Generalized Wigner-von Neumann entropy and its typicality

TL;DR

This paper introduces a generalized Wigner-von Neumann entropy applicable to pure and mixed quantum states, demonstrating its typicality, basis independence, and connection to classical thermodynamic entropies.

Contribution

It proposes a new generalized quantum entropy that unifies and extends previous definitions, with properties that emerge in large Hilbert spaces and relate to thermodynamic concepts.

Findings

GWvN entropy becomes basis-independent and approaches ln N for large N.

The entropy's evolution is typical and aligns with the quantum H theorem.

GWvN entropy matches Boltzmann and von Neumann entropies in specific ensembles.

Abstract

We propose a generalization of the quantum entropy introduced by Wigner and von Neumann in 1929 [Zeitschrift f\"ur Physik 57, 30 (1929)]. Our generalization is applicable to both quantum pure states and mixed states. When the dimension $N$ of the Hilbert space is large, as a result of typicality, this generalized Wigner-von Neumann (GWvN) entropy becomes independent of choices of basis and is asymptotically equal to $\ln N$. The dynamic evolution of our entropy is also typical, reminiscent of quantum H theorem proved by von Neumann. For the microcanonical ensemble, the GWvN entropy is equivalent to the Boltzmann entropy; for a subsystem in a mixed state, the GWvN entropy is equivalent to the familiar von Neumann entropy, which is zero for pure states. The GWvN entropy can be used to derive the Gibbs ensemble.

Figures

Equations

$$S(\phi )\equiv -\sum _{j=1}^N{\left|⟨\phi |{\varphi }_j⟩\right|}^2\ln {\left|⟨\phi |{\varphi }_j⟩\right|}^2.$$

$$\sum _{j=1}^N|z_j{|}^2=1,z_j=⟨{\varphi }_j⟩\phi$$

$$\mathrm{Pr}(s)=\frac{1}{1+\exp \left[-c_N\left(s-\frac{{\mu }_N}{\ln N}\right)\right]}$$

$${\mu }_N=\frac{{\Gamma }^{\prime }(N+1)}{\Gamma (N+1)}-\frac{{\Gamma }^{\prime }(2)}{\Gamma (2)},c_N={\mu }_N\sqrt{\frac{{\text{uppi}}^{2}N}{{\text{uppi}}^2-9}}.$$

$$S(\phi )={\frac{d{H}_{\lambda }}{d\lambda }|}_{\lambda =1}.$$

$$⟨{\left|z_j\right|}^{2\lambda }⟩=\frac{\Gamma \left(N\right)\Gamma \left(1+\lambda \right)}{\Gamma \left(N+\lambda \right)}.$$

$$⟨S⟩={-N\frac{d⟨{\left|z_i\right|}^{2\lambda }⟩}{d\lambda }|}_{\lambda =1}={\mu }_N.$$

$${\sigma }_S^2$$

$$=N(N-1){\frac{{\partial }^2⟨{\left|z_i\right|}^{2\lambda }{\left|z_j\right|}^{2\zeta }⟩}{\partial \lambda \partial \zeta }|}_{\lambda =\zeta =1,i\ne j}$$

$$+N{\frac{{\partial }^2⟨{\left|z_i\right|}^{2(\lambda +\zeta )}⟩}{\partial \lambda \partial \zeta }|}_{\lambda =\zeta =1}-⟨S{⟩}^2$$

$$=\frac{1}{N+1}\cdot \left(\frac{{\pi }^2}{3}-2\right)-{\Psi }_1\left(N+1\right),$$

$${\rho }_s={\text{Tr}}_e\rho =\sum _{ij}{p}_{ij}|{\varphi }_i⟩⟨{\varphi }_j|,$$

$$S({\rho }_s)$$

$$S({\rho }_s)={\frac{d{K}_{\lambda }}{d\lambda }|}_{\lambda =1}.$$

$$⟨p_{ii}^{\lambda }⟩=\frac{\Gamma \left(mn\right)\Gamma \left(m+\lambda \right)}{\Gamma \left(mn+\lambda \right)\Gamma \left(m\right)}.$$

