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 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.
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.
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.
We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most t samples. For sufficiently small ε, estimating an unknown state on Cd of rank at most r to trace norm error ε with constant success probability requires, and is achievable with, Θ(ε2drmax{1,tr}) samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most t samples improve the complexity of algorithms making single-sample measurements by at most a factor t. Further, measuring order r2 samples jointly is necessary and sufficient to attain the unrestricted collective rate.
For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on t samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.
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 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.
The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (positive operator valued measures) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.