# Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels

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## Abstract

**:**

## 1. Introduction

## 2. Studying Moments of Outputs of Random Quantum Channels: Techniques and First Examples

#### 2.1. Weingarten calculus

**Definition 2.1.**The unitary Weingarten function $Wg(n,\sigma )$ is a function of a dimension parameter n and of a permutation σ in the symmetric group ${\mathcal{S}}_{p}$ on p elements, defined as the pseudo-inverse of the function $\sigma \mapsto {n}^{\#\sigma}$ under the convolution for the symmetric group ($\#\sigma $ denotes the number of cycles of the permutation σ).

**Theorem 2.2.**Let n be a positive integer and $({i}_{1},\dots ,{i}_{p})$, $({i}_{1}^{\prime},\dots ,{i}_{p}^{\prime})$, $({j}_{1},\dots ,{j}_{p})$, $({j}_{1}^{\prime},\dots ,{j}_{p}^{\prime})$ be p-tuples of positive integers from $\{1,2,\dots ,n\}$. Then

**Theorem 2.3.**For a permutation $\sigma \in {\mathcal{S}}_{p}$, let $\mathrm{Cycles}\left(\sigma \right)$ denote the set of cycles of σ. Then

#### 2.2. Planar expansion

**Figure 1.**Basic diagrams and axioms: (a) diagram for a general tensor M; (b) trace of a $(1,1)$-tensor (matrix) M; (c) Scalar product $\langle y\phantom{\rule{0.277778em}{0ex}}\left|\phantom{\rule{0.277778em}{0ex}}M\phantom{\rule{0.277778em}{0ex}}\right|\phantom{\rule{0.277778em}{0ex}}x\rangle $; (d) tensor product of two diagrams. The round, square and diamond-shaped labels correspond to pairs of dual finite dimensional complex Hilbert spaces.

_{p}.

**Theorem 2.4.**The following holds true:

#### 2.3. Wishart matrices, Marchenko-Pastur distributions and their entropy

**Figure 2.**Densities for the Marchenko-Pastur measures of parameters $c=1/5$, $c=1$ and $c=5$. For $c=1/5$, only the absolutely continuous part of the measure was plotted; ${\pi}_{1/5}$ has a Dirac mass of $4/5$ at $x=0$ which is not represented.

#### 2.4. Application 1: Fixed ancilla space

^{m}):

**Theorem 2.5.**Almost surely, as $n\to \infty $, the ${k}^{2}$ non-zero eigenvalues of the random matrix $[\Phi \otimes \overline{\Phi}]\left({E}_{tnk}\right)\in {\mathcal{M}}_{{n}^{2}}\left(\right)$ converge towards the deterministic probability vector

#### 2.5. Application 2: Ancilla and input space of linear dimensions

**Figure 3.**Diagram for a quantum channel with equal input and output spaces. The state ${P}_{k}$ of the ancilla space is omitted, since it has no role to play in the computations. Round labels attached to boxes correspond to input/output spaces

^{n}and square symbols correspond to ancilla spaces

^{k}.

**Theorem 2.6.**Consider a pair of conjugate random quantum channels $\Phi ,\overline{\Phi}$ in the regime where $n,k\to \infty $, $k\sim cn$. The eigenvalues ${\lambda}_{1}\u2a7e\cdots \u2a7e{\lambda}_{{n}^{2}}$ of the random matrix ${Z}_{n}=[\Phi \otimes \overline{\Phi}]\left({E}_{n}\right)$ are such that:

- The first eigenvalue satisfies $cn{\lambda}_{1}\to 1$ (in probability).
- The distribution $\frac{1}{{n}^{2}-1}{\sum}_{i=2}^{{n}^{2}}{\delta}_{{c}^{2}{n}^{2}{\lambda}_{i}}$ converges a.s. to a free Poisson distribution of parameter ${c}^{2}$.

**Theorem 2.7.**In the regime $k\sim cn$, $n\to \infty $, let ${Z}_{n}=[\Phi \otimes \Psi ]\left({E}_{n}\right)$ be the output of the product of two independent quantum channels Φ and Ψ, when the input is a maximally entangled state ${E}_{n}$. Then, almost surely, the distribution of the rescaled output matrix ${c}^{2}{n}^{2}{Z}_{n}$ converges towards a free Poisson of parameter ${c}^{2}$.

## 3. Generalized Linear Setting—Input and Output Spaces of Different Dimension

**Figure 4.**Diagram for a quantum channel with different input and output spaces. Round labels attached to boxes correspond to output spaces

^{n}, square symbols correspond to ancilla spaces

^{k}and diamonds correspond to input spaces

^{m}. The rank-one projector ${P}_{l}$ is omitted.

