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arXiv:1507.00903 · Information and Inference A Journal of the IMA

Constrained quantum tomography of semi-algebraic sets with applications to low-rank matrix recovery

Michael Kech, Michael M. Wolf

Published Dec 26, 2016• 10 citations
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Abstract

We analyze quantum state tomography in scenarios where measurements and states are both constrained. States are assumed to live in a semi-algebraic subset of state space, and measurements are supposed to be rank-one positive operator-valued measures, possibly with additional constraints. Specifically, we consider sets of von Neumann measurements and sets of local observables. We provide upper bounds on the minimal number of measurement settings or outcomes that are required for discriminating all states within the given set. The bounds exploit tools from real-algebraic geometry and lead to generic results that do not only show the existence of good measurements, but guarantee that almost all measurements with the same dimension characteristic perform equally well. In particular, we show that on an |$n$|-dimensional Hilbert space any two states of a semi-algebraic subset can be discriminated by |$k$| generic von Neumann measurements if |$k(n-1)$| is larger than twice the dimension of the subset. In case the subset is given by states of rank at most |$r$|⁠, we show that |$k$| generic von Neumann measurements suffice to discriminate any two states provided that |$k(n-1)>4r(n-r)-2$|⁠. We obtain corresponding results for low-rank matrix recovery of hermitian matrices in the scenario where the linear measurement mapping is induced by tight frames.

DOI
10.1093/imaiai/iaw019

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This dissertation is concerned with quantum state tomography in scenarios where prior information constrains the set of relevant quantum states to a low dimensional subset of the state space. Frameworks to find both lower and upper bounds on the minimal number of measurement settings or outcomes needed to discriminate any two quantum states of such a subset are provided and applied to several concrete scenarios. Furthermore, measurement settings that allow a computationally tractable recovery of bounded rank quantum states are constructed.
Abstract. We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d − 4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log2(d) smaller than this upper bound. 1.
We present an algebraic algorithm for quantum state tomography that leverages measurements of certain observables to estimate structured entries of the underlying density matrix. Under low-rank assumptions, the remaining entries can be obtained solely using standard numerical linear algebra operations. The proposed algebraic matrix completion framework applies to a broad class of generic, low-rank mixed quantum states and, compared with state-of-the-art methods, is computationally efficient while providing deterministic recovery guarantees.
We investigate quantum tomography in scenarios where prior information restricts the state space to a smooth manifold of lower dimensionality. By considering stability we provide a general framework that relates the topology of the manifold to the minimal number of binary measurement settings that is necessary to discriminate any two states on the manifold. We apply these findings to cases where the subset of states under consideration is given by states with bounded rank, fixed spectrum, given unitary symmetry or taken from a unitary orbit. For all these cases we provide both upper and lower bounds on the minimal number of binary measurement settings necessary to discriminate any two states of these subsets.
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.
Recent work has shown that for one-dimensional quantum states that can be effectively approximated by matrix product operators (MPOs), a polynomial number of copies of the state suffices for reconstruction. Compared to MPOs in one dimension, projected entangled-pair states (PEPSs) and projected entangled-pair operators (PEPOs), which represent typical low-dimensional structures in two dimensions, are more prevalent as a looped tensor network. However, a formal analysis of the sample complexity required for estimating PEPS or PEPO has yet to be established. In this paper, we aim to address this gap by providing theoretical guarantees for the stable recovery of PEPS and PEPO. Our analysis primarily focuses on two quantum measurement schemes: (i)(i)(i) informationally complete positive operator valued measures (IC-POVMs), specifically the spherical ttt-designs (t≥3t \geq 3t≥3), and (ii)(ii)(ii) projective rank-one measurements, in particular Haar random projective measurements. We first establish stable embeddings for PEPSs (or PEPOs) to ensure that the information contained in the states can be preserved under these two measurement schemes. We then show that a constrained least-squares estimator achieves stable recovery for PEPSs (or PEPOs), with the recovery error bounded when the number of state copies scales linearly under spherical ttt-designs and polynomially under Haar-random projective measurements with respect to the number of qudits. These results provide theoretical support for the reliable use of PEPS and PEPO in practical quantum information processing.