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arXiv:quant-ph/0102027 · Physical Review A

Estimating the spectrum of a density operator

Michael Keyl, Reinhard F. Werner

Published Oct 15, 2001• 168 citations
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Abstract

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.

DOI
10.1103/physreva.64.052311

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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 .
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It has recently been established that, in a non-demolition measurement of an observable N\mathcal{N}N with a finite point spectrum, the density matrix of the system approaches an eigenstate of N\mathcal{N}N, i.e., it "purifies" over the spectrum of N\mathcal{N}N. We extend this result to observables with general spectra. It is shown that the spectral density of the state of the system converges to a delta function exponentially fast, in an appropriate sense. Furthermore, for observables with absolutely continuous spectra, we show that the spectral density approaches a Gaussian distribution over the spectrum of N\mathcal{N}N. Our methods highlight the connection between the theory of non-demolition measurements and classical estimation theory.
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