**Figure 5.**Diagram for the output of a product of two conjugate channels, when the input is the maximally entangled state. The complex Hilbert spaces associated to labels are as follows: $\u25e6\u2022{\u21dd}^{n}$, $\u25ab\blacksquare {\u21dd}^{k}$, $\u25c7\u25c6{\u21dd}^{m}$ and $\u25b5\u25b4{\u21dd}^{l}$.

**Proposition 3.1.**Consider a sequence of random quantum channels ${\Phi}_{n}$ where $m,n,k\to \infty $, $m/n\to b$ and $k/n\to c$. The asymptotic moments of the output matrix $Z=[\Phi \otimes \overline{\Phi}]\left({E}_{m}\right)$ are given by:

- for $p=1$, $f(\alpha ,\beta )\u2a7e0$, with equality iff. $\alpha =\beta =id$;
- for $p=2$, $f(\alpha ,\beta )\u2a7ep$, with equality iff. $\alpha =\beta \in \{id,\delta ,\gamma \}$;
- for $p\u2a7e3$, $f(\alpha ,\beta )\u2a7ep$, with equality iff. $\alpha =\beta =\delta $.

**Proposition 3.2.**Almost surely, the matrix ${c}^{2}{n}^{2}QZQ$ converges in distribution, to a free Poisson (or Marchenko-Pastur) law of parameter ${c}^{2}$.

**Remark 3.3.**By `almost surely’, we mean with probability one, in any probability space on which the whole sequence (indexed by the input dimension) of random quantum channels is defined. In this paper, we supply no proofs of almost sure convergence results, as they require further -not so enlightening- technicalities. We refer the interested reader to the appendix of [6] for details. Let us just mention that the proofs rely on the Borel-Cantelli lemma. More precisely, one proves that the covariance of any moment behaves as $O\left({n}^{-2}\right)$ as the dimension goes to infinity. The fact that $O\left({n}^{-2}\right)$ is summable over n makes it possible to use the Borel-Cantelli lemma.

**Theorem 3.4.**Consider a pair of conjugate random quantum channels $\Phi ,\overline{\Phi}$ in the regime where $m,n,k\to \infty $, $m\sim bn$ and $k\sim cn$. The eigenvalues ${\lambda}_{1}\u2a7e\cdots \u2a7e{\lambda}_{{n}^{2}}$ of the random matrix ${Z}_{n}=[\Phi \otimes \overline{\Phi}]\left({E}_{m}\right)$ are such that:

- The first eigenvalue satisfies $(c/b)n{\lambda}_{1}\to 1$ in probability.
- The distribution $\frac{1}{{n}^{2}-1}{\sum}_{i=2}^{{n}^{2}}{\delta}_{{c}^{2}{n}^{2}{\lambda}_{i}}$ converges almost surely to a free Poisson distribution of parameter ${c}^{2}$.

**Remark 3.5.**The almost sure convergence argument described in Remark 3.3 does not extend to the first item of Theorem 3.4, as the covariances tend to zero but are not summable.

## 4. Non-linear Output Dimension

^{k}scales with n in a non-linear fashion:

**Proposition 4.1.**The asymptotic moments of the random output matrix $Z=[\Phi \otimes \overline{\Phi}]\left({E}_{n}\right)$ are given by:

- If $d=0$ (see [5]):$$\mathbb{E}[Tr\left({Z}^{p}\right)]={\left(\frac{1}{c}+\frac{1}{{c}^{2}}-\frac{1}{{c}^{3}}\right)}^{p}+({c}^{2}-1){\left(\frac{1}{{c}^{2}}-\frac{1}{{c}^{3}}\right)}^{p}+o\left(1\right)\phantom{\rule{2.em}{0ex}}\forall p\u2a7e2$$
- If $d\in (0,1)$:$$\begin{array}{cc}& \mathbb{E}[Tr\left({Z}^{p}\right)]\sim 2{c}^{-2}{n}^{-2d}\sim 2{k}^{-2}\phantom{\rule{2.em}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}p=2\hfill \\ & \mathbb{E}[Tr\left({Z}^{p}\right)]\sim {c}^{-p}{n}^{-dp}\sim {k}^{-p}\phantom{\rule{2.em}{0ex}}\forall p\u2a7e3\hfill \end{array}$$
- If $d=1$ (see [6]):$$\begin{array}{cc}& \mathbb{E}[Tr\left({Z}^{p}\right)]\sim (1+2{c}^{-2}){n}^{-2}\phantom{\rule{2.em}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}p=2\hfill \\ & \mathbb{E}[Tr\left({Z}^{p}\right)]\sim {c}^{-p}{n}^{-dp}\sim {k}^{-p}\phantom{\rule{2.em}{0ex}}\forall p\u2a7e3\hfill \end{array}$$
- If $d\in (1,2)$:$$\begin{array}{cc}& \mathbb{E}[Tr\left({Z}^{p}\right)]\sim {n}^{-(2p-2)}\phantom{\rule{2.em}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}p<\frac{2}{2-d}\hfill \\ & \mathbb{E}[Tr\left({Z}^{p}\right)]\sim (1+{c}^{-p}){n}^{-dp}\sim (1+{c}^{-p}){n}^{-(2p-2)}\phantom{\rule{2.em}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}p=\frac{2}{2-d}\hfill \\ & \mathbb{E}[Tr\left({Z}^{p}\right)]\sim {c}^{-p}{n}^{-dp}\sim {k}^{-p}\phantom{\rule{2.em}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}p>\frac{2}{2-d}\hfill \end{array}$$
- If $d\u2a7e2$:$$\begin{array}{cc}& \mathbb{E}[Tr\left({Z}^{p}\right)]\sim {n}^{-(2p-2)}\phantom{\rule{2.em}{0ex}}\forall p\u2a7e2\hfill \end{array}$$

