Exact and approximate solutions for the quantum minimum-Kullback-entropy estimation problem

Carlo Sparaciari, Stefano Olivares, Francesco Ticozzi, and Matteo G. A. Paris

Phys. Rev. A 89, 042124 – Published 30 April 2014

Abstract

The minimum-Kullback-entropy principle (mKE) is a useful tool to estimate quantum states and operations from incomplete data and prior information. In general, the solution of an mKE problem is analytically challenging and an approximate solution has been proposed and employed in different contexts. Recently, the form and a way to compute the exact solution for finite dimensional systems has been found, and a question naturally arises on whether the approximate solution could be an effective substitute for the exact solution, and in which regimes this substitution can be performed. Here, we provide a systematic comparison between the exact and the approximate mKE solutions for a qubit system when average data from a single observable are available. We address both mKE estimation of states and weak Hamiltonians, and compare the two solutions in terms of state fidelity and operator distance. We find that the approximate solution is generally close to the exact one unless the initial state is near an eigenstate of the measured observable. Our results provide a rigorous justification for the use of the approximate solution whenever the above condition does not occur, and extend its range of application beyond those situations satisfying the assumptions used for its derivation.

Received 24 November 2013

DOI:https://doi.org/10.1103/PhysRevA.89.042124

Authors & Affiliations

Carlo Sparaciari^*

Dipartimento di Fisica dell'Università degli Studi di Milano, I-20133 Milan, Italy

Stefano Olivares^†

Dipartimento di Fisica dell'Università degli Studi di Milano, I-20133 Milan, Italy and CNISM UdR Milano Statale, I-20133 Milan, Italy

Francesco Ticozzi^‡

Dipartimento di Ingegneria dell'Informazione, Università di Padova, I-35131 Padova, Italy and Department of Physics and Astronomy, Dartmouth College, 6127 Wilder, Hanover, New Hampshire 03755, USA

Matteo G. A. Paris^§

Dipartimento di Fisica dell'Università degli Studi di Milano, I-20133 Milan, Italy and CNISM UdR Milano Statale, I-20133 Milan, Italy

^*carlo.sparaciari@studenti.unimi.it
^†stefano.olivares@mi.infn.it
^‡ticozzi@dei.unipd.it
^§matteo.paris@fisica.unimi.it

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Issue

Vol. 89, Iss. 4 — April 2014

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Images

Figure 1
Fidelity between the exact and the approximate mKE estimate as a function of the parameter $θ$ of the initial state and of the outcome of the measurement $〈 σ_{3} 〉$ . The plots are for two fixed values of the initial purity: $μ = 0.55$ (top left) and $μ = 0.7$ (top right). The lower panel shows the minimum of fidelity for $θ = π / 2$ (solid red), $θ = 5 π / 12$ (black dashed), and $θ = π / 3$ (blue dotted) as a function of purity $μ$ .
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Figure 2
Fidelity between the exact and the approximate mKE estimate for $μ = 0.9$ (left) and $μ = 0.9999$ (right). The lower left panel shows the behavior of the fidelity as a function of $θ$ for $〈 σ_{3} 〉 = 0$ and for different (close to unit) values of purity: $μ = 1 - 10^{- 4}$ (solid red line), $μ = 1 - 10^{- 5}$ (green dashed), and $μ = 1 - 10^{- 6}$ (blue dotted). The lower right panel shows the minimum of fidelity for for $〈 σ_{3} 〉 = 0$ (solid red line), $〈 σ_{3} 〉 = \pm 0.3$ (black dashed), and $〈 σ_{3} 〉 = \pm 0.5$ (blue dotted).
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Figure 3
The ratio $Z$ between the two fidelities $F (ρ_{apx}, τ)$ and $F (ρ_{exa}, τ)$ , as a function of $θ$ and $〈 σ_{3} 〉$ , for different values of initial purity $μ$ . In the first graphic we have $Z$ for $μ = 0.7$ . Then, from top to bottom, from left to right, we find graphics for $μ = 0.8$ , $μ = 0.9$ , and $μ = 0.99$ . It is evident that $Z$ is always greater (or equal) than one, which means that $ρ_{apx}$ is, in general, closer to $τ$ than $ρ_{exa}$ .
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Figure 4
Purities of the approximate and exact solutions of mKE estimation. The first three plots show the purity of the approximate solutions as a function of the exact one for the initial purity in the ranges $μ = 0.5$ –0.6 (top left), $μ = 0.7$ –0.8 (top right), and $μ = 0.9$ –1.0 (bottom left), respectively. In all the plots gray points correspond to purities of mKE solutions obtained by random selecting the initial states with $θ \in [0, π]$ and $μ$ in the given range, and simulating random measurements with $〈 σ_{3} 〉 \in [- 1, 1]$ . The right lower panels shows the ratio $R_{μ}$ as function of initial purity $μ$ . Here the gray points are obtained by randomly selecting the initial states in the full range of $θ$ and $μ$ and $〈 σ_{3} 〉 \in [- 1, 1]$ , the blue (square) points are obtained by sampling $θ$ in the range $θ \in [π / 2 - π / 10, π / 2 + π / 10$ ], and $| 〈 σ_{3} 〉 | \in [0.9, 1]$ , and the red (rhombus) points correspond to random sampling $θ \in [0, 0.5]$ and $〈 σ_{3} 〉 \in [- 0.1, 0.1]$ .
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Figure 5
Comparison between exact and approximate solution for the mKE estimation of weak Hamiltonians. On the left side, the trace distance $D$ between the $H_{exa}$ and the $H_{apx}$ is plotted for a purity of the initial state of $μ = 0.55$ . On the right, the trace distance $D$ is displayed for $μ = 0.7$ .
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Figure 6
Comparison between exact and approximate solution for the mKE estimation of weak Hamiltonians. On the left side, the trace distance $D$ between the $H_{exa}$ and the $H_{apx}$ is plotted for a purity of the initial state of $μ = 0.9$ . On the right, the trace distance $D$ is displayed for $μ = 0.9999$ .
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