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arXiv:0904.4663 · Physical Review A

Role of preparation in quantum process tomography

Kavan Kishore Modi, E. C. G. Sudarshan

Published May 27, 2010• 44 citations
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Abstract

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.

DOI
10.1103/physreva.81.052119

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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.
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.....
We study the effects of the preparation of input states in a quantum tomography experiment. We show that maps arising from a quantum process tomography experiment (called process maps) differ from the well-known dynamical maps. The difference between the two is due to the preparation procedure that is necessary for any quantum experiment. We study two preparation procedures: stochastic preparation and preparation by measurements. The stochastic preparation procedure yields process maps that are linear, while the preparations using von Neumann measurements lead to nonlinear processes and can only be consistently described by a bilinear process map. A process tomography recipe is derived for preparation by measurement for qubits. The difference between the two methods is analyzed in terms of a quantum process tomography experiment. A verification protocol is proposed to differentiate between linear processes and bilinear processes. We also emphasize that the preparation procedure will have a nontrivial effect for any quantum experiment in which the system of interest interacts with its environment.
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.
Quantum systems are never perfectly isolated; they will generally be correlated with some environment before being initialized for an experiment. As a consequence, the choice of initial system preparation can drastically influence the subsequent evolution, which may yield seemingly unphysical results. Previous work showed that some information about the initial system-environment state could be extracted from system measurements. However, the scope and operational significance of these results is limited.
For a quantum system interacting with its environment, the role of state preparation is nontrivial. The reason is that before the state preparation procedure, the system and the environment are correlated. Consequently, the state preparation procedure (which acts on the system) indirectly influences the state of the environment depending on the state preparation. In this paper, we use an experimentally realizable model describing a collection of N two-level atoms coupled to a common environment to investigate the influence of the state preparation procedure. We show that the dynamical map describing the evolution of the open quantum system can depend appreciably on the state preparation procedure. Moreover, this effect can be enhanced by increasing N. Our results should be useful for quantum control and quantum tomography.
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.