d | Minorant for ${S}_{2}\left(\beta \right)$ | Equality cases |

0 | $-1$ | $\{\delta \to \tilde{\gamma}\}$ |

$(0,2)$ | $dp-1$ | δ |

2 | $2p-1$ | $\{id\to \delta \}$ |

$(2,\infty )$ | $2p-1$ | id |

- $\beta \notin \{id\to \tilde{\gamma}\}$. Since β is not a geodesic permutation, we have that $\left|\beta \right|+|\beta {\tilde{\gamma}}^{-1}|\u2a7e2p+1$, $\left|\beta \right|+\left|\beta \delta \right|\u2a7ep+2$, $\left|\beta \right|\u2a7e1$ and $\left|\beta \delta \right|\u2a7e1$. It follows that$$\begin{array}{cc}\hfill {S}_{1}\left(\beta \right)& \u2a7e{S}_{2}\left(\beta \right)-1\u2a7e\left|\beta \right|+|\beta {\tilde{\gamma}}^{-1}|+(d-1)(\left|\beta \right|+\left|\beta \delta \right|)+(2-d)|\beta \delta |-p-1\hfill \\ & \u2a7e2p+1+(d-1)(p+2)+(2-d)-p-1=dp+d>dp={S}_{1}\left(\delta \right)\hfill \end{array}$$
- $\beta \in \{id\to \gamma \}$. In this situation, ${S}_{1}\left(\beta \right)={S}_{2}\left(\beta \right)-1$ and thus (we use the fact that $\left|\beta \delta \right|\u2a7ep$ in this case)$$\begin{array}{cc}\hfill {S}_{1}\left(\beta \right)& ={S}_{2}\left(\beta \right)-1\u2a7e\left|\beta \right|+|\beta {\tilde{\gamma}}^{-1}|+(d-1)|\beta |+|\beta \delta |-p-1\hfill \\ & \u2a7e2p-1+0+p-p-1\u2a7e2p-2={S}_{1}(id)\hfill \end{array}$$
- $\beta \in \{id\to \tilde{\gamma}\},\phantom{\rule{0.277778em}{0ex}}\beta \notin \{id\to \gamma \}$. In this situation, ${S}_{1}\left(\beta \right)={S}_{2}\left(\beta \right)+1$ and thus$$\begin{array}{cc}\hfill {S}_{1}\left(\beta \right)& ={S}_{2}\left(\beta \right)+1\u2a7e\left|\beta \right|+|\beta {\tilde{\gamma}}^{-1}|+(d-1)(\left|\beta \right|+\left|\beta \delta \right|)+(2-d)|\beta \delta |-p+1\hfill \\ & \u2a7e2p-1+(d-1)p+(d-2)0-p+1\u2a7edp={S}_{1}\left(\delta \right)\hfill \end{array}$$

d | Minorant for ${S}_{1}\left(\beta \right)$ | Equality cases |

0 | 0 | $\{\delta \to \gamma \}$ |

$(0,1)\phantom{\rule{1.em}{0ex}}(p=2)$ | $2d$ | $\delta ,\gamma $ |

$1\phantom{\rule{1.em}{0ex}}(p=2)$ | 2 | $id,\delta ,\gamma $ |

$(0,1]\phantom{\rule{1.em}{0ex}}(p\u2a7e3)$ | $dp$ | δ |

$(1,2)\phantom{\rule{1.em}{0ex}}(p<\frac{2}{2-d})$ | $2p-2$ | id |

$(1,2)\phantom{\rule{1.em}{0ex}}(p=\frac{2}{2-d})$ | $2p-2=dp$ | $id,\delta $ |

$(1,2)\phantom{\rule{1.em}{0ex}}(p>\frac{2}{2-d})$ | $dp$ | δ |

$[2,\infty )$ | $2p-2$ | id |

**Proposition 4.2.**In the regime $d=0$ (which corresponds to considering a fixed ancilla dimension $k=c$), the eigenvalues of the random matrix Z are such that, almost surely, in the limit $n\to \infty $,

