector in L ( R ). It is easy to check that all valuesof Wigner’s function W ψ ( x, p ) are real, but some values may be negative. Inany nice state ψ , W ψ gives rise to a unique quasiprobability distribution W ψ , Wigner’s quasiprobability distribution , for which it is a quasiprobability densityfunction.
Remark 2.
The Wigner function W ψ is also known as the Wigner-Ville functionbecause it was introduced in 1948 by Jean-Andr´e Ville in the context of signalprocessing [2, 10]. Signal processing is beyond the scope of this paper.For any real numbers a, b , the physical property z = ax + bp of Q is rep-resented by the self-adjoint operator Z = aX + bP . In any nice state ψ ofquantum system Q , let w zψ , or w a,bψ , be the probability distribution Prob Z givenby formula ( ∗ ) with A = Z . (The alternative notation w a,bψ is more explicit, butwe will use the more succinct notation w zψ .)The following proposition was presented in [1] and rigorously proved in [3]. Proposition 1.
In every nice state ψ , Wigner’s quasiprobability distribution W ψ is the unique quasiprobability distribution on R whose image, under anylinear mapping ( x, p ) z = ax + bp , is exactly w zψ . Although W ψ often has negative values, its images w zψ are genuine nonneg-ative probability distributions. The proposition allows us to refine Protocol 1as follows. Protocol 2 (Wigner-style quantum measurement) . K := 1.FOR n = 1 , , . . . ,1. Experimenter prepares and announces a nice state ψ n .2. Quantum Mechanics announces a quasiprobability distribution, namely W ψ n .3. Experimenter chooses a physical property z n = a n x + b n p and an-nounces the observable Z n = a n X + b n P .4. Skeptic announces a measurable function f n ∈ [0 , ∞ ] R such that Z f n dw z n ψ n = 1.5. Reality announces the result r n ∈ R of the measurement of observable Z n in state ψ n .6. K n := K n − f n ( r n ).Protocol 2 shows how we can test Wigner’s quasiprobability distribution.Notice that Skeptic gambles only against nonnegative probability distributions w z n ψ n . This is a coherent testing protocol in the sense of [8, 9]. Recall that eachprobability distribution w z n ψ n is an image of quasiprobability distribution W ψ n ;negative probabilities are used only to generate nonnegative probabilities whichare tested as usual. 4 orollary 2. Let F : R → R be a bounded measurable function. Skeptic canforce the event lim N →∞ N N X n =1 (cid:18) F ( r n ) − Z Z F ( a n x + b n p ) W ψ n ( x, p ) dx dp (cid:19) = 0 in the sense of having a strategy ensuring K n → ∞ whenever the equality fails. Corollary 2 is stronger than Corollary 1 in the sense that, in Protocol 2,Quantum Mechanics makes its choice before
Experimenter announces an ob-servable. By Proposition 1, this is impossible to achieve without negative prob-abilities.
Wigner’s function is a simple and concise description of probabilistic predic-tions for a wide range of observables. It can be tested using the usual approachof game-theoretic probability. The function has found useful applications inphysics [3] and signal processing [2], and we expect that the role of quasiprob-ability distributions will only grow both in practice and in the foundations ofprobability and statistics.
Acknowledgement
We thank Andreas Blass for useful comments.
References [1] Jacqueline Bertrand and Pierre Bertrand, “A tomographic approach toWigner’s function,”
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2A (1948) 61–74. En-glish translation: “Theory and Applications of the Notion ofComplex Signal,” RAND Corporation, Santa Monica, CA, 1958, .[11] Eugene P. Wigner, “On the quantum correction for thermodynamic equi-librium,”