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Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States

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Abstract

We consider a problem in random matrix theory that is inspired by quantum information theory: determining the largest eigenvalue of a sum of p random product states in \({(\mathbb {C}^d)^{\otimes k}}\), where k and p/d k are fixed while d → ∞. When k = 1, the Marčenko-Pastur law determines (up to small corrections) not only the largest eigenvalue (\({(1+\sqrt{p/d^k})^2}\)) but the smallest eigenvalue \({(\min(0,1-\sqrt{p/d^k})^2)}\) and the spectral density in between. We use the method of moments to show that for k > 1 the largest eigenvalue is still approximately \({(1+\sqrt{p/d^k})^2}\) and the spectral density approaches that of the Marčenko-Pastur law, generalizing the random matrix theory result to the random tensor case. Our bound on the largest eigenvalue has implications both for sampling from a particular heavy-tailed distribution and for a recently proposed quantum data-hiding and correlation-locking scheme due to Leung and Winter.

Since the matrices we consider have neither independent entries nor unitary invariance, we need to develop new techniques for their analysis. The main contribution of this paper is to give three different methods for analyzing mixtures of random product states: a diagrammatic approach based on Gaussian integrals, a combinatorial method that looks at the cycle decompositions of permutations and a recursive method that uses a variant of the Schwinger-Dyson equations.

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Authors and Affiliations

  1. Faculty of Computing, University of Latvia, Riga, Latvia

    Andris Ambainis

  2. Department of Mathematics, University of Bristol, Bristol, UK

    Aram W. Harrow

  3. Department of Computer Science & Engineering, University of Washington, Seattle, WA, USA

    Aram W. Harrow

  4. Microsoft Research, University of California, Station Q, CNSI Building, Santa Barbara, CA, USA

    Matthew B. Hastings

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  1. Andris Ambainis
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  2. Aram W. Harrow
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Correspondence to Aram W. Harrow.

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Communicated by M.B. Ruskai

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Ambainis, A., Harrow, A.W. & Hastings, M.B. Random Tensor Theory: Extending Random Matrix Theory to Mixtures of Random Product States. Commun. Math. Phys. 310, 25–74 (2012). https://doi.org/10.1007/s00220-011-1411-x

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  • DOI: https://doi.org/10.1007/s00220-011-1411-x

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