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Register a new userPurity and Gaussianity bounded uncertainty relation
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Added by
Aikaterini Mandilara
Publication date
2009
fields
Physics
and research's language is
English
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Bounded uncertainty relations provide the minimum value of the uncertainty assuming some additional information on the state. We derive analytically an uncertainty relation bounded by a pair of constraints, those of purity and Gaussianity. In a limiting case this uncertainty relation reproduces the purity-bounded derived by V I Manko and V V Dodonov and the Gaussianity-bounded one [Phys. Rev. A 86, 030102R (2012)].
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We address truncated states of continuous variable systems and analyze their statistical properties numerically by generating random states in finite-dimensional Hilbert spaces. In particular, we focus to the distribution of purity and non-Gaussianity for dimension up to d=21. We found that both quantities are distributed around typical values with variances that decrease for increasing dimension. Approximate formulas for typical purity and non-Gaussianity as a function of the dimension are derived.
We experimentally verify uncertainty relations for mixed states in the tomographic representation by measuring the radiation field tomograms, i.e. homodyne distributions. Thermal states of single-mode radiation field are discussed in details as paradigm of mixed quantum state. By considering the connection between generalised uncertainty relations and optical tomograms is seen that the purity of the states can be retrieved by statistical analysis of the homodyne data. The purity parameter assumes a relevant role in quantum information where the effective fidelities of protocols depend critically on the purity of the information carrier states. In this contest the homodyne detector becomes an easy to handle purity-meter for the state on-line with a running quantum information protocol.
We suggest an improved version of Robertson-Schrodinger uncertainty relation for canonically conjugate variables by taking into account a pair of characteristics of states: non-Gaussianity and mixedness quantified by using fidelity and entropy, respectively. This relation is saturated by both Gaussian and Fock states, and provides strictly improved bound for any non-Gaussian states or mixed states. For the case of Gaussian states, it is reduced to the entropy-bounded uncertainty relation derived by Dodonov. Furthermore, we consider readily computable measures of both characteristics, and find weaker but more readily accessible bound. With its generalization to the case of two-mode states, we show applicability of the relation to detect entanglement of non-Gaussian states.
Local pure states are an important resource for quantum computing. The problem of distilling local pure states from mixed ones can be cast in an information theoretic paradigm. The bipartite version of this problem where local purity must be distilled from an arbitrary quantum state shared between two parties, Alice and Bob, is closely related to the problem of separating quantum and classical correlations in the state and in particular, to a measure of classical correlations called the one-way distillable common randomness. In Phys. Rev. A 71, 062303 (2005), the optimal rate of local purity distillation is derived when many copies of a bipartite quantum state are shared between Alice and Bob, and the parties are allowed unlimited use of a unidirectional dephasing channel. In the present paper, we extend this result to the setting in which the use of the channel is bounded. We demonstrate that in the case of a classical-quantum system, the expression for the local purity distilled is efficiently computable and provide examples with their tradeoff curves.
Quantum uncertainty relations are formulated in terms of relative entropy between distributions of measurement outcomes and suitable reference distributions with maximum entropy. This type of entropic uncertainty relation can be applied directly to observables with either discrete or continuous spectra. We find that a sum of relative entropies is bounded from above in a nontrivial way, which we illustrate with some examples.
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