Optimal entropic uncertainty relation for successive measurements in quantum information theory

Srinivas, M.

doi:10.1007/bf02704281

Cited by 35 publications

(55 citation statements)

References 18 publications

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“…It may be noted that uncertainty relations have also been studied in the successive measurement scenario, both in the form of entropic relations [20,21], and in the form of error-disturbance relations [4,[22][23][24] that are in line with Heisenberg's original interpretation of the uncertainty principle. In contrast, here we look at the disturbances associated with distinct measurements of in-compatible observables on identically prepared ensembles of systems.…”

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confidence: 78%

Disturbance trade-off principle for quantum measurements

Mandayam

Srinivas

2014

Phys. Rev. A

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We demonstrate a fundamental principle of disturbance tradeoff for quantum measurements, along the lines of the celebrated uncertainty principle: The disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. Indeed, we show that the average of the disturbances associated with a set of projective measurements is strictly greater than zero whenever the associated observables do not have a common eigenvector. For such measurements, we show an equivalence between disturbance tradeoff measured in terms of fidelity and the entropic uncertainty tradeoff formulated in terms of the Tsallis entropy (T2). We also investigate the disturbances associated with the class of nonprojective measurements, where the difference between the disturbance tradeoff and the uncertainty tradeoff manifests quite clearly.The uncertainty principle, which is one of the cornerstones of quantum theory, has had a long history. In its original formulation by Heisenberg for canonically conjugate variables [1], the uncertainty principle was stated as an effect of the disturbance caused due to a measurement of one observable on a succeeding measurement of another. However, the subsequent mathematical formulation due to Robertson [2] and Schrödinger [3] in terms of variances departed from this original interpretation. Rather, they obtained a non-trivial lower bound on the product of the variances associated with the measurement of a pair of incompatible observables, performed on distinct yet identically prepared copies of a given system. The Robertson-Schrödinger inequality thus expresses a fundamental limitation as regards the preparation of an ensemble of systems in identical states, and is therefore a manifestation of the so-called preparation uncertainty [4].Along the same lines, the more recent entropic formulations of the uncertainty principle [5] also demonstrate the existence of a fundamental tradeoff for the uncertainties associated with independent measurements of incompatible observables on identically prepared ensemble of systems. Entropic uncertainty relations (EURs) have been obtained for specific classes of observables for both the Shannon and Rényi entropies [6][7][8][9][10][11][12][13][14], as well as for the Tsallis entropies [15][16][17].Here, we prove the existence of a similar principle of tradeoff for the disturbances associated with the measurements of a set of observables. It is a fundamental feature of quantum theory that when an observable is measured on an ensemble of systems, the density operator of the resulting ensemble is in general different from that prior to the measurement. The distance between these two density operators is indeed a measure of the disturbance due to measurement. Different measures of distance between density operators [18] give rise to different measures of disturbance. We are thus lead to a class of disturbance measures which have been used recently in the context of quantifying incompatibilit...

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confidence: 78%

Disturbance trade-off principle for quantum measurements

Mandayam

Srinivas

2014

Phys. Rev. A

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“…Following-on from Heisenberg's original insight, M. D. Srinivas derived the EUR for successive measurements in 2002 [22] as follows. Consider observ-ables X and Y with non-degenerate spectra,…”

Section: Entropic Uncertainty Relation For Successive Measurementsmentioning

confidence: 99%

“…Using relations (21), (22) and (23), we consider a successive measurement of observables X(0) = σ x and Y (0) = σ y with an input state vector |ψ . Then the probabilities of outcomes obtained by successive measurement of X(0), Y (0) are restricted by the uncertainty relations.…”

Section: Comparison Between the Eur For Successive Measurements Amentioning

confidence: 99%

Optimized entropic uncertainty for successive projective measurements

2014

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In the history of quantum mechanics, various types of uncertainty relationships have been introduced to accommodate different operational meanings of Heisenberg uncertainty principle. We derive an optimized entropic uncertainty relation (EUR) that quantifies an amount of quantum uncertainty in the scenario of successive measurements. The EUR characterizes the limitation in the measurability of two different quantities of a quantum state when they are measured through successive measurements. We find that the bound quantifies the information between the two measurements and imposes a condition that is consistent with the recently-derived error-disturbance relationship.

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“…[31][32][33][34][35][36] Brillouin appears to be the first one to think of information theoretic approach to the uncertainty relations. 30 As stressed first by Everett, 31 these relations are stronger than HUR.…”

Section: Discussionmentioning

confidence: 99%

Minimum Uncertainty and Entanglement

Dass

Qureshi

Sheel

2013

Int. J. Mod. Phys. B

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We address the question, does a system A being entangled with another system B, put any constraints on the Heisenberg uncertainty relation (or the Schrödinger-Robertson inequality)? We find that the equality of the uncertainty relation cannot be reached for any two noncommuting observables, for finite dimensional Hilbert spaces if the Schmidt rank of the entangled state is maximal. One consequence is that the lower bound of the uncertainty relation can never be attained for any two observables for qubits, if the state is entangled. For infinite-dimensional Hilbert space too, we show that there is a class of physically interesting entangled states for which no two noncommuting observables can attain the minimum uncertainty equality.

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Optimal entropic uncertainty relation for successive measurements in quantum information theory

Cited by 35 publications

References 18 publications

Disturbance trade-off principle for quantum measurements

Disturbance trade-off principle for quantum measurements

Optimized entropic uncertainty for successive projective measurements

Minimum Uncertainty and Entanglement

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