Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

Collins, Benoı̂t; Śniady, Piotr

doi:10.1007/s00220-006-1554-3

Cited by 536 publications

(683 citation statements)

References 13 publications

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“…Notice that the function σ → n #σ is invertible when n is large, as it behaves like n p δ e as n → ∞. Actually, if n < p the function is not invertible any more, but we can keep this definition by taking the pseudo inverse and the theorems below will still hold true (we refer to [8] for historical references and further details). We shall use the shorthand notation Wg(σ) = Wg(n, σ) when the dimension parameter n is clear from context.…”

Section: Review On Random Quantum Channels and Unitary Integrationmentioning

confidence: 99%

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Towards a state minimizing the output entropy of a tensor product of random quantum channels

Collins

Fukuda

Nechita

2012

Journal of Mathematical Physics

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Abstract. 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 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.1. Introduction 1.1. Background. One of the most important questions in quantum communication theory is whether a quantum channel has additive properties or not [1,4,9,10,11,12,13,14,18]. If a channel Φ is additive for the Holevo capacity χ(·) in the sense that ∃N ∈ N, ∀n N χ(Φ ⊗n ) = nχ(Φ) (1) then the classical capacity of the quantum channel equals the Holevo capacity, giving a one-shot (non-asymptotic) formula for the classical capacity C(·):where, {p i , ρ i } are ensembles and S(·) is the von Neumann entropy. The above additive property was conjectured to be true for all quantum channels until Hastings showed [13] existence of channels such thatHere, S min (·) is the minimal output entropy (MOE). Indeed, this result also gives counterexamples to (1) by [18,12,9] and as a result C(Φ) = χ(Φ) in general. Note thatwhere the minimum is take over all pure inputs (rank-one projections).One of the most important results in the additivity theory of quantum channels is Hastings' proof of the additivity of the minimal output entropy [13]. The proof contains two disjoint parts: establishing a lower bound for the minimum output entropy for one channel, which was the main 2000 Mathematics Subject Classification. Primary 15A52; Secondary 94A17, 94A40.

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Section: Review On Random Quantum Channels and Unitary Integrationmentioning

confidence: 99%

“…the minimal number of transpositions that multiply to σ. We refer to [8] for details about the function Mob but what we have to know in this paper is that…”

Section: Review On Random Quantum Channels and Unitary Integrationmentioning

confidence: 99%

Towards a state minimizing the output entropy of a tensor product of random quantum channels

Collins

Fukuda

Nechita

2012

Journal of Mathematical Physics

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“…In this paper, the following hypothesis and notation about the sequence of matrices A N ∈ M N (C) will be used frequently. Using the results presented in our previous article [CŚ04] it is possible to prove that the following quite general statement holds true:…”

Section: Introduction and The Main Resultsmentioning

confidence: 83%

“…We shall not prove this theorem since it is an immediate consequence of Theorem 5.5 of [CŚ04]. On the other hand, it has been proved recently in Theorem 7 of [GM05], that if the L ∞ -norm of A N is bounded by a constant depending on B N (see Theorem 4 in this paper), under Hypothesis 3 for B N , and provided that M(N) = O(N 1/2−ε ) for some ε > 0, one has for β = 1, 2:…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

New scaling of Itzykson–Zuber integrals

Collins

Śniady

2007

Annales de l'Institut Henri Poincare (B) Probability and Statis

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ABSTRACT. We study asymptotics of the Itzykson-Zuber integrals in the scaling when one of the matrices has a small rank compared to the full rank. We show that the result is basically the same as in the case when one of the matrices has a fixed rank. In this way we extend the recent results of Guionnet and Maïda who showed that for a latter scaling the Itzykson-Zuber integral is given in terms of the Voiculescu's Rtransform of the full rank matrix.

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“…We also give upper bounds on the size of a unitary code. In order to verify directly that a set of unitary matrices forms a t-design, the RHS of (1) must be evaluated explicitly; this has been done by Collins [4] and Collins andŚniady [5]. In particular, X is a unitary 1-design if and only if 1 |X|…”

Section: Introductionmentioning

confidence: 99%

Unitary designs and codes

Roy

Scott

2009

Des. Codes Cryptogr.

100

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A unitary design is a collection of unitary matrices that approximate the entire unitary group, much like a spherical design approximates the entire unit sphere. In this paper, we use irreducible representations of the unitary group to find a general lower bound on the size of a unitary t-design in U(d), for any d and t. We also introduce the notion of a unitary code -a subset of U(d) in which the trace inner product of any pair of matrices is restricted to only a small number of distinct values -and give an upper bound for the size of a code of degree s in U(d) for any d and s. These bounds can be strengthened when the particular inner product values that occur in the code or design are known. Finally, we describe some constructions of designs: we give an upper bound on the size of the smallest weighted unitary tdesign in U(d), and we catalogue some t-designs that arise from finite groups. *

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Integration with Respect to the Haar Measure on Unitary, Orthogonal and Symplectic Group

Cited by 536 publications

References 13 publications

Towards a state minimizing the output entropy of a tensor product of random quantum channels

Towards a state minimizing the output entropy of a tensor product of random quantum channels

New scaling of Itzykson–Zuber integrals

Unitary designs and codes

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