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arXiv:2009.05062 · Physical Review Research

Discretized continuous quantum-mechanical observables that are neither continuous nor discrete

Thais de Lima Silva, Łukasz Rudnicki, D. S. Tasca, Stephen Patrick Walborn

Published Jan 28, 2022• 3 citations
arXivHTMLPDF

Abstract

Most of the fundamental characteristics of quantum mechanics, such as nonlocality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number, $d$, of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on $d$, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with $d=3$ outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.

DOI
10.1103/physrevresearch.4.013060

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Most of the fundamental characteristics of quantum mechanics, such as nonlocality and contextuality, are manifest in discrete, finite-dimensional systems. However, many quantum information tasks that exploit these properties cannot be directly adapted to continuous variable systems. To access these quantum features, continuous quantum variables can be made discrete by binning together their different values, resulting in observables with a finite number, ddd, of outcomes. While direct measurement indeed confirms their manifestly discrete character, here we employ a salient feature of quantum physics known as mutual unbiasedness to show that such coarse-grained observables are in a sense neither continuous nor discrete. Depending on ddd, the observables can reproduce either the discrete or the continuous behavior, or neither. To illustrate these results, we present an example for the construction of such measurements and employ it in an optical experiment confirming the existence of four mutually unbiased measurements with d=3d=3d=3 outcomes in a continuous variable system, surpassing the number of mutually unbiased continuous variable observables.
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.
Mutual unbiasedness of the eigenstates of phase-space operators---such as position and momentum, or their standard coarse-grained versions---exists only in the limiting case of infinite squeezing. In Phys. Rev. Lett. 120, 040403 (2018), it was shown that mutual unbiasedness can be recovered for periodic coarse graining of these two operators. Here we investigate mutual unbiasedness of coarse-grained measurements for more than two phase-space variables. We show that mutual unbiasedness can be recovered between periodic coarse graining of any two nonparallel phase-space operators. We illustrate these results through optics experiments, using the fractional Fourier transform to prepare and measure mutually unbiased phase-space variables. The differences between two and three mutually unbiased measurements is discussed. Our results contribute to bridging the gap between continuous and discrete quantum mechanics, and they could be useful in quantum-information protocols.
Observables with a continuous spectrum are known to fundamentally differ from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for discrete variables have been translated to continuous ones, this is not the case in general. Here we show that it is possible to manipulate, detect, and classify continuous-variable states using observables with a continuous spectrum revealing properties and symmetries which are analogous to finite discrete systems. Our approach leads to an operational way to define, and adapt, to arbitrary continuous quantum systems, quantum protocols, and algorithms developed for discrete systems.
Quantum information can be processed in two fundamentally different ways, using either discrete or continuous variable implementations. Each one of them provides different practical advantages and drawbacks. In this thesis we study theoretical means allowing to implement quantum information protocols originally formulated for discrete quantum systems in physical objects characterized by continuous variables. At the heart of our considerations is the use of modular variables as helpful technique to reveal discrete structure in continuous-variable states, operations and observables. The present work is strongly guided by the experimental applicability of our ideas in quantum optics experiments, with a particular focus on the transverse degrees of freedom of single photons.One of the main themes of this thesis is the formulation of a framework for quantum information processing in phase-space based on the use of modular variables. The term modular variables refers to a specific class of observables that are periodic with respect to some pair of conjugate variables. In our framework we use these periodic observables in order to encode discrete quantum information in Hilbert spaces of infinite dimension. In particular, we consider protocols that involve measurements of judiciously chosen logical observables enabling the readout of the encoded discrete quantum information from the corresponding logical states. Using this framework we show how to perform tests of fundamental properties of quantum mechanics, such as entanglement, Bell nonlocality and contextuality, in Hilbert spaces of various dimensions. Particularly, we generalize known tests of each of these properties, that were originally formulated for measurements with discrete outcomes, to more general measurement contexts comprising the case of bounded continuous outcomes. Concerning experimental implementations of the presented theoretical elaborations we discuss the transverse degrees of freedom of single photons as a natural platform to manipulate and measure modular variables. In particular, we demonstrate how to process discrete quantum information encoded in the spatial distribution of single photons via the optical Talbot effect - a near-field interference effect. Finally, we show how to produce d-dimensional entangled Talbot photon pairs without post-selection, using spontaneous parametric down-conversion and linear optical elements only. As last topic we explore the nonlocal properties of a specific class of hybrid entanglement between particle-like and wave-like optical qubits. Using a hybrid measurement scheme of Pauli and displaced parity measurements we show that, even after including realistic experimental losses, a violation of local-realism is theoretically possible.
Contextuality is a non-classical behaviour that can be exhibited by quantum systems. It is increasingly studied for its relationship to quantum-over-classical advantages in informatic tasks. To date, it has largely been studied in discrete variable scenarios, where observables take values in discrete and usually finite sets. Practically, on the other hand, continuous-variable scenarios offer some of the most promising candidates for implementing quantum computations and informatic protocols. Here we set out a framework for treating contextuality in continuous-variable scenarios. It is shown that the Fine--Abramsky--Brandenburger theorem extends to this setting, an important consequence of which is that nonlocality can be viewed as a special case of contextuality, as in the discrete case. The contextual fraction, a quantifiable measure of contextuality that bears a precise relationship to Bell inequality violations and quantum advantages, can also be defined in this setting. It is shown to be a non-increasing monotone with respect to classical operations that include binning to dis cretise data. Finally, we consider how the contextual fraction can be formulated as an infinite linear program, and calculated with increasing accuracy using semi-definite programming approximations.
Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.