$${-\frac{d⟨p_{ii}^{\lambda }⟩}{d\lambda }|}_{\lambda =1}=\frac{1}{n}\left[\frac{{\Gamma }^{\prime }(mn+1)}{\Gamma (mn+1)}-\frac{{\Gamma }^{\prime }(m+1)}{\Gamma (m+1)}\right].$$

$$⟨S\left({\rho }_s\right)⟩=\frac{{\Gamma }^{\prime }(mn+1)}{\Gamma (mn+1)}-\frac{{\Gamma }^{\prime }(m+1)}{\Gamma (m+1)}\approx \ln n,$$

$${\sigma }_{{\rho }_s}^2$$

$$=n(n-1){\frac{{\partial }^2⟨p_{ii}^{\lambda }{p}_{jj}^{\zeta }⟩}{\partial \lambda \partial \zeta }|}_{\lambda =\zeta =1,i\ne j}$$

$$+n{\frac{{\partial }^2⟨p_{ii}^{\lambda +\zeta }⟩}{\partial \lambda \partial \zeta }|}_{\lambda =\zeta =1}-⟨S\left({\rho }_s\right){⟩}^2$$

$$=\frac{m+1}{N+1}{\Psi }_1\left(m+1\right)-{\Psi }_1\left(N+1\right),$$

$${\sigma }_{{\rho }_s}^2\approx \frac{1}{2mN}+O(\frac{1}{N^2}).$$

$$⟨S\left({\rho }_s\right)+S\left({\rho }_e\right)-S\left(\rho \right)⟩$$

$$|\phi (t)⟩=\sum _j{a}_j{e}^{-i{E}_{j}t/\hslash }|E_j⟩,$$

$$\overline{S(t)}=-\frac{d}{d\beta }{\left[p^{\beta }\left(t\right)+\sum _{i=1}^{N-1}\overline{{\left|⟨\psi \left(t\right)|{\xi }_i⟩\right|}^{2\beta }}\right]}_{\beta =1},$$

$$\overline{{\left|⟨\psi \left(t\right)|{\xi }_i⟩\right|}^{2\beta }}={\left(1-p\left(t\right)\right)}^{\beta }\frac{\Gamma \left(N-1\right)\Gamma \left(1+\beta \right)}{\Gamma \left(N-1+\beta \right)}.$$

$$\overline{S(t)}=f\left(p\left(t\right)\right)+\left(1-p\left(t\right)\right)\left[\frac{{\Gamma }^{\prime }(N)}{\Gamma (N)}-\frac{{\Gamma }^{\prime }(2)}{\Gamma (2)}\right],$$

$${\frac{d}{d\epsilon }\sum _i\text{Tr}\left(\left(\rho +\epsilon \Omega \right)P_{E_i}\right)\ln \left(\text{Tr}\left(\left(\rho +\epsilon \Omega \right)P_{E_i}\right)\right)|}_{\epsilon =0}=0,$$

$$\text{Tr}\left(\Omega \sum _i\ln \left(\text{Tr}\left(\rho P_{E_i}\right)\right)P_{E_i}\right)=0.$$

$$\rho =\alpha \exp \left(-\beta H\right),$$

$$\alpha =\frac{1}{\text{Tr}\left(\exp \left(-\beta H\right)\right)},\beta =\frac{\delta S}{\delta ⟨H⟩}.$$

$$\Delta ⟨H⟩=\frac{\delta ⟨H⟩}{\delta S}\Delta S=T\Delta S,$$

$$\mathrm{Pr}\left(\left|S\left(\phi \right)-⟨S⟩\right|\ge \delta \right)\le 2\exp \left(\frac{-2N{\delta }^2}{9{\pi }^3{\eta }^2}\right),$$

