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Quantum Process Estimation with Unknown Measurements

Brian J. Smith, Merlin Cooper, Michał Karpiński

Published Jan 1, 2015• 0 citations
Publisher

Abstract

Quantum process estimation without assumptions about measurements used is presented. The detector response is characterized with a finite set of probe states. Numerical simulations for quantum optical processes demonstrate the technique for complex systems.
DOI
10.1364/cleo_qels.2015.ftu4a.3

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Identifying an unknown unitary quantum process is a fundamental task in the development of quantum technology. In this paper, we propose an efficient identification algorithm for estimating unitary processes. In the method, input pure states are used and a fast quantum state tomography algorithm is developed to reconstruct the output states. Then the information of the output states is used to estimate the unitary process. The identification algorithm has computational complexity O(d3) for a d-dimensional system. Numerical results show that the proposed identification algorithm is much more efficient than the maximum likelihood estimation method and works well for input mixed states with high purity. An analytical upper bound for the identification error is also provided, and numerical simulations and experimental results on quantum optical systems verify the theoretical results.
Characterization of quantum processes is a preliminary step necessary in the development of quantum technology. The conventional method uses standard quantum process tomography, which requires d2{d}^{2}d2 input states and d4{d}^{4}d4 quantum measurements for a ddd-dimensional Hilbert space. These experimental requirements are compounded by the complexity of processing the collected data, which can take several orders of magnitude longer than the experiment itself. In this paper we propose an alternative self-guided algorithm for quantum process tomography, tuned for the task of finding an unknown unitary process. Our algorithm is a fully automated and adaptive process characterization technique. The advantages of our algorithm are the following: it has an inherent robustness to both statistical and technical noise; it requires less space and time since there is no postprocessing of the data; it requires only a single input state and measurement; and it provides on-the-fly diagnostic information while the experiment is running. Numerical results show our algorithm achieves the same 1/n1/n1/n scaling as standard quantum process tomography when nnn uses of the unknown process are used. We also present experimental results wherein the algorithm and its advantages are realized for the task of finding an element of SU(2).
The subject of this thesis, quantum estimation theory, is inspired by recent developments, both practical and theoretical, in the new field of quantum information and computation. The main task of quantum estimation theory, the subject of this thesis, is that given many identical copies of an unknown quantum system one wants to guess or estimate what the system is. This is done by performing an measurement and analyzing the data obtained from it. A quantum state is represented by a density matrix. In some cases one does not need to know the whole density matrix, sometimes a partial characterization is enough. Chapter 2 of this thesis is about optimally estimating the spectrum, i.e. the eigenvalues, of a density matrix. This is useful, for example, for estimating the entanglement of a bipartite pure state. The results of chapter 2 tell us that one can optimally estimate bipartite entanglement with LOCC measurements on one of the two parties. Unitary operations transform density matrices to density matrices. They are the gates of quantum information processing. Suppose one has one such unitary operation but one does not know what it does. In order to characterize it one needs to know how it transforms any input state. In order to know how a particular state is transformed, one can prepare that state, feed it to the unitary many times, and measure the output. One state is, of course, not enough. It turns out that the unitary can be completely characterized with a finite set of states, this is known as quantum process tomography. Unfortunately, in many cases it is difficult to have available in the lab all the input states needed to perform quantum process tomography. It is here where entanglement comes to the rescue. One pair of maximally entangled qubits, where one of them is used as input for the unitary gate and the other is left untouched, is sufficient. All the information about the unknown gate is contained in the joint output state. This is the problem treated in chapter 3. Now the output state needs to be measured. Again, one could measure the two qubits separately (LOCC) or jointly (collective measurement). Is there anything to gain by performing joint measurements? The answer is yes, optimal collective measurements are at least 3 times more accurate than any LOCC measurement. Chapter 4 treats the problem of estimating commuting unitary gates. There it is shown that this extra information makes it unnecessary to use an entangled input state and to use collective measurements at the output. In chapters 3 and 4 it is assumed that only one copy of the unitary gate is available. If, however, more than one copy (say N ) are available at the same time one could use an N-fold entangled state as input. If no entanglement is used, classical statistics tells us that the error of the estimation is proportional to 1/√N. In the literature, it has been shown that by using an entangled state in the input one can achieve an error proportional to 1/N. However, this has been shown only for qubit gates. In chapter 5 important steps are taken in the direction of showing the same thing for a general unitary gate, but this remains an open problem.
Improved measurement techniques are central to technological development and foundational scientific exploration. Quantum physics relies on detectors sensitive to non-classical features of systems, enabling precise tests of physical laws and quantum-enhanced technologies including precision measurement and secure communications. Accurate detector response calibration for quantum-scale inputs is key to future research and development in these cognate areas. To address this requirement, quantum detector tomography has been recently introduced. However, this technique becomes increasingly challenging as the complexity of the detector response and input space grow in a number of measurement outcomes and required probe states, leading to further demands on experiments and data analysis. Here we present an experimental implementation of a versatile, alternative characterization technique to address many-outcome quantum detectors that limits the input calibration region and does not involve numerical post processing. To demonstrate the applicability of this approach, the calibrated detector is subsequently used to estimate non-classical photon number states.