A note on Bianchi-Don\`a's proof to the variance formula of von Neumann entropy

TL;DR

This paper clarifies that Bianchi and Dona's recent proof of the von Neumann entropy variance formula is fundamentally similar to the original proof by Alicki and Fannes, despite different initial approaches.

Contribution

It demonstrates that the new proof by Bianchi and Dona aligns with the original proof's core calculations, providing insight into the proof's structure.

Findings

The proof's core calculations are essentially the same as in the original work.

Different starting points lead to similar results in the proof.

Clarifies the relationship between recent and original proofs of the variance formula.

Abstract

Bianchi and Don\`{a} [1] have recently reported a proof to the variance formula of von Neumann entropy, which was conjectured in [2] and firstly proved in [3]. The purpose of this short note is to show that, despite having a different starting point, the subsequent calculations (omitted in [1]) leading to the result are essentially the same as in [3].

Equations

$$f\left(\mathbf{\lambda }\right)\propto \delta \left(1-\sum _{i=1}^m{\lambda }_i\right)\prod _{1\le i<j\le m}{\left({\lambda }_i-{\lambda }_j\right)}^2\prod _{i=1}^m{\lambda }_i^{n-m},$$

$${\mathbb{E}}_f\left[S^k\right],k=1,2,\dots ,$$

$$S=-\sum _{i=1}^m{\lambda }_i\ln {\lambda }_i,S\in \left[0,\ln m\right],$$

$${\mathbb{E}}_f\left[S\right]={\psi }_0(mn+1)-{\psi }_0(n)-\frac{m+1}{2n},$$

$$\mathbb{V}{}_f\left[S\right]=-{\psi }_1\left(mn+1\right)+\frac{m+n}{mn+1}{\psi }_1\left(n\right)-\frac{(m+1)(m+2n+1)}{4n^2(mn+1)},$$

$$\displaystyle\mathbb{E}_{f}\!\left[\left(\sum_{i=1}^{m}\lambda_{i}^{r_{1}}\right)\left(\sum_{i=1}^{m}\lambda_{i}^{r_{2}}\right)\right]=\frac{\Gamma(mn)}{\Gamma(mn+r_{1}+r_{2})}\big{(}{\mathrm{tr}}\mathbf{X}(r_{1}+r_{2})+$$

$$\displaystyle{\mathrm{tr}}\mathbf{X}(r_{1}){\mathrm{tr}}\mathbf{X}(r_{2})-{\mathrm{tr}}\left(\mathbf{X}(r_{1})\mathbf{X}(r_{2})\right)\big{)}$$

$$\frac{{\Gamma }^2(r+1)\Gamma (j+1)}{\Gamma (n-m+i+1)}\sum _{k=0}^{m-1}\frac{{\left(k!\Gamma (i-k+1)\Gamma (j-k+1)\right)}^{-1}\Gamma (n-m+r+k+1)}{\Gamma (r+k-i+1)\Gamma (r+k-j+1)}.$$

$${\mathbb{E}}_f\left[S^2\right]=\underset{\begin{array}{c}{r}_1\to 1\\ r_2\to 1\end{array}}{\lim }\frac{{\partial }^2}{\partial r_2\partial r_1}{\mathbb{E}}_f\left[\left(\sum _{i=1}^m{\lambda }_i^{r_1}\right)\left(\sum _{i=1}^m{\lambda }_i^{r_2}\right)\right].$$

$${\mathbb{E}}_f\left[S^2\right]=T_a+T_b+T_c,$$

$$T_a$$

$$T_b$$

$$T_c$$

$$T_a$$

$$c_1$$

$$c_2$$

$$c_3$$

$$c_{2}=\frac{\partial}{\partial r_{1}}{\mathrm{tr}}\mathbf{X}(r_{1}+r_{2})\bigg{|}_{\begin{subarray}{c}r_{1}=1\\ r_{2}=1\end{subarray}}:={\mathrm{tr}}^{\prime}\mathbf{X}(2).$$

$$\Gamma (-l+ϵ)$$

$${\psi }_0(-l+ϵ)$$

$${\psi }_1(-l+ϵ)$$

$$c_1$$

$$c_2$$

$$c_3$$

$$T_b$$

$$c_4$$

$$c_5$$

$$c_4$$

$$c_5$$

$$T_c$$

$$c_6$$

$$c_7$$

$$c_8$$

$$c_6$$

$$c_7$$

$$c_8$$

$$\mathbb{V}{}_f\left[S\right]$$

$$=$$

Full text

1 Introduction

We concisely outline the mathematical formulation and notations of the considered problem, where the physical background can be found in [1, 2]. Consider a bipartite quantum system that consists of two subsystems $A$ and $B$ of Hilbert space dimensions $m$ and $n$. For such a system, the eigenvalue density of the reduced density matrix is

