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arXiv:1811.08847

On the spectral gap of random quantum channels

Carlos E. González-Guillén, Marius Junge, Ion Nechita

Published Nov 21, 2018• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1811.08847

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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≥169k\geq 169k≥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.
Abstract In this work we investigate how quantum expanders (i.e., quantum channels with few Kraus operators but a large spectral gap) can be constructed from unitary designs. Concretely, we prove that a random quantum channel whose Kraus operators are independent unitaries sampled from a 222-design measure is with high probability an optimal expander (in the sense that its spectral gap is as large as possible). More generally, we show that, if these Kraus operators are independent unitaries of the form U⊗kU^{\otimes k}U⊗k, with UUU sampled from a 2k2k2k-design measure, then the corresponding random quantum channel is typically an optimal kkk-copy tensor product expander, a concept introduced by Harrow and Hastings (Quant. Inf. Comput. 2009).
We consider a random quantum channel obtained by taking a selection of ddd independent and Haar distributed NNN dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values \cite{hastings2007random,harrow2007quantum}. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon-Boppana bound for ddd-regular graphs. The upper bound is obtained using Schwinger-Dyson equations.
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We consider a random quantum channel obtained by taking a selection of [Formula: see text] independent and Haar distributed [Formula: see text]-dimensional unitaries. We follow the argument of Hastings to bound the spectral gap in terms of eigenvalues and adapt it to give an exact estimate of the spectral gap in terms of singular values [7, 8]. This shows that we have constructed a random quantum expander in terms of both singular values and eigenvalues. The lower bound is an analog of the Alon–Boppana bound for [Formula: see text]-regular graphs. The upper bound is obtained using Schwinger–Dyson equations.
Abstract. In this study, we generate quantum channels with random Kraus operators to typically obtain almost twirling quantum channels and quantum expanders. To prove the concentration phenomena, we use matrix Bernstein’s inequality. In this way, our random models do not utilize Haar-distributed unitary matrices or Gaussian matrices. Rather, as in the preceding research, we use unitary t-designs to generate mixed tensor-product unitary channels acting on Cdt. Although our bounds in Schatten p-norm are valid only for 1 ≤p≤ 2, we show that they are typically quantum ϵ-twirling channels with the tail bound proportional to 1/poly(dt), while such bounds were previously constants. The number of required Kraus operators was also improved by powers of logdandt, to be proportional todtlogd/ϵ2. Such random quantum channels are also typically quantum expanders, but the number of Kraus operators must grow proportionally totlogdin our case. Finally, a new non-unital model of super-operators generated by bounded and isotropic random Kraus operators was introduced, which can be typically rectified to yield almost randomizing quantum channels and quantum expanders.
We consider the image of some classes of bipartite quantum states under a tensor product of random quantum channels. Depending on natural assumptions that we make on the states, the eigenvalues of their outputs have new properties which we describe. Our motivation is provided by the additivity questions in quantum information theory, and we build on the idea that a Bell state sent through a product of conjugated random channels has at least one large eigenvalue. We generalize this setting in two directions. First, we investigate general entangled pure inputs and show that Bell states give the least entropy among those inputs in the asymptotic limit. We then study mixed input states, and obtain new multi-scale random matrix models that allow to quantify the difference of the outputs’ eigenvalues between a quantum channel and its complementary version in the case of a non-pure input.