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arXiv:1702.01680 · Physical Review A

Combined-probability space and certainty or uncertainty relations for a finite-level quantum system

Arun Sehrawat

Published Aug 7, 2017• 1 citation
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Abstract

The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys certain quantum constraints, which are triangle inequalities in our case. Such a restricted set of combined vectors, called the combined-probability space, is presented here for a $d$-level quantum system (qudit). The combined space is a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum on the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) over the whole space; it will be on the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. We also demonstrate that many known tight certainty or uncertainty relations for a qubit can be obtained with the triangle inequalities.

DOI
10.1103/physreva.96.022111

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The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys certain quantum constraints, which are triangle inequalities in our case. Such a restricted set of combined vectors, called the combined-probability space, is presented here for a ddd-level quantum system (qudit). The combined space is a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum on the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) over the whole space; it will be on the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. We also demonstrate that many known tight certainty or uncertainty relations for a qubit can be obtained with the triangle inequalities.
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.
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.
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.
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For a qudit, a quantum system on a finite dimensional Hilbert space, we construct minimum uncertainty states and a non-negative quasi probability distribution. They are the discrete version of the coherent states and the Husimi function in the continuous quantum mechanics. The two bases related by the discrete Fourier transformation are identified with the position and momentum bases. We prove an inequality concerning uncertainties of the position and momentum of a qudit. The minimum uncertainty states are the ones which saturate the inequality. Like the coherent states, they are not mutually orthogonal but comprise an overcomplete set in the state space. As the dimension of the Hilbert space goes to infinity, these states approach the coherent states. Using the minimum uncertain states, we construct a non-negative quasi probability distribution for a qudit. Its marginal distributions are smeared out, but we show that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance is assumed.
A geometric approach to formulate the uncertainty principle between quantum observables acting on an NNN-dimensional Hilbert space is proposed. We consider the fidelity between a density operator associated with a quantum system and a projector associated with an observable, and interpret it as the probability of obtaining the outcome corresponding to that projector. We make use of fidelity-based metrics such as angle, Bures, and root infidelity to propose a measure of uncertainty. The triangle inequality allows us to derive a family of uncertainty relations. In the case of the angle metric, we recover the Landau-Pollak inequality for pure states and show, in a natural way, how to extend it to the case of mixed states in arbitrary dimension. In addition, we derive and compare alternative uncertainty relations when using other known fidelity-based metrics.