Quasiprobability Representations of Quantum Mechanics with Minimal Negativity

Abstract

Quasiprobability representations, such as the Wigner function, play an important role in various research areas. The inevitable appearance of negativity in such representations is often regarded as a signature of nonclassicality, which has profound implications for quantum computation. However, little is known about the minimal negativity that is necessary in general quasiprobability representations. Here we focus on a natural class of quasiprobability representations that is distinguished by simplicity and economy. We introduce three measures of negativity concerning the representations of quantum states, unitary transformations, and quantum channels, respectively. Quite surprisingly, all three measures lead to the same representations with minimal negativity, which are in one-to-one correspondence with the elusive symmetric informationally complete measurements. In addition, most representations with minimal negativity are automatically covariant with respect to the Heisenberg-Weyl groups. Furthermore, our study reveals an interesting tradeoff between negativity and symmetry in quasiprobability representations. [2][3][4][5][6][7], which are very useful in many research areas, such as quantum optics, quantum tomography, and quantum computing. The inevitable appearance of negativity in such representations is often regarded as a signature of nonclassicality, which is closely related to other nonclassical features [5,6,[8][9][10]. In particular, it was shown by Spekkens that negativity and contextuality are equivalent notions of nonclassicality [9]. Moreover, negativity has been recognized as a resource in quantum computation and has attracted increasing attention in the quantum information community [11][12][13][14][15][16][17]. To be specific, in the paradigm of magic state quantum computation [18], negativity in the Wigner function is necessary for computational speedup and is directly related to the efficiency of classical simulation [13,14,17].To fully understand the distinction between quantum theory and classical theory in terms of negativity, one should not only consider the representation of quantum states, but also representations of measurements and transformations. In addition, it is crucial to consider the whole class of QPRs instead of a particular choice. This advice is also helpful for understanding the power of quantum information processing as well as the connection between negativity and contextuality [5, 9, 10]. However, little is known about the degree of negativity in any QPR other than the Wigner function and its discrete analogs.In this paper we initiate a systematic study of the degree of negativity in general QPRs. Our focus is an important family of QPRs in finite dimensions that is distinguished by simplicity and economy. It includes most examples proposed in the literature, such as many discrete Wigner functions [2,3,5,6]. In these representations, information is encoded without redundancy; the analogy to classical probability theory is preserved as far as possible except for the app...

Thermal rectification and spin-spin coupling of nonreciprocal localized and surface modes

“…[53] for a non-reciprocal medium this contribution persist even in global equilibrium. This persistent heat flux at an interface has also been observed for a single or several nanoparticles [1,6]. Here, in particular the non-reciprocal surface modes in the InSb substrate will produce a persistent heat flux in positive or negative x-direction described by this third term.…”

“…As shown in Ref. [6] these resonances are connected with a clockwise (counterclockwise) radiative heat flux for m = −1 (m = +1) as well as an angular momentum and spin which also persist in global thermal equilibrium and which are at the heart of the persistent heat current and thermal Hall effect in many-particle assemblies [1,[5][6][7]. As shown in Ref.…”

“…Since, the second and third term persists in global equilibrium they describe the persistent heat flux of the interface [53] and the nanoparticles [1,6], but they do not describe any heat transfer between the nanoparticles. Therefore the heat transfer between the particles is fully determined by the first term of the Poynting vector…”

“…In the last few years, it could be shown that the non-reciprocal behaviour of magneto-optical materials like InSb has very interesting consequences for nanoscale thermal radiation. For example, fundamental effects like a persistent heat-current [1,2], giant magnetoresistance [3,4], thermal Hall effect [5] as well as a circular heat flux, angular momentum, and spin which do also persist in global equilibrium [6] were highlighted. As reviewed and discussed in detail in Ref.…”

Aims and Scope of the Special Issue, “Quantum Foundations: Informational Perspective”

“…Their work is an extension of theorems proven by Zhu [30], which indicate that by one measure, the SIC representation is an optimal one. Modifying Zhu's measure, and presenting a natural class of measures to which it is seen to belong, DeBrota and Fuchs find new discrete structures in quantum state space with significant information-theoretic properties.…”