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arXiv:2306.11436 · Frontiers in Quantum Science and Technology

A tomographic approach to the sum uncertainty relation and quantum entanglement in continuous variable systems

Soumyabrata Paul, S. Lakshmibala, V. Balakrishnan, S. Ramanan

Published Jun 12, 2023• 4 citations
arXivHTMLPDF

Abstract

Entropic uncertainty relations (EURs) have been examined in various contexts, primarily in qubit systems, including their links with entanglement, as they subsume the Heisenberg uncertainty principle. With their genesis in the Shannon entropy, EURs find applications in quantum information and quantum optics. EURs are state-dependent, and the state has to be reconstructed from tomograms (which are histograms readily available from experiments). This is a challenge when the Hilbert space is large, as in continuous variable (CV) systems and certain hybrid quantum (HQ) systems. A viable alternative approach therefore is to extract as much information as possible about the unknown quantum state directly from appropriate tomograms. Many variants of EURs can be extracted from tomograms, even for CV systems. In earlier work we have defined many tomographic entanglement indicators (TEIs) that can be readily calculated from tomograms without knowledge of the density matrix, and have reported on their efficacy as entanglement indicators in various contexts in both CV and HQ systems. The specific objectives of the present work are as follows: (i) To use the tomographic approach to investigate the links between EURs and TEIs in CV and HQ systems as they evolve in time. (ii) To identify the TEI that most closely tracks the temporal evolution of EURs. We consider two generic systems. The first is a multilevel atom modeled as a nonlinear oscillator interacting with a quantized radiation field. The second is the Λ-atom interacting with two radiation fields. The former model accomodates investigations on the role of the initial state of the field and the ratio of the strengths of interaction and nonlinearity in the connection between TEIs and EURs. The second model opens up the possibility of examining the connection between mixed state bipartite entanglement and EURs, when the number of atomic levels is finite. Since the tomogram respects the requirements of classical probability theory, this effort also sheds light on the extent to which TEIs reflect the temporal behaviour of those EURs which are rooted in the Shannon entropy.

