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IEEE Transactions on Quantum Engineering

Linear Quantum State Observers

Maison Clouâtré, Mark J. Balas, Vinod P. Gehlot, John Valasek

Published Jan 1, 2022• 9 citations
Publisher

Abstract

This paper considers the use of linear state observers to infer the unknown state (density matrix) of a closed quantum system using generalized quantum measurement—positive operator-valued measures (POVMs). An efficient test for observability is described in terms of two quantum observability matrices. The main contribution of this work is the development of a canonical quantum state observer, which may be utilized to estimate the state of any closed quantum system using POVM measurements. The observer ensures the state estimation error tends to the unobservable space of the system, which is exactly zero when the system is observable. The space of Hermitian matrices is shown to be invariant with respect to the dynamics of the canonical observer, which allows one to project the observer's state onto the space of valid quantum density operators while retaining convergence to the true state.

DOI
10.1109/tqe.2022.3209927

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This paper considers the use of linear state observers to infer the unknown state (density matrix) of a closed quantum system using generalized quantum measurement—positive operator-valued measures (POVMs). An efficient test for observability is described in terms of two quantum observability matrices. The main contribution of this work is the development of a canonical quantum state observer, which may be utilized to estimate the state of any closed quantum system using POVM measurements. The observer ensures the state estimation error tends to the unobservable space of the system, which is exactly zero when the system is observable. The space of Hermitian matrices is shown to be invariant with respect to the dynamics of the canonical observer, which allows one to project the observer's state onto the space of valid quantum density operators while retaining convergence to the true state.
Designing an observer of an unknown quantum density operator is difficult because the operator must be Hermitian positive semidefinite with unit trace. In this paper, we derive a closed-form solution for projecting an arbitrary matrix onto the set of valid density operators. This allows us to design linear quantum state observers and retract the observer’s state back to this set while retaining the exponential convergence rate of the linear observer. The derived closed-form projection can be used alongside any quantum state estimation technique to produce a valid state estimate without increasing the state estimation error.
Abstract In this paper, we introduce an analytical framework for the reconstruction of quantum states. The reconstruction of an unknown quantum state requires the information of a complete set of observables, obtained through experimental measurements of Hermitian operators usually defined as positive-operator-valued measures (POVMs). The scheme involves a single-qubit unambiguous state discrimination POVM, which can be generalized to perform n-qubit measurements. We also use maximum likelihood estimation as a method in the reconstruction of the density matrix from experimental data and show that the expected value of the cleaner is independent of the parameter of the density operator.
Generalized quantum measurements, described by positive operator-valued measures (POVMs), are essential for modeling realistic processes in open quantum systems. While quantum process tomography can fully characterize a POVM, it is resource-intensive and impractical when only specific POVM elements or matrix elements of a particular POVM element are of interest. Direct quantum measurement tomography offers a more efficient alternative but typically relies on weak interactions and complex structures of the system, environment, and probe as the dimension of the system increases, limiting its precision and scalability. Furthermore, characterizing a POVM element alone is insufficient to determine the underlying physical mechanism, as multiple Kraus operators can yield the same measurement statistics. In this work, we present a unified framework for the direct characterization of individual matrix elements of Kraus operators associated with specific POVM elements and arbitrary input states without requiring weak interaction, complex structures of the system-environment-probe or full process and state tomography. This framework naturally extends to projective measurements, enabling direct observable tomography, and to the characterization of unitary operations. Our method also captures modular and weak values of observables and Kraus operators, without invoking weak interaction approximations. We demonstrate potential implementations in optical systems, highlighting the experimental feasibility of our approach.
In the Hilbert space formulation of quantum mechanics, ideal measurements of physical variables are discussed using the spectral theory of Hermitian operators and the corresponding projector-valued measures (PVMs). However, more general types of measurements require the treatment in terms of positive-operator-valued measures (POVMs). In the Hamiltonian formulation of quantum mechanics, canonical coordinates are related to PVM. In this paper the results of an analysis of various aspects of applications of POVMs in the Hamiltonian formulation are reported. Several properties of state parameters and quantum observables given by POVMs or represented in an overcomplete basis, including the general Hamiltonian treatment of the Neumark extension, are presented. An analysis of the phase operator, given by the corresponding POVMs, in the Hilbert space and the Hamiltonian frameworks is also given.
Positive operator valued measurements (POVMs) play an important role in efficient quantum communication and computation. While optical systems are one of the strongest candidates for long distance quantum communication and information processing, efficient methods to implement POVMs in these systems are scarce. Here we propose an all-optical scheme to implement an arbitrary POVM using linear optical components on m-dimensional Hilbert space of internal degrees of freedom. Linear optical nature of the proposed scheme makes it efficient and robust. We show how the scheme can be applied for state tomography and for preparing arbitrary mixed states.
We discuss the problem of implementing generalized measurements [positive operator-valued measures (POVMs)] with linear optics, either based upon a static linear array or including conditional dynamics. In our approach, a given POVM shall be identified as a solution to an optimization problem for a chosen cost function. We formulate a general principle: the implementation is only possible if a linear-optics circuit exists for which the quantum mechanical optimum (minimum) is still attainable after dephasing the corresponding quantum states. The general principle enables us, for instance, to derive a set of necessary conditions for the linear-optics implementation of the POVM that realizes the quantum mechanically optimal unambiguous discrimination of two pure nonorthogonal states. This extends our previous results on projection measurements and the exact discrimination of orthogonal states.