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arXiv:1509.09318 · Open Systems & Information Dynamics

Dynamic Quantum Tomography Model for Phase-Damping Channels

Artur Czerwiński, Andrzej Jamiołkowski

Published Dec 1, 2016• 6 citations
arXivHTMLPDF

Abstract

In this paper we propose a dynamic quantum tomography model for open quantum systems with evolution given by phase-damping channels. Mathematically, these channels correspond to completely positive trace-preserving maps defined by the Hadamard product of the initial density matrix with a time-dependent matrix which carries the knowledge about the evolution. Physically, there is a strong motivation for considering this kind of evolution because such channels appear naturally in the theory of open quantum systems. The main idea behind a dynamic approach to quantum tomography claims that by performing the same kind of measurement at some time instants one can obtain new data for state reconstruction. Thus, this approach leads to a decrease in the number of distinct observables which are required for quantum tomography; however, the exact benefit for employing the dynamic approach depends strictly on how the quantum system evolves in time. Algebraic analysis of phase-damping channels allows one to determine criteria for quantum tomography of systems in question. General theorems and observations presented in the paper are accompanied by a specific example, which shows step by step how the theory works. The results introduced in this paper can potentially be applied in experiments where there is a tendency to look at quantum tomography from the point of view of economy of measurements, because each distinct kind of measurement requires, in general, preparing a separate setup.

DOI
10.1142/s1230161216500190

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In this paper we propose a dynamic quantum tomography model for open quantum systems with evolution given by phase-damping channels. Mathematically, these channels correspond to completely positive trace-preserving maps defined by the Hadamard product of the initial density matrix with a time-dependent matrix which carries the knowledge about the evolution. Physically, there is a strong motivation for considering this kind of evolution because such channels appear naturally in the theory of open quantum systems. The main idea behind a dynamic approach to quantum tomography claims that by performing the same kind of measurement at some time instants one can obtain new data for state reconstruction. Thus, this approach leads to a decrease in the number of distinct observables which are required for quantum tomography; however, the exact benefit for employing the dynamic approach depends strictly on how the quantum system evolves in time. Algebraic analysis of phase-damping channels allows one to determine criteria for quantum tomography of systems in question. General theorems and observations presented in the paper are accompanied by a specific example, which shows step by step how the theory works. The results introduced in this paper can potentially be applied in experiments where there is a tendency to look at quantum tomography from the point of view of economy of measurements, because each distinct kind of measurement requires, in general, preparing a separate setup.
Abstract The article introduces efficient quantum state tomography schemes for qutrits and entangled qubits subject to pure decoherence. We implement the dynamic state reconstruction method for open systems sent through phase-damping channels, which was proposed in: Czerwinski and Jamiolkowski Open Syst. Inf. Dyn.23, 1650019 (2016). In the present article we prove that two distinct observables measured at four different time instants suffice to reconstruct the initial density matrix of a qutrit with evolution given by a phase-damping channel. Furthermore, we generalize the approach in order to determine criteria for quantum tomography of entangled qubits. Finally, we prove two universal theorems concerning the number of observables required for quantum state tomography of qudits subject to pure decoherence. We believe that dynamic state reconstruction schemes bring advancement and novelty to quantum tomography since they utilize the Heisenberg representation and allow to define the measurements in time domain.
Abstract The research presented in this article concerns the stroboscopic approach to quantum tomography, which is an area of science where quantum physics and linear algebra overlap. In this article we introduce the algebraic structure of the parametric-dependent quantum channels for 2-level and 3-level systems such that the generator of evolution corresponding with the Kraus operators has no degenerate eigenvalues. In such cases the index of cyclicity of the generator is equal to 1, which physically means that there exists one observable the measurement of which performed a sufficient number of times at distinct instants provides enough data to reconstruct the initial density matrix and, consequently, the trajectory of the state. The necessary conditions for the parameters and relations between them are introduced. The results presented in this paper seem to have considerable potential applications in experiments due to the fact that one can perform quantum tomography by conducting only one kind of measurement. Therefore, the analyzed evolution models can be considered optimal in the context of quantum tomography. Finally, we introduce some remarks concerning optimal evolution models in the case of n -dimensional Hilbert space.
In this article we propose a dynamic approach to complex vector reconstruction in the context of quantum tomography. There are two underlying assumptions behind our reasoning. The first one claims that the evolution of a d-level pure quantum system is given by the Schrödinger equation with a time-independent Hamiltonian and the other states that the knowledge about the quantum state is provided from projective measurements, also called intensity measurements. The problem of quantum state reconstruction is connected with the notion known as phase retrieval -- recovering a complex vector from modulus of inner product with frame vectors. Phase retrieval is widely studied in many areas of science but still there is a number of problems that remain to be answered. We believe that the dynamic approach can significantly improve the effectiveness of the vector reconstruction as it aims to decrease the number of distinct projectors by taking advantage of the knowledge about the evolution. General conditions and observations are applied to a specific unitary evolution model.
Dynamical maps are the principal subject of the open system theory. Formally, the dynamical map of a given open quantum system is a density matrix transformation that takes any initial state and sends it to the state at a later time. Physically, it encapsulates the system’s evolution due to coupling with its environment. Hence, the theory provides a flexible and accurate framework for computing expectation values of open system observables. However, expectation values-or more generally, single-time correlation functions-capture only the simplest aspects of a quantum system’s dynamics. A complete characterization of the dynamics requires access to multi-time correlation functions as well: phenomena like detailed balance, linear and non-linear response, non-equilibrium transport in general, or even sequential measurements of system observables are all described in terms of multi-time correlations. For closed systems, such correlations are well-defined, even though knowledge of the system’s state alone is insufficient to determine them fully. In contrast, the standard dynamical map formalism for open systems does not account for multi-time correlations, as it is fundamentally limited to describing state evolution. Here, we extend the scope of open quantum system theory by developing a systematic perturbation theory for computing multi-time correlation functions.
Abstract Quantum process tomography conventionally uses a multitude of initial quantum states and then performs state tomography on the process output. Here we propose and study an alternative approach which requires only a single (or few) known initial states together with time-delayed measurements for reconstructing the unitary map and corresponding Hamiltonian of the time dynamics. The overarching mathematical framework and feasibility guarantee of our method is provided by the Takens embedding theorem. We explain in detail how the reconstruction of a single-qubit Hamiltonian works in this setting and provide numerical methods and experiments for general few-qubit and lattice systems with local interactions. In particular, the method allows to find the Hamiltonian of a two qubit system by observing only one of the qubits.
We consider quantum state tomography with measurement procedures of the following type: First, we subject the quantum state we aim to identify to a known time evolution for a desired period of time. Afterwards we perform a measurement with a fixed measurement setup. This procedure can then be repeated for other periods of time, the measurement setup however remains unaltered. Given an n-dimensional system with suitable unitary dynamics, we show that any two states can be discriminated by performing a measurement with a setup that has n outcomes at n + 1 points in time. Furthermore, we consider scenarios where prior information restricts the set of states to a subset of lower dimensionality. Given an n-dimensional system with suitable unitary dynamics and a semi-algebraic subset R of its state space, we show that any two states of the subset can be discriminated by performing a measurement with a setup that has n outcomes at l steps of the time evolution if (n−1)l≥2dimR. In addition, by going beyond unitary dynamics, we show that one can in fact reduce to a setup with the minimal number of two outcomes.