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arXiv:quant-ph/0411067 · Journal of Physics A Mathematical and General

Purity-bounded uncertainty relations in multidimensional space—generalized purity

Mikalai U. Karelin

Published Jun 29, 2005• 3 citations
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Abstract

Uncertainty relations for mixed quantum states (precisely, purity-bounded position–momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in the general multidimensional case. An expression for a family of mixed states at the lower bound of uncertainty relation is obtained. It is shown that in the case of entropy-bounded uncertainty relations, the lower-bound state is thermal, and a transition from the one-dimensional problem to the multidimensional one is trivial. Results of numerical calculation of the relation of the lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.

DOI
10.1088/0305-4470/38/28/012

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Uncertainty relations for mixed quantum states (precisely, purity-bounded position–momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in the general multidimensional case. An expression for a family of mixed states at the lower bound of uncertainty relation is obtained. It is shown that in the case of entropy-bounded uncertainty relations, the lower-bound state is thermal, and a transition from the one-dimensional problem to the multidimensional one is trivial. Results of numerical calculation of the relation of the lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.
In the space of mixed states the Schrödinger-Robertson uncertainty relation holds though it can never be saturated. Two tight extensions of this relation in the space of mixed states exist; one proposed by Dodonov and Man'ko, where the lower limit on the uncertainty depends on the purity of the state, and another where the uncertainty is bounded by the von Neumann entropy of the state proposed by Bastiaans. Driven by the needs that have emerged in the field of quantum information, in a recent work we have extended the puritybounded uncertainty relation by adding an additional parameter characterizing the state, namely its degree of non-Gaussianity. In this work we alternatively present a extension of the entropy-bounded uncertainty relation. The common points and differences between the two extensions of the uncertainty relation help us to draw more general conclusions concerning the bounds on the non-Gaussianity of mixed states.
We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert-Schmidt distance between the minimizing states and the `most classical' thermal states is used as a `measure of nonclassicality' together with the Mandel parameter. For highly mixed minimizing states (whose `purities' are very small), the normalized Hilbert-Schmidt distance tends to a finite limit, which depends on the nature of the state (15% of the maximal possible distance if the deviation from pure states is characterized by the `standard purity' Tr 2 and 37% if the `superpurity' lim r →0 [Tr ( 1+1/ r )] r is chosen as a measure of deviation).
We give a review of different forms of uncertainty relations for mixed quantum states obtained over the last two decades and present many new results. The nonclassical properties of mixed states minimizing the purity-bounded uncertainty relations (a possibility of sub-Poissonian statistics, squeezing etc) are considered. The normalized Hilbert–Schmidt distance between the minimizing states and the ‘most classical ’ thermal states is used as a ‘measure of nonclassicality ’ together with the Mandel parameter. For highly mixed minimizing states (whose ‘purities ’ are very small), the normalized Hilbert–Schmidt distance tends to a finite limit, which depends on the nature of the state (15 % of the maximal possible distance if the deviation from pure states is characterized by the ‘standard purity ’ Tr ρ̂2 and 37 % if the ‘superpurity ’ limr→0[Tr(ρ̂1+1/r)]r is chosen as a measure of deviation).
Abstract Majorization uncertainty relations are generalized for an arbitrary mixed quantum state ρ of a finite size N . In particular, a lower bound for the sum of two entropies characterizing the probability distributions corresponding to measurements with respect to two arbitrary orthogonal bases is derived in terms of the spectrum of ρ and the entries of a unitary matrix U relating both bases. The results obtained can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as an uncertainty relation for the sum of conditional entropies.