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International Conference on Computer Science and Service System

Entanglement condition using Schrodinger-Robertson uncertainty relation

Zhongsheng Pu, Kangkang Fan, Guanqiang Song, Peng Yang

Published Jun 1, 2011• 0 citations
Publisher

Abstract

Compared with the Heisenberg uncertainty relation, Schrodinger-Robertson uncertain relation generally provides stronger bound on product of uncertainties for two noncommutation observables, so the stricter separability condition can be provided by using Schrodinger-Robertson uncertainty relation. In this paper, using the Schrodinger Robertson uncertainty relation, the separability condition previously proposed by L. M. Duan is improved into the stricter separability condition. Then we extend the above ideas to the general two-mode situation and multi-mode terms, some new separation conditions are obtained.

DOI
10.1109/csss.2011.5974886

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Compared with the Heisenberg uncertainty relation, Schrodinger-Robertson uncertain relation generally provides stronger bound on product of uncertainties for two noncommutation observables, so the stricter separability condition can be provided by using Schrodinger-Robertson uncertainty relation. In this paper, using the Schrodinger Robertson uncertainty relation, the separability condition previously proposed by L. M. Duan is improved into the stricter separability condition. Then we extend the above ideas to the general two-mode situation and multi-mode terms, some new separation conditions are obtained.
The Schr\"odinger-Robertson inequality generally provides a stronger bound on the product of uncertainties for two noncommuting observables than the Heisenberg uncertainty relation, and as such it can yield a stricter separability condition in conjunction with partial transposition. In this paper, using the Schr\"odinger-Robertson uncertainty relation, the separability condition previously derived from the su(2) and su(1,1) algebra is made stricter and refined to a form invariant with respect to local phase shifts. Furthermore, a linear optical scheme is proposed to test this invariant separability condition.
Abstract We report a universal improvement to the standard Robertson–Schrödinger uncertainty relation. Our result shows that the Robertson–Schrödinger lower bound can be supplemented by a new noncommutativity-induced term. This term represents a previously overlooked quantum contribution and becomes more pronounced as the state becomes more mixed. Moreover, it is expressed as the expectation value of a positive observable, namely the squared modulus of the commutator, and therefore preserves the direct, experimentally accessible character of the Robertson–Schrödinger relation. For two-level quantum systems, our relation becomes an exact equality for any state and any pair of observables, thereby ensuring the tightness of the bound in the strongest possible sense. The relation also yields, as a corollary, a complete proof of a general uncertainty bound that had previously been supported only by numerical evidence.
The Heisenberg-Robertson uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial; i.e., the lower bound can be null even for two noncompatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.
Heisenberg-Robertson's uncertainty relation expresses a limitation in the possible preparations of the system by giving a lower bound to the product of the variances of two observables in terms of their commutator. Notably, it does not capture the concept of incompatible observables because it can be trivial, i.e., the lower bound can be null even for two non-compatible observables. Here we give two stronger uncertainty relations, relating to the sum of variances, whose lower bound is guaranteed to be nontrivial whenever the two observables are incompatible on the state of the system.