Estimating the spectrum of a density operator

By Michael Keyl, Reinhard F. Werner. 2001

180 citations recorded by the source corpus.

Given N quantum systems prepared according to the same density operator \rho, we propose a measurement on the N-fold system which approximately yields the spectrum of \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 \rho in decreasing order. We show convergence of these estimates in the limit N\to\infty, and that the probability for errors decreases exponentially with a rate we compute explicitly.

Subfields: Applied Mathematics, Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/299ce380a7a9fc704a4d98498564902b769cc28e

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Citations in the corpus

On quantum statistical inference (2003, background): “Some key references are the book Leonhardt (1997) and the survey papers D’Ariano (997a,b), though there has been much further progress since then.”
Fundamentals of quantum information theory (2002, background): “We will review here a result from [99] which concerns the subproblem of estimating the spectrum.”
Entanglement detection (2008, methodology): “Alternatively, one may use the spectrum estimation technique from [484].”
Sample-Optimal Tomography of Quantum States (2015, methodology): “Here, the optimal covariant measurement was described by Keyl and Werner [26], its large-deviation properties were derived in [27] (see also [28]), and it was analyzed further in [29, 30].”
Efficient quantum tomography (2015, unknown): “bability distribution, and let λ ∼ SWn(α). Then for all k ∈ [d], as n → ∞ we have λk/n → αk in probability. This theorem has been reproven in a variety of works, includingindependently by [ARS88] and [KW01] in the quantum computing literature. Subsequent work determined the lower-order asymptotics o”
The Spectra of Quantum States and the Kronecker Coefficients of the Symmetric Group (2004, background): “A remarkable connection between Young frames and density operators was discovered by Keyl and Werner [7] (see also R.”
Nonzero Kronecker Coefficients and What They Tell us about Spectra (2007, methodology): “dparts; i.e. the Young diagrams with no more than drows and a total of kboxes. 2 Consider a density operator ρon Cd. We can take kcopies of it and measure the label λon (Cd)⊗k. The estimation theorem [16, 11] states that, as kincreases, the spectrum rof ρis increasingly well approximated by the norm”