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arXiv:1410.5674

Estimating the eigenstates of an unknown density operator without damaging it

Jingliang Gao, Feng Cai

Published Oct 21, 2014• 0 citations
arXivHTMLPDF

Abstract

Given n qubits prepared according to the same unknown density operator, we propose a nondestructive measuring method which approximately yields the eigenstates. It is shown that, for any plane which passes through the center point of the Bloch sphere, there exists corresponding projective measurement. By performing these measurements, we can scan the whole Bloch sphere like radar to search for the orientation of and determine the eigenstates. We show the convergency of the measurements. This result actually reveals a mathematical structure of the n-fold Hilbert space .

DOI
10.48550/arxiv.1410.5674

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Given n qubits prepared according to the same unknown density operator, we propose a nondestructive measuring method which approximately yields the eigenstates. It is shown that, for any plane which passes through the center point of the Bloch sphere, there exists corresponding projective measurement. By performing these measurements, we can scan the whole Bloch sphere like radar to search for the orientation of and determine the eigenstates. We show the convergency of the measurements. This result actually reveals a mathematical structure of the n-fold Hilbert space .
Given N quantum systems prepared according to the same density operator $\ensuremath{\rho},$ we propose a measurement on the N-fold system that approximately yields the spectrum of $\ensuremath{\rho}.$ The projections of the proposed observable decompose the Hilbert space according to the irreducible representations of the permutations on N points, and are labeled by Young frames, whose relative row lengths estimate the eigenvalues of $\ensuremath{\rho}$ in decreasing order. We show convergence of these estimates in the limit $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty},$ and that the probability for errors decreases exponentially with a rate we compute explicitly.
We obtain the optimal scheme for estimating unknown qubit mixed states when an arbitrary number NNN of identically prepared copies is available. We discuss the case of states in the whole Bloch sphere as well as the restricted situation where these states are known to lie on the equatorial plane. For the former case we obtain that the optimal measurement does not depend on the prior probability distribution provided it is isotropic. Although the equatorial-plane case does not have this property for arbitrary NNN, we give a prior-independent scheme which becomes optimal in the asymptotic limit of large NNN. We compute the maximum mean fidelity in this asymptotic regime for the two cases. We show that within the pointwise estimation approach these limits can be obtained in a rather easy and rapid way. This derivation is based on heuristic arguments that are made rigorous by using van Trees inequalities. The interrelation between the estimation of the purity and the direction of the state is also discussed. In the general case we show that they correspond to independent estimations whereas for the equatorial-plane states this is only true asymptotically.
In the context of measurement-based quantum computation a way of maintaining the coherence of a graph state is to measure its stabilizer operators. Aside from performing quantum error correction, it is possible to exploit the information gained from these measurements to characterize and then counteract a coherent source of errors; that is, to determine all the parameters of an error channel that applies a fixed---but unknown---unitary operation to the physical qubits. Such a channel is generated, e.g., by local stray fields that act on the qubits. We study the case in which each qubit of a given graph state may see a different error channel and we focus on channels given by a rotation on the Bloch sphere around either the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbit{x}}$, the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbit{y}}$, or the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\mathbit{z}}$ axis, for which analytical results can be given in a compact form. The possibility of reconstructing the channels at all qubits depends nontrivially on the topology of the graph state. We prove via perturbation methods that the reconstruction process is robust and supplement the analytic results with numerical evidence.
Abstract We study an analogous Bloch sphere representation of higher-level quantum systems using the Heisenberg-Weyl operator basis. We introduce a parametrization method that will allow us to identify a real-valued Bloch vector for an arbitrary density operator. Before going into arbitrary d-level (d ≥ 3) quantum systems (qudits), we start our analysis with three-level ones (qutrits). It is well known that we need at least eight real parameters in the Bloch vector to describe arbitrary three-level quantum systems (qutrits). However, using our method we can divide these parameters into four weight, and four angular parameters, and find that the weight parameters are inducing a unit sphere in four-dimension. And, the four angular parameters determine whether a Bloch vector is physical. Therefore, unlike its qubit counterpart, the qutrit Bloch sphere does not exhibit a solid structure. Importantly, this construction allows us to define different properties of qutrits in terms of Bloch vector components. We also examine the two and three-dimensional sections of the sphere, which reveal a non-convex yet closed structure for physical qutrit states. Further, we apply our representation to derive mutually unbiased bases (MUBs), characterize unital maps for qutrits, and assess ensembles using the Hilbert-Schmidt and Bures metrics. Moreover, we extend this construction to qudits, showcasing its potential applicability beyond the qutrit scenario.