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Physical Review A

Complete and incomplete state estimation via the simultaneous unsharp measurement of two incompatible qubit operators

Daniel Saavedra, K. M. Fonseca-Romero

Published Apr 30, 2019• 4 citations
Publisher

Abstract

We consider the simultaneous coupling of a qubit system to two qubit probes, designed to measure noncommuting qubit observables when working in isolation. While the single meter model usually corresponds to an unsharp measurement of a qubit operator, the double meter model usually describes an informationally complete measurement. The double meter positive operator measurement can be characterized by three vectors associated with the space of Hermitian tracesless observables of the system. The dimension of the linear space generated by these vectors (Bloch rank) is equal to the number of independent operators (or the number of independent Bloch-vector components) which can be estimated. $\mathbb{S}$, the subspace of parameters associated with informationally incomplete measurements (IICs), is characterized either analytically or numerically, and their Bloch rank is used to classify the IICs. Bloch rank of the ``simultaneous measurement'' of two Pauli matrices is shown to be 2 if the isolated measurements are projective and 3 if they are weak; however, two-meter weak measurements are expected to converge more slowly than their single-meter counterparts.

DOI
10.1103/physreva.99.042130

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We consider the simultaneous coupling of a qubit system to two qubit probes, designed to measure noncommuting qubit observables when working in isolation. While the single meter model usually corresponds to an unsharp measurement of a qubit operator, the double meter model usually describes an informationally complete measurement. The double meter positive operator measurement can be characterized by three vectors associated with the space of Hermitian tracesless observables of the system. The dimension of the linear space generated by these vectors (Bloch rank) is equal to the number of independent operators (or the number of independent Bloch-vector components) which can be estimated. S\mathbb{S}S, the subspace of parameters associated with informationally incomplete measurements (IICs), is characterized either analytically or numerically, and their Bloch rank is used to classify the IICs. Bloch rank of the ``simultaneous measurement'' of two Pauli matrices is shown to be 2 if the isolated measurements are projective and 3 if they are weak; however, two-meter weak measurements are expected to converge more slowly than their single-meter counterparts.
We consider the evolution of a qubit (spin 1/21/21/2) under the simultaneous continuous measurement of three noncommuting qubit operators ${\stackrel{^}{\ensuremath{\sigma}}}_{x}$, ${\stackrel{^}{\ensuremath{\sigma}}}_{y}$, and ${\stackrel{^}{\ensuremath{\sigma}}}_{z}$. For identical ideal detectors, the qubit state evolves by approaching a pure state with a random direction in the Bloch vector space and by undergoing locally isotropic diffusion in the perpendicular directions. The quantum state conditioned on the complete detector record is used to assess the fidelity of classically inspired estimates based on running time averages and discrete time bin detector outputs.
Abstract Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of two parameters that are respectively associate with the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating N qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy—first, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values j , then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show that when the state is nearly pure, for sufficiently but not arbitrarily large N , most information will be captured in the largest three j -subspaces. Then, we study how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N = 2, 3, or approximately when the qubits are nearly pure for other N values. We also obtain numerical results that suggest that using a Bell multiport setup, one can distinguish between projection onto the j = N /2 and j = N /2 − 1 subspaces from their respective interference signatures at the output.
Quantum estimation involving multiple parameters remains an important problem of both theoretical and practical interest. In this work, we study the problem of simultaneous estimation of the length and direction of the Bloch vector for identically prepared qubit states that is confined to a plane, where in order to obtain the optimal estimation precision for both parameters, collective measurements on multiple qubits are necessary. Upon treating NNN qubits as an ensemble of spin-1/2 systems, we show that simultaneous optimal estimation for both parameters can be attained asymptotically with a simple collective measurement strategy -- First, we estimate the length parameter by measuring the populations in spaces corresponding to different total angular momentum values jjj, then we estimate the direction parameter by performing a spin projection onto an optimal basis. Furthermore, we show how the total angular-momentum measurement can be realized by observing output signatures from a Bell multiport setup, either exactly for N=2,3N=2,3N=2,3, or approximately when the qubits are nearly pure for other NNN values. Our results point to the conjecture that using a Bell multiport setup, one can distinguish between projection onto the j=N/2j=N/2j=N/2 and j=N/2−1j=N/2-1j=N/2−1 subspaces from their respective interference signatures at the output.
Starting from a simple estimation problem, here we propose a general approach for decoding quantum measurements from the perspective of information extraction.By virtue of the estimation fidelity only, we provide surprisingly simple characterizations of rank-1 projective measurements, mutually unbiased measurements, and symmetric informationally complete measurements.Notably, our conclusions do not rely on any assumption on the rank, purity, or the number of measurement outcomes, and we do not need bases to start with.Our work demonstrates that all these elementary quantum measurements are uniquely determined by their information-extraction capabilities, which are not even anticipated before.In addition, we offer a new perspective for understanding noncommutativity and incompatibility from tomographic performances, which also leads to a universal criterion for detecting quantum incompatibility.Furthermore, we show that the estimation fidelity can be used to distinguish inequivalent mutually unbiased bases and symmetric informationally complete measurements.In the course of study, we introduce the concept of (weighted complex projective) 1/2-designs and show that all 1/2-designs are tied to symmetric informationally complete measurements, and vice versa.
We consider a joint measurement of two and three unsharp qubit observables through an Arthur–Kelly-type joint measurement model for qubits. We investigate the effect of the initial state of the detectors on the unsharpness of the measurement as well as the post-measurement state of the system. Particular emphasis is given on a physical understanding of the POVM to PVM transition in the model and entanglement between the system and the detectors. Two approaches for characterizing the unsharpness of the measurement and the resulting measurement uncertainty relations are considered. The corresponding measures of unsharpness are connected for the case where both the measurements are equally unsharp. The connection between the POVM elements and symmetries of the underlying Hamiltonian of the measurement interaction is made explicit and used to perform the joint measurement in arbitrary directions. Finally, in the case of three observables, we derive a necessary condition for the approximate joint measurement and use it to show the relative freedom available when the observables are non-orthogonal.
We formulate the accuracy of a quantum measurement for a qubit (spin-1/2) system in terms of a $3\ifmmode\times\else\texttimes\fi{}3$ matrix. This matrix, which we refer to as the accuracy matrix, can be calculated from a positive operator-valued measure (POVM) corresponding to the quantum measurement. Based on the accuracy matrix, we derive trade-off relations between the measurement accuracy of two or three noncommuting observables of a qubit system. These trade-off relations offer a quantitative information-theoretic representation of Bohr's principle of complementarity. They can be interpreted as the uncertainty relations between measurement errors in simultaneous measurements and also as the trade-off relations between the measurement error and back-action of the measurement. A no-cloning inequality is derived from the trade-off relations. Furthermore, our formulation and the results obtained can be applied to analyze quantum-state tomography. We also show that the accuracy matrix is closely related to the maximum-likelihood estimation and the Fisher information matrix for a finite number of samples; the accuracy matrix tells us how accurately we can estimate the probability distributions of observables of an unknown state by a finite number of quantum measurements.