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arXiv:1403.0673

Density matrix and fidelity estimation of multiphoton entanglement via phaselift

Lu, Yiping, Qing Bo Zhao

Published Mar 4, 2014• 0 citations
arXivHTMLPDF

Abstract

The experiments of multi-photon entanglements have been made by some groups, including Pan's group (Ref.[2],[3],[5]). Obviously, the increase number of the photon would cause a dramatically increase in the dimension of the measurement matrix, which result in a great consumption of time in the measurements. From a practical view, we wish to gain the most information through as little measurements as possible for the multi-photon entanglements. The low rank matrix recovery (LRMR) provides such a possibility to resolve all the issues of the measurement matrix based on less data. In this paper, we would like to verify that whether the LRMR works for six qubits and eight photons in comparison to the data given by Pan's group, i.e. we input a fraction of the data to calculate all of others. Through exploring their density matrix, fidelity and visibility, we find that the results remain consistent with the data provided by Pan's group, which allows us to confirm that the LRMR can simplify experimental measure- ments for more photons. In particular, we find that very limited data would also give excellent support to the experiment for fidelity when low rank, pure state, sparse or position information are utilized. Our analytical calculations confirm that LRMR would generalize to multi-photon state entanglement.

DOI
10.48550/arxiv.1403.0673

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The experiments of multi-photon entanglements have been made by some groups, including Pan's group (Ref.[2],[3],[5]). Obviously, the increase number of the photon would cause a dramatically increase in the dimension of the measurement matrix, which result in a great consumption of time in the measurements. From a practical view, we wish to gain the most information through as little measurements as possible for the multi-photon entanglements. The low rank matrix recovery (LRMR) provides such a possibility to resolve all the issues of the measurement matrix based on less data. In this paper, we would like to verify that whether the LRMR works for six qubits and eight photons in comparison to the data given by Pan's group, i.e. we input a fraction of the data to calculate all of others. Through exploring their density matrix, fidelity and visibility, we find that the results remain consistent with the data provided by Pan's group, which allows us to confirm that the LRMR can simplify experimental measure- ments for more photons. In particular, we find that very limited data would also give excellent support to the experiment for fidelity when low rank, pure state, sparse or position information are utilized. Our analytical calculations confirm that LRMR would generalize to multi-photon state entanglement.
Multi-photon entanglement has been successfully made by experimental groups. As the increase of photon number, several problems are encountered, say, greater number of copies, longer time, the error of fidelity and so on. In this paper, we present a new scheme based on Lagrange multiplier and feedback to save the measure copies in multi-photon experiment and five percent of measuring time, also guarantee the acceptable error of fidelity. All the results have been supported by the data of eight photon experiment. Furthermore, same approach is applied in the simulation for ten photon entanglement, and 22.45 percent of copies are saved, optimized copy distribution gives better estimation of fidelity than the average copy distribution.
Multi-photon system has been studied by many groups. The challenge facing by most groups is the limited samples are far from enough for detecting quantum entanglement. The problem is even more serious for the quantum state tomography. Adaptive quantum state tomography is one possible way to solve the problem. We develop a distribution scheme of samples and extend it into three step to improve the performance of adaptive quantum state tomography. Mean Square Error of reconstructed density matrix is improved for several states being tested numerically. In addition, Phaselift is applied to save the storage space of measurement operator.
Quantum multiphoton interferometry has now reached the six-photon stage. Thus far, the observed fidelities of entangled states never reached 2/3. We report a high fidelity (estimated at 88%) experiment in which six-qubit singlet correlations were observed. With such a high fidelity we are able to demonstrate the central property of these "singlet" correlations, their "rotational invariance," by performing a full set of measurements in three complementary polarization bases. The patterns are almost indistinguishable. The data reveal genuine six-photon entanglement. We also study several five-photon states, which result upon detection of one of the photons. Multiphoton singlet states survive some types of depolarization and are thus important in quantum communication schemes.
Recent insights concerning the PhaseLift algorithm for retrieving phases have furthered our understanding of low rank matrix recovery from rank-one projective measurements. Motivated by the structure of certain quantum mechanical experiments, we introduce a particular class of such rank-one measurements: orthonormal basis measurements. One such measurement corresponds to choosing an orthonormal basis and treating all the rank-one projectors onto different basis elements as a series of consecutive measurement matrices. We elaborate on performing low-rank matrix recovery from few, sufficiently random orthonormal basis measurements and sketch applications of such a procedure in quantum physics. We conclude this article by presenting numerical experiments testing such an approach.
Many experiments in quantum information aim at creating multi-partite entangled states. Quantifying the amount of entanglement that was actually generated can, in principle, be accomplished using full-state tomography. This method requires the determination of a parameter set that is growing exponentially with the number of qubits and becomes infeasible even for moderate numbers of particles. Non-trivial bounds on experimentally prepared entanglement can however be obtained from partial information on the density matrix. As introduced in the paper of K.M.R. Audenaert and M.B. Plenio [New J. Phys. 2006, 8, 266], the fundamental question is then formulated as: What is the entanglement content of the least entangled quantum state that is compatible with the available measurement data? We formulate the problem mathematically [Audenaert, K.M.R.; Plenio, M.B. New J. Phys. 2006, 8, 266] employing methods from the theory of semi-definite programming and then address this problem for the case, where the goal of the experiment is the creation of graph states. The observables that we consider are the generators of the stabilizer group, thus the number of measurement settings grows only linearly in the number of qubits. We provide analytical solutions as well as numerical methods that may be applied directly to experiments, and compare the obtained bounds with results from full-state tomography for simulated data.
Quantum state tomography is a fundamental technique for estimating the state of a quantum system from measured data and plays a crucial role in evaluating the performance of quantum devices. However, standard estimation methods become computationally prohibitive as the system size increases due to the exponential growth of the density matrix, describing a quantum state, with the number of qubits. We propose a low-rank tensor-network framework for mixed-state quantum state tomography based on a block tensor train (Block-TT) factorization. Specifically, the density matrix is represented as the contraction of a Block-TT with its Hermitian transpose, yielding a TT analogue of the Burer-Monteiro factorization. This parameterization guarantees Hermiticity and positive semidefiniteness by construction while compressing the number of optimization variables from exponential to linear in the number of qubits. Building on this representation, we develop single-site and two-site density matrix renormalization group (DMRG) algorithms for estimating quantum states from compressed measurements. The resulting methods operate directly on the compressed parameterization, support adaptive rank refinement, and exploit efficient tensor-network contractions for expectation-value evaluation. The framework is applicable to a broad class of low-rank quantum states, including pure states, nearly pure states, and ground states that admit accurate tensor-network approximations. Numerical experiments demonstrate accurate state reconstruction from limited measurements together with substantial reductions in memory requirements and computational cost compared with conventional low-rank tomography methods.