Some trends and problems in quantum probability

Accardi, Luigi

doi:10.1007/bfb0071706

Cited by 28 publications

(51 citation statements)

References 15 publications

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“…One however shall have in mind that mathematical underpinning of this observation is highly nontrivial because the operators A k B l are all of trace class, even though neither A k nor B l have this property. The fact that the overlap does not depend on indices k and l suggests that the operators in (1) are interrelated in a special way -socalled Accardi complementarity [43]. Note, however, that the constant-trace condition alone is not even enough to assure that original variables are connected by the Fourier transformation [44].…”

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confidence: 99%

Mutual Unbiasedness in Coarse-Grained Continuous Variables

et al. 2018

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The notion of mutual unbiasedness for coarse-grained measurements of quantum continuous variable systems is considered. It is shown that while the procedure of "standard" coarse graining breaks the mutual unbiasedness between conjugate variables, this desired feature can be theoretically established and experimentally observed in periodic coarse graining. We illustrate our results in an optics experiment implementing Fraunhofer diffraction through a periodic diffraction grating, finding excellent agreement with the derived theory. Our results are an important step in developing a formal connection between discrete and continuous variable quantum mechanics.Introduction. The ability to measure a system in an infinite number of non-commuting bases distinguishes the quantum world from classical physics. Wave-particle duality and more generally the complementarity principle are directly rooted in this feature of quantum mechanics. Though one can measure a quantum system in several distinct bases, uncertainty relations limit the amount of information that can be obtained. It is well known that projection onto an eigenstate of one basis reduces the information that can be obtained through or inferred about subsequent measurement in a different basis. The information is minimum for mutually unbiased bases (MUBs), for which all outcomes of the second measurement are equally likely, so that total uncertainty is always substantial (the sharpest uncertainty relations [1]) and most insensitive to input states [2]. MUBs play an important role in complementarity [3], quantum cryptography [4] and quantum tomography [5,6], are useful for certifying quantum randomness [7], and for detecting quantum correlations such as entanglement [8][9][10] and steering [11][12][13][14][15][16][17][18].

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Mutual Unbiasedness in Coarse-Grained Continuous Variables

et al. 2018

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“…Also of these years is the solution of another traditional problem on the foundations of quantum theory, namely the deduction of the mathematical model of classical and quantum probability from a set of simple, plausible and physically meaningful axioms [Ac82].…”

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confidence: 99%

Quantum probability: an historical survey

Accardi¹

2000

Probability on Algebraic Structures

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1What is quantum probability Quantum Probability (QP) is a new branch of mathematics interconnecting classical probability, functional analysis, pure algebra, quantum physics and information and communication engineering. The mid seventies is the period that marks the beginning of QP as an autonomous discipline. Since then this cross disciplinary nature has accompanied the development of QP and still now it is one of its points of strength, making it an original new trend in contemporary mathematics as well as one of the earliest pioneers of non-commutative mathematics: a field now flourishing with the more recent development of quantum groups, non-commutative geometry, quantum computer, . . .The developments in this area are taking place at such an high pace and such a broad horizon to make impossible any attempt of a detailed synthesis in a single paper. In what follows I shall give a quick historical survey of QP pointing out some of the main problems which have driven these developments and of the main applications to quantum physics. The early periodSeveral notions, tools and single results, now stably entered in the domain of QP, were developed simultaneously with the birth of quantum mechanics. However these developments occurred independently one another, in several different fields such as functional analysis, the foundations of quantum theory, 1 Abstract of an invited talk to the Annual meeting of the Japan Mathematical Society ,

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“…The main emphasis of this exposition will not be on proofs (for these one can look at [Ac82c] or [Ac95a]) but on the global picture and on the questions which are still open.…”

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