DOI
10.3389/frqst.2023.1131798

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Entropic uncertainty relations (EURs) have been examined in various contexts, primarily in qubit systems, including their links with entanglement, as they subsume the Heisenberg uncertainty principle. With their genesis in the Shannon entropy, EURs find applications in quantum information and quantum optics. EURs are state-dependent, and the state has to be reconstructed from tomograms (which are histograms readily available from experiments). This is a challenge when the Hilbert space is large, as in continuous variable (CV) systems and certain hybrid quantum (HQ) systems. A viable alternative approach therefore is to extract as much information as possible about the unknown quantum state directly from appropriate tomograms. Many variants of EURs can be extracted from tomograms, even for CV systems. In earlier work we have defined many tomographic entanglement indicators (TEIs) that can be readily calculated from tomograms without knowledge of the density matrix, and have reported on their efficacy as entanglement indicators in various contexts in both CV and HQ systems. The specific objectives of the present work are as follows: (i) To use the tomographic approach to investigate the links between EURs and TEIs in CV and HQ systems as they evolve in time. (ii) To identify the TEI that most closely tracks the temporal evolution of EURs. We consider two generic systems. The first is a multilevel atom modeled as a nonlinear oscillator interacting with a quantized radiation field. The second is the Λ-atom interacting with two radiation fields. The former model accomodates investigations on the role of the initial state of the field and the ratio of the strengths of interaction and nonlinearity in the connection between TEIs and EURs. The second model opens up the possibility of examining the connection between mixed state bipartite entanglement and EURs, when the number of atomic levels is finite. Since the tomogram respects the requirements of classical probability theory, this effort also sheds light on the extent to which TEIs reflect the temporal behaviour of those EURs which are rooted in the Shannon entropy.
A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomograms of qudits are given. A new universal integral inequality for generic wave function is obtained on the base of tomographic entropic uncertainty relations.
Abstract Extensive theoretical and experimental investigations on multipartite systems close to an avoided energy-level crossing reveal interesting features such as the extremisation of entanglement. Conventionally, the estimation of entanglement directly from experimental observation involves either one of two approaches: uncertainty-relation-based estimation that captures the linear correlation between relevant observables, or rigorous but error-prone quantum state reconstruction on tomograms obtained from homodyne measurements. We investigate the behaviour, close to avoided crossings, of entanglement indicators that can be calculated directly from a numerically-generated tomogram. The systems we study are two generic bipartite continuous-variable systems: a Bose–Einstein condensate trapped in a double-well potential, and a multi-level atom interacting with a radiation field. We also consider a multipartite hybrid quantum system of superconducting qubits interacting with microwave photons. We carry out a quantitative comparison of the indicators with a standard measure of entanglement, the subsystem von Neumann entropy (SVNE). It is shown that the indicators that capture the nonlinear correlation between relevant subsystem observables are in excellent agreement with the SVNE.
Uncertainty principle, which was first introduced by Werner Heisenberg in 1927, forms a fundamental component of quantum mechanics. A graceful aspect of quantum mechanics is that the uncertainty relations between incompatible observables allow for succinct quan- titative formulations of this revolutionary idea: it is impossible to simultaneously measure two complementary variables of a particle in precision. In particular, information theory offers two basic ways to express the Heisenberg’s principle: variance-based uncertainty relations and entropic uncertainty relations. We first investigate the uncertainty relations based on the sum of variances and derive a family of weighted uncertainty relations to provide an optimal lower bound for all situations. Our work indicates that it seems unreasonable to assume a priori that incompatible observables have equal contribution to the variance-based sum form uncertainty relations. We also study the role of mutually exclusive physical states in the recent work and generalize the variance-based uncertainty relations to mutually exclusive uncertainty relations. Next, we develop a new kind of entanglement detection criteria within the framework of marjorization theory and its matrix representation. By virtue of majorization uncertainty bounds, we are able to construct the entanglement criteria which have advantage over the scalar detect- ing algorithms as they are often stronger and tighter. Furthermore, we explore various expression of entropic uncertainty relations, including sum of Shannon entropies, majorization uncer- tainty relations and uncertainty relations in presence of quantum memory. For entropic uncertainty relations without quantum side information, we provide several tighter bounds for multi-measurements, with some of them also valid for Rényi and Tsallis entropies besides the Shannon entropy. We employ majorization theory and actions of the symmetric group to obtain an admixture bound for entropic uncertainty relations with multi-measurements. Comparisons among existing bounds for multi-measurements are also given. However,classical entropic uncertainty relations assume there has only classical side information. For modern uncertainty relations, those who allowed for non-trivial amount of quantum side information, their bounds have been strengthened by our recent result for both two and multi- measurements. Finally, we propose an approach which can extend all uncertainty relations on Shannon entropies to allow for quantum side information and discuss the applications of our entropic framework. Combined with our uniform entanglement frames, it is possible to detect entanglement via entropic uncertainty relations even if there is no quantum side in- formation. With the rising of quantum information theory, uncertainty relations have been established as important tools for a wide range of applications, such as quantum cryptography, quantum key distribution, entanglement detection, quantum metrology, quantum speed limit and so on. It is thus necessary to focus on the study of uncertainty relations.
Abstract Uncertainty relations are central to quantum physics. While they were originally formulated in terms of variances, they have later been successfully expressed with entropies following the advent of Shannon information theory. Here, we review recent results on entropic uncertainty relations involving continuous variables, such as position x and momentum p . This includes the generalization to arbitrary (not necessarily canonically-conjugate) variables as well as entropic uncertainty relations that take x - p correlations into account and admit all Gaussian pure states as minimum uncertainty states. We emphasize that these continuous-variable uncertainty relations can be conveniently reformulated in terms of entropy power, a central quantity in the information-theoretic description of random signals, which makes a bridge with variance-based uncertainty relations. In this review, we take the quantum optics viewpoint and consider uncertainties on the amplitude and phase quadratures of the electromagnetic field, which are isomorphic to x and p , but the formalism applies to all such variables (and linear combinations thereof) regardless of their physical meaning. Then, in the second part of this paper, we move on to new results and introduce a tighter entropic uncertainty relation for two arbitrary vectors of intercommuting continuous variables that takes correlations into account. It is proven conditionally on reasonable assumptions. Finally, we present some conjectures for new entropic uncertainty relations involving more than two continuous variables.
We discuss some applications of various versions of uncertainty relations for both discrete and continuous variables in the context of quantum information theory. The Heisenberg uncertainty relation enables demonstration of the Einstein, Podolsky and Rosen (EPR) paradox. Entropic uncertainty relations (EURs) are used to reveal quantum steering for non-Gaussian continuous variable states. EURs for discrete variables are studied in the context of quantum memory where fine-graining yields the optimum lower bound of uncertainty. The fine-grained uncertainty relation is used to obtain connections between uncertainty and the nonlocality of retrieval games for bipartite and tripartite systems. The Robertson-Schrödinger (RS) uncertainty relation is applied for distinguishing pure and mixed states of discrete variables.