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On the spectral gap of random quantum channels

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arxiv 1811.08847 v1 pith:PWNLEQWX submitted 2018-11-21 quant-ph math-phmath.MP

On the spectral gap of random quantum channels

classification quant-ph math-phmath.MP
keywords channelsquantumrandombounddifferencefirstkrausnumber
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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In this work, we prove a lower bound on the difference between the first and second singular values of quantum channels induced by random isometries, that is tight in the scaling of the number of Kraus operators. This allows us to give an upper bound on the difference between the first and second largest (in modulus) eigenvalues of random channels with same large input and output dimensions for finite number of Kraus operators $k\geq 169$. Moreover, we show that these random quantum channels are quantum expanders, answering a question posed by Hastings. As an application, we show that ground states of infinite 1D spin chains, which are well-approximated by matrix product states, fulfill a principle of maximum entropy.

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