The sum of square root of spectrum of density matrices is defined as where m is the dimension of the density matrix and the set ${\left\{{\lambda }_i\right\}}_{i=1}^m$ is its spectrum. The random variable (1) is closely related to the negativity (2) and fidelity (3) as discussed below.

Negativity is introduced in [1] as a computable measure of entanglement. For a pure state $\rho =\left|\psi \right.〉\left.〈\psi \right|$ with $\left|\psi \right.〉={\sum }_{j=1}^m\sqrt{{\lambda }_j}\left|jj\right.〉$ and ${\sum }_{j=1}^m{\lambda }_j=1$, the negativity is defined as where $\left|\right|·{\left|\right|}_1$ is the trace norm (also known as the Schatten 1-norm) and ${\rho }^{T_A}$ refers to the partial transpose of $\rho$. Moreover, it has a uniqueness property that suppose $E\left(\rho \right)$ is a weak entanglement monotone that is a symmetric function of negative eigenvalues of ${\rho }^{{\mathrm{T}}_{\mathrm{A}}}$, then $E\left(\rho \right)$ is a nondecreasing function of $\mathcal{N}\left(\rho \right)$, and $E\left(\rho \right)=c\log (1+2\mathcal{N}(\rho \left)\right)$ for some constant $c\ge 0$ in the case that it is additive, see [2].

Fidelity [3] refers to a measure of the similarity or overlap between two quantum states. It quantifies how closely one quantum state resembles another. It is defined as: In this work, we only consider the case that $\sigma =\frac{1}{m}{\mathbb{I}}_m$, which is the maximum mixed state, and $\rho$ is the the density matrix corresponding to Bures-Hall ensemble. In this case, we have The case of Hilbert-Schmidt is computed in [4].

The Bures-Hall ensemble is described as follows [5,6]. Consider a composite(bipartite) system that consists of two subsystems A and B of Hilbert space(complex vector space) with dimensions m and n, respectively. The Hilbert space ${\mathcal{H}}_{A+B}={\mathcal{H}}_A\otimes {\mathcal{H}}_B$. A random pure state of the composite system ${\mathcal{H}}_{A+B}$ is defined as a linear combination of the random coefficients $z_{i,j}$ and the complete basis $\left|i^A\right.〉$ and $\left|j^B\right.〉$ of ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ [5], where each $z_{i,j}$ follows the standard Gaussian distribution. We now consider a superposition of the state (5), where U is an $m\times m$ unitary random matrix with the measure proportional to $\det {({\mathbf{I}}_m+\mathbf{U})}^{2\alpha +1}$ [7]. The corresponding density matrix of the pure state (6) is which has the natural probability constraint Without loss of generality, we assume that $m\le n$. The reduced density matrix ${\rho }_A$ of the smaller subsystem A is computed by partial tracing (purification) of the full density matrix (7) over the other subsystem B (environment) as The resulting density of the eigenvalues of ${\rho }_A$ (${\lambda }_i\in [0,1],i=1,...,m$) is the (generalized) complex Bures-Hall measure [7,8,9,10], where the parameter $\alpha$ takes half-integer values, and the constant C is For convenience, we need to define the random variable below:Then, the negativity and fidelity are defined, respectively, as

Proposition 1.1 The exact mean of the random variable $\mathsf{\Lambda }$ defined in (13) valid for any subsystem dimensions $m\le n$ under the Bures-Hall ensemble (10) is obtained as where d is The proof of Proposition 1.1 is given in Sec. 2.2.

Proposition 1.2 The exact second moment of $\mathsf{\Lambda }$ in (13) valid for any subsystem dimensions $m\le n$ under the Bures-Hall ensemble (10) is obtained as where we denote Therefore, the mean of negativity and fidelity, valid for any subsystem dimensions $m\le n$, are obtained, respectively, as where the expectation ${\mathbb{E}}_f[.]$ is taken over the Bures-Hall ensemble (10). The exact variance of the first moment $\mathsf{\Lambda }$ under the Bures-Hall ensemble is given by The proof of these results is given in Sec. 2.3.

Now we can understand the distribution of the $\mathsf{\Lambda }$ with the expressions of the mean (15) and variance (20). For convenience, we standardize $\mathsf{\Lambda }$ as which make the random variable X be supported in $X\in (-\infty ,\infty )$ with zero mean and unit variance. We obtain the simulation with comparison to Gaussian distribution by Matlab.

As we can see from the Figure 1 and Figure 2, while the distribution of von Neumann entropy which is conjectured to be the same as Gaussian distribution [11,12], the distribution of $\mathsf{\Lambda }$ is also similar to Gaussian distribution.

We calculate the random variables under the original ensemble by covering it to unconstrained ensemble. The unconstrained ensemble of the Bures-Hall measure is where $x_i\in [0,\infty )$, i=1,...,m, and the constant C’ depends on the constant (12) as with d denoting The density of trace is obtained as where we have applied the change of variables implies that $h\left(x\right)$ is factored as [13] which shows $\theta$ is independent of each ${\lambda }_i$.

By multiplying an appropriate constant (26), denote that we have The variance is defined as We have Similarly, by multiplying a proper constant (29), we obtain

Applying (31) and (34), we have

Following the formulas for ${\mathbb{E}}_h\left[T_P\right]$ [14, Eq. (26) to (48)], with the same notation, letting letting $\beta =\frac{1}{2}$ instead of $\beta =2$, we obtain Applying the identity of Gamma function: we are able to write the result as Therefore, the mean of $\mathsf{\Lambda }$ is given by

By (35), now it suffices to calculate ${\mathbb{E}}_h\left[X^2\right]$. where where where we denote The kernal functions above((39) and (42) to (44)) are obtained in [15,16] , which were successfully used in calculating the mean and variance of von Neumann entropy under Bures-Hall ensemble [11].

So we can calculate five integrals separately to get the result: