Optimized entropic uncertainty for successive projective measurements

Kyunghyun Baek, Tristan Farrow, and Wonmin Son

Phys. Rev. A 89, 032108 – Published 10 March 2014

Abstract

We focus here on the uncertainty of an observable $Y$ caused by a precise measurement of $X$ . We illustrate the effect by analyzing the general scenario of two successive measurements of spin components $X$ and $Y$ . We derive an optimized entropic uncertainty limit that quantifies the necessary amount of uncertainty observed in a subsequent measurement of $Y$ . We compare this bound to recently derived error-disturbance relations and discuss how the bound quantifies the information of successive quantum measurements.

Received 4 December 2013

DOI:https://doi.org/10.1103/PhysRevA.89.032108

Authors & Affiliations

Kyunghyun Baek¹, Tristan Farrow^2,3, and Wonmin Son^1,*

¹Department of Physics, Sogang University, Mapo-gu, Shinsu-dong, Seoul 121-742, Korea
²Atomic & Laser Physics, Clarendon Laboratory, University of Oxford, OX1 3PU, United Kingdom
³Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543, Singapore

^*sonwm71@sogang.ac.kr

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Vol. 89, Iss. 3 — March 2014

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Figure 1
Pure state representation in a Bloch sphere. A prepared state is denoted by $| ψ 〉 = cos θ | 0 〉 + sin θ e^{i φ} | 1 〉$ , with polar angle $0 \leq θ \leq π$ and azimuthal angle $0 \leq φ \leq 2 π$ . The north and south poles are chosen to correspond to eigenvectors of $σ_{z}$ . Varying $φ$ from 0 to $π / 2$ with a fixed $θ$ , we can consider state vectors on a circle (dashed blue). When $θ = π / 2$ , it becomes a circle on the $x$ - $y$ plane (dashed red).
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Figure 2
Graphs illustrating how EUR and Robertson's uncertainty relations behave with the probabilities of outcomes of observables $X (0)$ and $Y (0)$ . To compare two relations, we plot the LHSs of the inequalities (11) against the azimuthal angle $φ$ for chosen values of polar angle $θ$ : (a) 0, (b) $3 π / 8$ , (c) $4 π / 9$ , and (d) $π / 2$ . The EUR is optimized as $θ$ goes to $π / 2$ , whereas the Roberson uncertainty relation diverges. Relations (2) and (4) have a minimum value when $θ = 0$ and $π / 2$ for fixed $ϕ$ , respectively.
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Figure 3
Schematic of a successive measurement scheme for observables $X$ and $Y$ , used to clarify and compare EURs ( $12$ ), ( $16$ ), and ( $17$ ). After preparing the input states $| x_{+} 〉$ and $| y_{+} 〉$ , which are eigenstates of $σ_{x}$ and $σ_{y}$ , respectively, the successive measurement is assumed to measure observables $X = σ_{ϕ}$ and $Y = σ_{y}$ . It results in four possible outcomes.
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Figure 4
Graphs showing the LHSs of relations (12) (blue), (16) (purple), and (17) (orange) and the lower bound (dashed red) with respect to the detuning angle $ϕ$ . Two input states $| x_{+} 〉$ and $| y_{+} 〉$ are considered. By inspection, we find that the LHS of the EUR for successive measurement (12) always has a larger value than the LHS of (16) due to subadditivity (14). The conditional entropy has a smaller value. Three relations have zero value only when two measurements are the same at $ϕ = π / 2$ , but only the conditional entropy is saturated when two observables are mutually unbiased at $ϕ = 0$ . This means that the inner product of all pairs of each eigenstate is given by $1 / \sqrt{d}$ , where $d$ is a dimension of the Hilbert space [21].
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Figure 5
Graphs illustrating how the different EURs for successive measurements (16), (17), and the Ozawa's relation (3) impose restrictions on the probabilities of outcomes of observables $X (0)$ and $Y (0)$ . The LHSs of relations (21) [joint entropy (JE) (blue)], (22) [conditional entropy (CE) (purple)], and (23) [Ozawa's relation (orange)] are plotted against the azimuthal angle $φ$ for fixed values of the polar angle $θ$ (0, $π / 4$ , $3 π / 8$ , and $π / 2$ ). By inspection, we find that the JE (21) decreases with respect to increasing $θ$ , whereas the Ozawa relation (23) diverges. However, the conditional entropy of CE (22) is given by a constant 1 for all values of $θ$ and $ϕ$ .
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