[TABLE]

where $0<\lambda_{i}<1$ and $\sum_{i=1}^{m}\lambda_{i}=1$. The entanglement of the bipartite system can be understood from the moments

[TABLE]

of von Neumann entropy

[TABLE]

over the environment (1). It is known that the mean value of von Neumann entropy is given by [4]

[TABLE]

where $\psi_{0}(x)=\,\mathrm{d}\ln\Gamma(x)/\,\mathrm{d}x$ is the digamma function.

This note concerns the variance of von Neumann entropy, a formula of which was conjectured by Vivo, Pato, and Oshanin [2] as

[TABLE]

where $\psi_{1}(x)=\,\mathrm{d}^{2}\ln\Gamma(x)/\,\mathrm{d}x^{2}$ is the trigamma function. The above formula was firstly proved in [3] by computing directly the second moment $\mathbb{E}_{f}\!\left[S^{2}\right]$. Recently, Bianchi and Donà also reported a proof [1] to the variance formula (5), where the starting point is their newly discovered representation [1, Eq. (S29)]

[TABLE]

with the entry $x_{ij}$ ($i,j=0,\dots,m-1$) of the $m\times m$ matrix $\mathbf{X}(r)$ being

[TABLE]

Clearly, the second moment will be recovered by taking the limits of the derivatives of the representation (1), i.e.,

[TABLE]

The variance formula (5) was then immediately declared in [1], where the authors described the calculation procedure of (8) as “We take the derivatives and the limits with the help of Wolfram’s Mathematica”.

The aim of this note is to reveal the details of the calculations in (8). We find that the computation of (8) necessarily involves, albeit in a different order, the bulk of calculations and simplifications as reported in [3]. Moreover, the computation relies on new summation identities (derived in [3]) that in fact have not been implemented in Mathematica [5].

2 Calculations of (8)

Inserting (1) into (8), the computation can be divided into three parts

[TABLE]

where

[TABLE]

2.1 Calculating $T_{a}$

[TABLE]

where

[TABLE]

Here, for example, we have used the shorthand notation

[TABLE]

Computing the coefficients $c_{1}$, $c_{2}$, and $c_{3}$ consists of taking up to the second derivatives of (7), resolving the indeterminacy of limits by using series expansion of gamma and polygamma functions around negative integers

[TABLE]

and evaluating the summations of resulting polygamma functions. These types of computation have been performed in [3], which similarly lead to the results

[TABLE]

Note that in obtaining the above results one has to make use of sum identities of powers of digamma functions, e.g., [3, Eqs. (A4)-(A6)] and a semi closed-form sum identity [3, Eq. (A12)]. Mathematica is currently unable to evaluate these two types of summations [5].

2.2 Calculating $T_{b}$

[TABLE]

where

[TABLE]

Similarly as for $T_{a}$, the coefficients $c_{4}$ and $c_{5}$ are computed as

[TABLE]

2.3 Calculating $T_{c}$

[TABLE]

where the coefficients

[TABLE]

are similarly computed as

[TABLE]

2.4 Recovering $\mathbb{V}\!_{f}\!\left[S\right]$

Putting the results together, we arrive at the claimed formula (5),

[TABLE]

after some necessary simplification by the identities

[TABLE]

The essential derivation of the variance resides in the tedious calculations of the coefficients $c_{1}$ to $c_{8}$, which were computed in a different order in the proof [3]. In particular, one of the key ingredients in the proof was to capture the cancellations of an unsimplifiable term

[TABLE]

which is also observed in the present note, cf. (23), (35), and (37). Capturing the cancellations requires certain tailor-made simplification tools derived in [3], which are unavailable in the computer algebra system Mathematica [5]. It is therefore unclear the statement in [1] that the calculations of (8) were performed with the help of Mathematica to yield the final result (5).