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arXiv:1103.2803

Quantum state tomography in the presence of an unknown heat bath

Jochen Rau

Published Mar 14, 2011• 0 citations
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Abstract

In quantum state tomography, one potential source of error is uncontrolled contact of the system with a heat bath whose detailed properties are not known, and whose impact on the system moreover varies between different runs of the experiment. Precisely these variations provide a handle for reconstructing the system's effective relaxation dynamics. I propose a pertinent estimation scheme which is based on a steepest-descent ansatz and maximum likelihood. After reconstructing the relaxation dynamics, the original quantum state of the system can be constrained to a curve in state space.

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In quantum state tomography, one potential source of error is uncontrolled contact of the system with a heat bath whose detailed properties are not known, and whose impact on the system moreover varies between different runs of the experiment. Precisely these variations provide a handle for reconstructing the system's effective relaxation dynamics. I propose a pertinent estimation scheme which is based on a steepest-descent ansatz and maximum likelihood. After reconstructing the relaxation dynamics, the original quantum state of the system can be constrained to a curve in state space.
The principle of maximum likelihood reconstruction has proven to yield satisfactory results in the context of quantum state tomography for many-body systems of moderate system sizes. Until recently, however, quantum state tomography has been considered to be infeasible for systems consisting of a large number of subsystems due to the exponential growth of the Hilbert space dimension with the number of constituents. Several reconstruction schemes have been proposed since then to overcome the two main obstacles in quantum many-body tomography: experiment time and post-processing resources. Here we discuss one strategy to address these limitations for the maximum likelihood principle by adopting a particular state representation to merge a well established reconstruction algorithm maximizing the likelihood with techniques known from quantum many-body theory.
A simple yet efficient state reconstruction algorithm of linear regression estimation (LRE) is presented for quantum state tomography. In this method, quantum state reconstruction is converted into a parameter estimation problem of a linear regression model and the least-squares method is employed to estimate the unknown parameters. An asymptotic mean squared error (MSE) upper bound for all possible states to be estimated is given analytically, which depends explicitly upon the involved measurement bases. This analytical MSE upper bound can guide one to choose optimal measurement sets. The computational complexity of LRE is O(d(4)) where d is the dimension of the quantum state. Numerical examples show that LRE is much faster than maximum-likelihood estimation for quantum state tomography.
Quantum state tomography aims at reconstructing the state of a quantum system. The standard approach is based on a tomographically complete set of observables and a maximum likelihood estimation of the density matrix. However, the number of measurements scales exponentially with the number of qubits or qudits of the system. By leveraging some properties of any physical density matrix, we propose a method in which the number of necessary measurements depends on the state under investigation but without the need for any a priori information. In some cases, this can lead to a significant reduction of the resources necessary for state characterization.
Abstract Modern day quantum simulators can prepare a wide variety of quantum states but the accurate estimation of observables from tomographic measurement data often poses a challenge. We tackle this problem by developing a quantum state tomography scheme which relies on approximating the probability distribution over the outcomes of an informationally complete measurement in a variational manifold represented by a convolutional neural network. We show an excellent representability of prototypical ground- and steady states with this ansatz using a number of variational parameters that scales polynomially in system size. This compressed representation allows us to reconstruct states with high classical fidelities outperforming standard methods such as maximum likelihood estimation. Furthermore, it achieves a reduction of the estimation error of observables by up to an order of magnitude compared to their direct estimation from experimental data.
Quantum state tomography is the experimental procedure of determining an unknown state. It is not only essential for the verification of resources and processors of quantum information but is also important in its own right with regard to the foundation of quantum mechanics. Standard methods have been elusive for large systems because of the enormous number of observables to be measured and the exponential complexity of data post-processing. Here, we propose a new scheme of quantum state tomography that requires the measurement of only three observables (acting jointly on the system and pointer) regardless of the size of the system. The system is coupled to a "pointer" of single qubit, and the wavefunction of the system is "reaped" onto the pointer upon the measurement of the system. Subsequently, standard two-state tomography on the pointer and classical post-processing are used to reconstruct the quantum state of the system. We also developed an efficient and scalable iterative maximum likelihood algorithm to estimate states from statistically incomplete data.
Modern day simulators can prepare a wide variety of states but extracting observables from the resulting quantum data often poses a challenge. We tackle this problem by developing a state tomography scheme which relies on approximating the probability distribution over the outcomes of an informationally complete measurement in a variational manifold represented by a convolutional neural network. We show an excellent representability of prototypical ground- and steady states with this ansatz using a number of variational parameters that scales polynomially in system size. This compressed representation allows us to reconstruct states with high classical fidelities outperforming standard methods such as maximum likelihood estimation. Furthermore, it achieves a reduction of the root mean square errors of observables by up to an order of magnitude compared to their direct estimation from experimental data.