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Physical Review A

Heisenberg uncertainty relation for mixed states

Shunlong Luo

Published Oct 20, 2005• 192 citations
Publisher

Abstract

The Heisenberg uncertainty relation sets a fundamental limit for quantum measurement of incompatible observables. Its standard form derived by Weyl and Robertson is of purely quantum nature when the state is pure. However, for mixed states, because the variance involving a mixed state is a hybrid of classical and quantum uncertainty, the conventional uncertainty relation is of a ``mixed'' flavor. It is desirable to seek some decomposition of variance into classical and quantum parts and to cast the Heisenberg uncertainty relation for mixed states in a more quantum form. By use of the skew information introduced by Wigner and Yanase in 1963, we make such an attempt and establish a different uncertainty relation which is stronger than the conventional one.

DOI
10.1103/physreva.72.042110

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The Heisenberg uncertainty relation sets a fundamental limit for quantum measurement of incompatible observables. Its standard form derived by Weyl and Robertson is of purely quantum nature when the state is pure. However, for mixed states, because the variance involving a mixed state is a hybrid of classical and quantum uncertainty, the conventional uncertainty relation is of a ``mixed'' flavor. It is desirable to seek some decomposition of variance into classical and quantum parts and to cast the Heisenberg uncertainty relation for mixed states in a more quantum form. By use of the skew information introduced by Wigner and Yanase in 1963, we make such an attempt and establish a different uncertainty relation which is stronger than the conventional one.
The variance of an observable in a quantum state is usually used to describe Heisenberg uncertainty relation. For mixed states, the variance includes quantum and classical uncertainties. By means of the skew information and the decomposition of the variance, a stronger uncertainty relation was presented by Luo [Phys. Rev. A 72, 042110 (2005)]. In this paper, by using Wigner-Yanase-Dyson information which is a generalization of the skew information, we propose a general uncertainty relation of mixed states.
In this paper, we show that skew information introduced by Wigner and Yanase, which is a natural informational extension of variance for pure states, can be interpreted as a measure of quantum uncertainty. By virtue of skew information, we establish a new uncertainty relation in the spirit of Schrodinger, which incorporates both incompatibility (encoded in the commutator) and correlations (encoded in a new correlation measure related to skew information) between observables, and moreover is stronger than the conventional ones.
Abstract Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases (MUBs), and general symmetric informationally complete positive operator valued measurements (SIC-POVMs), respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance, which are of hybrid nature (mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement (reference basis) such that the coherence of these states can be simultaneously small .
We propose an entanglement criterion, specially designed for mixed states, based on uncertainty relation and the Wigner-Yanase skew information. The variances in this uncertainty relation does not involve any classical mixing uncertainty, and thus turns out to be purely of quantum mechanical nature. We show that any mixed entangled state can be characterized by our criterion. We demonstrate its utility for several generalized mixed entangled state including Werner states and it turns out to be stronger than any other known criterion in identifying the correct domain of relevant parameters for entanglement. The proposed criterion reduces to the Schrodinger-Robertson inequality for pure states.
We report a refinement of Robertson-Schroedinger uncertainty relation via Wigner-Yanase skew information. Besides the well known quantum uncertainty arising from the noncommutativity of observables, there is classical uncertainty arising from the mixedness of the states that is quantified by the difference between the variance and the skew information. Our refined uncertainty relation for canonical observables is saturated by all the Gaussian states, pure or mixed, and thus provides an alternative measure for the non-Gaussianity of quantum states. Generalizations to the case of metric adjusted skew information are presented, unifying and refining most of previous results.
Abstract We propose an entanglement criterion, specially designed for mixed states, based on uncertainty relation and the Wigner–Yanase skew information. The variances in this uncertainty relation do not involve any classical mixing uncertainty, and is therefore purely of quantum mechanical nature. We show that a large class of mixed entangled states can be characterized by our criterion. We demonstrate its utility for several generalized mixed entangled state including two-qubit and two-qutrit Werner states and it turns out, for the states discussed in this paper, to be stronger than any other known criterion in identifying the correct domain of relevant parameters for entanglement. The relevant uncertainty relation reduces to the Schrodinger–Robertson inequality for pure states.