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arXiv:quant-ph/0603166

Quantum process tomography: the role of initial correlations

Mário Ziman

Published Mar 19, 2006• 3 citations
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Abstract

We address the problem of quantum process tomography with the preparators producing states correlated with the environmental degrees of freedom that play role in the system-environment interactions. We discuss the physical situations, in which the dynamics is described by nonlinear, or noncompletely positive transformations. In particular, we show that arbitrary mapping $\varrho_{\rm in}\to\varrho_{\rm out}$ can be realized by using appropriate set of preparators and applying the unitary operation SWAP. The experimental ``realization'' of perfect NOT operation is presented. We address the problem of the verification of the compatibility of the preparator devices with the estimating process. The evolution map describing the dynamics in arbitrary time interval is known not to be completely positive, but still linear. The tomography and general properties of these maps are discussed.

DOI
10.48550/arxiv.quant-ph/0603166

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We address the problem of quantum process tomography with the preparators producing states correlated with the environmental degrees of freedom that play role in the system-environment interactions. We discuss the physical situations, in which the dynamics is described by nonlinear, or noncompletely positive transformations. In particular, we show that arbitrary mapping ϱin→ϱout\varrho_{\rm in}\to\varrho_{\rm out}ϱin​→ϱout​ can be realized by using appropriate set of preparators and applying the unitary operation SWAP. The experimental ``realization'' of perfect NOT operation is presented. We address the problem of the verification of the compatibility of the preparator devices with the estimating process. The evolution map describing the dynamics in arbitrary time interval is known not to be completely positive, but still linear. The tomography and general properties of these maps are discussed.
It was recently pointed out how differences in preparation procedures for quantum experiments can lead to nontrivial differences in the results of the experiment. The differences arise from the initial correlations between the system and the environment. Therefore, any quantum experiment that is prone to influences from the environment must be prepared carefully. We study quantum-process tomography in light of this. We suggest several experimental setups, where preparation of the initial state plays a role in the final outcome of the experiment. By studying the linearity and the positivity of the resulting maps, the experimenter can determine the nature of the initial correlations between the system and the environment.
We propose and evaluate experimentally an approach to quantum process tomography that completely removes the scaling problem plaguing the standard approach. The key to this simplification is the incorporation of prior knowledge of the class of physical interactions involved in generating the dynamics, which reduces the problem to one of parameter estimation. This allows part of the problem to be tackled using efficient convex methods, which, when coupled with a constraint on some parameters, allows globally optimal estimates for the Krauss operators to be determined from experimental data. Parameterizing the maps provides further advantages: it allows the incorporation of mixed states of the environment as well as some initial correlation between the system and environment, both of which are common physical situations following excitation of the system away from thermal equilibrium. Although the approach is not universal, in cases where it is valid it returns a complete set of positive maps for the dynamical evolution of a quantum system at all times.
We investigate the evolution of open quantum systems in the presence of initial correlations with an environment. Here the standard formalism of describing evolution by completely positive trace preserving (CPTP) quantum operations can fail and non-completely positive (non-CP) maps may be observed. A new classification of correlations between a system and environment using quantum discord is explored. However, we find quantum discord is not a symmetric quantity between exchange of systems and this leads to ambiguity in classifications - states which are both quantum and classically correlated depending on the order of the two systems. State preparation in quantum process tomography is investigated with regard to non-CP maps. In SQPT the preparation procedure can influence the complete-positivity of the reconstructed quantum operation if our system is initially correlated with an environment. We examine a recently proposed preparation procedures using projective measurements, and propose our own protocol that uses a single measurement followed by unitary rotations. The former can give rise to non-CP evolution while the later will always give rise to a CP map. State preparation in AAPT was found always to give rise to CP evolution. We examine the effect of statistical noise in process tomography and find it can result in the identification of a non-CP when the evolution should be CP. The variance of the distribution for reconstructed processes is found to be inversely proportional to the number of copies of a state used to perform tomography. Finally, we detail an experiment using currently available linear optics QC devices to demonstrate non-CP maps arising in SQPT.
In recent years there has been a significant development of the dynamical map formalism for initially correlated states of a system and its environment. Based on some of these results, we study quantum process tomography for initially correlated states of the system and the environment. This is beyond the usual assumption that the state of the system and the environment are initially uncorrelated. Since quantum process tomography is an experimental procedure, we wind up having to study the role of preparation of input states for open quantum experiments. We work out a theory for the general preparation procedure, and study two preparation procedures in detail. In specific, we study the stochastic preparation procedure and the projective preparation procedure and apply them to quantum process tomography. The two preparation procedures describe the ways to uncorrelate.....
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