**Proposition 4.3.**The matrix ${k}^{2}QZQ$ converges, in moments, to the Dirac mass at 1, ${\delta}_{1}$.

**Theorem 4.4.**In the regime $k\sim c{n}^{d}$, with $d\in (0,1)$, the eigenvalues ${\lambda}_{1}\u2a7e\cdots \u2a7e{\lambda}_{{n}^{2}}$ of $Z=[\Phi \otimes \overline{\Phi}]\left({E}_{n}\right)$ satisfy:

- In probability, $k{\lambda}_{1}\to 1$.
- Almost surely, $\frac{1}{{k}^{2}-1}{\sum}_{i=2}^{{k}^{2}}{\delta}_{{k}^{2}{\lambda}_{i}}$ converges in distribution to the Dirac mass at 1, ${\delta}_{1}$.
- The remaining ${n}^{2}-{k}^{2}$ eigenvalues are null: ${\lambda}_{{k}^{2}+1}=\cdots ={\lambda}_{{n}^{2}}=0$.

**Proposition 4.5.**In the regime $d\in (1,2)$, the matrix ${n}^{2}QZQ$ converges, in moments, to the Dirac mass at 1, ${\delta}_{1}$.

**Theorem 4.6.**In the regime $k\sim c{n}^{d}$, with $d\in (1,2)$, the eigenvalues ${\lambda}_{1}\u2a7e\cdots \u2a7e{\lambda}_{{n}^{2}}$ of $Z=[\Phi \otimes \overline{\Phi}]\left({E}_{n}\right)$ satisfy:

- In probability, $k{\lambda}_{1}\to 1$.
- Almost surely, $\frac{1}{{n}^{2}-1}{\sum}_{i=2}^{{n}^{2}}{\delta}_{{n}^{2}{\lambda}_{i}}$ converges in distribution to the Dirac mass at 1, ${\delta}_{1}$.

## 5. Asymptotics of the Von Neumann Entropy

**Theorem 5.1.**The asymptotic von Neumann entropy of the random output matrix $Z=[\Phi \otimes \overline{\Phi}]\left({E}_{n}\right)$ is given by:

- If $d=0$ ($k=c$ is an integer):$$H\left(Z\right)=-\left(\frac{1}{c}+\frac{1}{{c}^{2}}-\frac{1}{{c}^{3}}\right)log\left(\frac{1}{c}+\frac{1}{{c}^{2}}-\frac{1}{{c}^{3}}\right)-({c}^{2}-1)\left(\frac{1}{{c}^{2}}-\frac{1}{{c}^{3}}\right)log\left(\frac{1}{{c}^{2}}-\frac{1}{{c}^{3}}\right)+o\left(1\right)$$
- If $d\in (0,1)$:$$\begin{array}{cc}& H\left(Z\right)=2logk+o\left(1\right)\hfill \end{array}$$
- If $d=1$ (see [6]):$$H\left(Z\right)=\left\{\begin{array}{cc}2logk-\frac{{c}^{2}}{2}+o\left(1\right)\phantom{\rule{1.em}{0ex}}\hfill & \phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}0<c<1,\hfill \\ 2logn-\frac{1}{2{c}^{2}}+o\left(1\right)\phantom{\rule{1.em}{0ex}}\hfill & \phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{1.em}{0ex}}c\u2a7e1\hfill \end{array}\right.$$
- If $d\in (1,2)$:$$\begin{array}{cc}& H\left(Z\right)=2logn+o\left(1\right)\hfill \end{array}$$
- If $d\u2a7e2$:$$\begin{array}{cc}& H\left(Z\right)=2logn+o\left(1\right)\hfill \end{array}$$

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Collins, B.; Nechita, I.
Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels. *Entropy* **2010**, *12*, 1612-1631.
https://doi.org/10.3390/e12061612

**AMA Style**

Collins B, Nechita I.
Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels. *Entropy*. 2010; 12(6):1612-1631.
https://doi.org/10.3390/e12061612

**Chicago/Turabian Style**

Collins, Benoît, and Ion Nechita.
2010. "Eigenvalue and Entropy Statistics for Products of Conjugate Random Quantum Channels" *Entropy* 12, no. 6: 1612-1631.
https://doi.org/10.3390/e12061612