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Journal of Physics A Mathematical and Theoretical

Uncertainty relations on the joint numerical range of operators

Arun Sehrawat

Published Aug 19, 2019• 2 citations
Publisher

Abstract

Abstract The expectation values of operators drawn from a single quantum state cannot be outside of a particular region, called their allowed region or the joint numerical range of the operators. We present a method to obtain all necessary and sufficient constraints—from Hermiticity, normalization, and positivity of a state and through the Born rule—that analytically defines the allowed region. Then, we present the allowed regions for the Heisenberg–Weyl operators, the angular momentum operators, and for their functions in dimension two to infinity. Especially, we consider three kinds of functions—combinations of powers of the ladder operators, powers of the angular momentum operators, and their anticommutators—and discover the allowed regions of different shapes. Here we also introduce uncertainty measures on the joint numerical range that are different from the standard deviation and the Shannon entropy. With the measures, we achieve a new kind of tight uncertainty relations for the Weyl- and the angular-momentum-operators. Overall, we demonstrate how the joint numerical range and the uncertainty relations change as the dimension grows. We apply the quantum de Finetti theorem to attain the allowed regions and tight uncertainty relations in the limit where the dimension goes to infinity.

DOI
10.1088/1751-8121/ab3ca6

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Abstract The expectation values of operators drawn from a single quantum state cannot be outside of a particular region, called their allowed region or the joint numerical range of the operators. We present a method to obtain all necessary and sufficient constraints—from Hermiticity, normalization, and positivity of a state and through the Born rule—that analytically defines the allowed region. Then, we present the allowed regions for the Heisenberg–Weyl operators, the angular momentum operators, and for their functions in dimension two to infinity. Especially, we consider three kinds of functions—combinations of powers of the ladder operators, powers of the angular momentum operators, and their anticommutators—and discover the allowed regions of different shapes. Here we also introduce uncertainty measures on the joint numerical range that are different from the standard deviation and the Shannon entropy. With the measures, we achieve a new kind of tight uncertainty relations for the Weyl- and the angular-momentum-operators. Overall, we demonstrate how the joint numerical range and the uncertainty relations change as the dimension grows. We apply the quantum de Finetti theorem to attain the allowed regions and tight uncertainty relations in the limit where the dimension goes to infinity.
The expectation values of operators drawn from a single quantum state cannot be outside of a particular region, called their allowed region or the joint numerical range of the operators. Basically, the allowed region is an image of the state space under the Born rule. The maximum-eigenvalue-states---of every linear combination of the operators of interest---are sufficient to generate boundary of the allowed region. In this way, we obtain the numerical range of certain Hermitian operators (observables) that are functions of the angular momentum operators. Especially, we consider here three kinds of functions---combinations of powers of the ladder operators, powers of the angular momentum operators and their anticommutators---and discover the allowed regions of different shapes. By defining some specific concave (and convex) functions on the joint numerical range, we also achieve tight uncertainty (and certainty) relations for the observables. Overall, we demonstrate how the numerical range and uncertainty relations change as the angular momentum quantum number grows. Finally, we apply the quantum de Finetti theorem by taking a multi-qubit system and attain the allowed regions and tight uncertainty relations in the limit where the quantum number goes to infinity.
We present a systematic procedure to obtain all necessary and sufficient (quantum) constraints on the expectation values for any set of qudit's operators. These constraints---arise form Hermiticity, normalization, and positivity of a statistical operator and through Born's rule---analytically define an allowed region. A point outside the admissible region does not correspond to any quantum state, whereas every point in it come from a quantum state. For a set of observables, the allowed region is a compact and convex set in a real space, and all its extreme points come from pure quantum states. By defining appropriate concave functions on the permitted region and then finding their absolute minimum at the extreme points, we obtain different tight uncertainty relations for qubit's and spin observables. In addition, quantum constraints are explicitly given for the Weyl operators and the spin observables.
Uncertainty relations are usually stated as bounds on selected combinations of variances, but the full covariance matrix contains substantially richer information about the geometry of quantum state space and about the operational capabilities of quantum systems. Here we characterize the covariance matrices attainable by a finite-dimensional canonical pair of observables related by the discrete Fourier transform, the natural analogue of position and momentum in a finite Hilbert space. We combine analytic arguments with convex-geometric and semidefinite-programming methods based on joint numerical ranges to describe the admissible region through unitary invariants, in particular the trace and determinant of the covariance matrix. This provides a systematic way to identify extremal states, generalizing the notion of minimum-uncertainty states, and to quantify how the discrete uncertainty geometry approaches its continuous counterpart with increasing dimension. We further show that the resulting covariance-matrix characterization has direct consequences for applications: it yields accuracy bounds for multi parameter estimation protocols and separability criteria for finite-dimensional bipartite systems, including discrete analogues of continuous-variable EPR-type witnesses. Our results establish a systematic and versatile platform for connecting uncertainty relations, convex quantum geometry, metrology, and entanglement detection in finite-dimensional systems.
Abstract Constructive techniques to establish state-independent uncertainty relations for the sum of variances of arbitrary two observables are presented. We investigate the range of simultaneously attainable pairs of variances, which can be applied to a wide variety of problems including finding exact bound for the sum of variances of two components of angular momentum operator for any total angular momentum quantum number j and detection of quantum entanglement. Resulting uncertainty relations are state-independent, semianalytical, bounded-error and can be made arbitrarily tight. The advocated approach, based on the notion of joint numerical range of a number of observables and uncertainty range, allows us to improve earlier numerical works and to derive semianalytical tight bounds for the uncertainty relation for the sum of variances expressed as roots of a polynomial of a single real variable.
Abstract In standard formulations of the uncertainty principle, two fundamental features are typically cast as impossibility statements: two noncommuting observables cannot in general both be sharply defined (for the same state), nor can they be measured jointly. The pioneers of quantum mechanics were acutely aware and puzzled by this fact, and it motivated Heisenberg to seek a mitigation, which he formulated in his seminal paper of 1927. He provided intuitive arguments to show that the values of, say, the position and momentum of a particle can at least be unsharply defined, and they can be measured together provided some approximation errors are allowed. Only now, nine decades later, a working theory of approximate joint measurements is taking shape, leading to rigorous and experimentally testable formulations of associated error tradeoff relations. Here we briefly review this new development, explaining the concepts and steps taken in the construction of optimal joint approximations of pairs of incompatible observables. As a case study, we deduce measurement uncertainty relations for qubit observables using two distinct error measures. We provide an operational interpretation of the error bounds and discuss some of the first experimental tests of such relations.
As a foundation of quantum physics, uncertainty relations describe ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertainty relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position and momentum operators, Heisenberg’s uncertainty principle points exactly to the attainable area of the variances of position and momentum. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of (a) a pair of one-qubit observables and (b) a pair of projective observables for arbitrary dimension, and give the first experimental observation of such areas in a photonic system.