Full text

I Introduction

As an empirical understanding of macroscopic irreversible processes, the second law of thermodynamics has stood the test of time. However, its microscopic origin is still perplexing physicists. Boltzmann gave us a clear understanding how the second law could arise from the time-reversible Newton’s laws of motion. Unfortunately, we now know that macroscopic objects are made of microscopic particles that obey the laws of quantum mechanics. It is therefore desirable and imperative to understand the second law in the perspective of quantum dynamics  v. Neumann (1929); von Neumann (2010); Srednicki (1994); Gemmer and Mahler (2003); Popescu et al. (2006); Rigol et al. (2008); Reimann (2008); Gogolin (2010); Goldstein et al. (2013, 2015); Han and Wu (2015); Tian et al. (2016); Gogolin and Eisert (2016); Fang et al. (2018a); Tian et al. (2018).

In 1929, von Neumann made the first attempt to understand the second law quantum mechanically by proving quantum ergodic theorem and quantum H theorem v. Neumann (1929); von Neumann (2010); Goldstein et al. (2010). In proving the H theorem, von Neumann introduced an entropy for quantum pure states but acknowledged that this definition came from Wigner’s unpublished workv. Neumann (1929); von Neumann (2010). Therefore, we call such a quantum entropy Wigner-von Neumann (WvN) entropy. The well-known von Neumann entropy was not used because it is always zero for quantum pure states and can not describe the relaxation and fluctuations in isolated macroscopic quantum systems. After introducing WvN entropy, von Neumann showed with the help of typicality arguments that for overwhelming majority of basis the H-theorem holds without exception for all states. The typicality argument Lloyd (2013); Goldstein et al. (2006); Reimann (2007, 2010); Tasaki (2016) is mathematically known as measure concentration Milman VD (1986) and Levy’s lemma Popescu et al. (2006).

However, WvN entropy involves vaguely-defined coarse graining and it does not apply to systems with spins. It was shown in Ref. Han and Wu (2015) that the quantum H theorem still holds for a generalized WvN entropy that does not involve coarse graining. In this work we generalize WvN entropy further so that it applies to any quantum systems including spin systems. We choose the eigenstates of a given observable as a complete set of basis. The generalized WvN (GWvN) entropy for a quantum state is defined with the probability distribution of this state over the chosen basis. We show analytically that the GWvN entropy of almost all quantum states in a Hilbert space of dimension $N$ lies around $\ln N$ with a variance of order $1/N$. This means that when $N$ is large there is typicality for the GWvN entropy. The GWvN entropy can also be readily defined for a mixed state, for which it has a similar typical behavior. When it applies to a system entangled with environment, the GWvN entropy is consistent with the familiar von Neumann entropy. When the quantum state can sample adequately the Hilbert space during its dynamical evolution, the GWvN entropy will typically change quickly from its initial value and saturate around $\ln N$, reminiscent of the quantum H theorem v. Neumann (1929); von Neumann (2010); Han and Wu (2015). In the sense of typicality, the GWvN entropy is additive. Therefore, when it is applied a microcanonical ensemble of $N$ quantum states in an energy shell, the GWvN entropy is not only identical to the Boltzmann entropy but also share its property of being additive. In the end, we show that the Gibbs ensemble can be derived from the GWvN entropy with the maximal entropy principle Jaynes (1957); Neumann (1927).

II Generalized Wigner-von Neumann Entropy and its typicality

In 1929, von Neumann introduced an entropy for pure quantum state but he acknowledged that the idea was from Wigner’s unpublished work v. Neumann (1929); von Neumann (2010). To define the entropy, Wigner and von Neumann chose a pair of macroscopic position and macroscopic momentum operators that commute. Their common eigenstates are wave functions localized on individual Planck cells in the phase space and form a complete set of basis. A wave function is then be mapped unitarily to the phase space with this basis. The resulted probability distribution in the phase space is coarse grained and used to define the WvN entropy. This WvN entropy was generalized in Ref.Han and Wu (2015) with two improvements: (1) there is no more coarse graining; (2) an efficient way is found to compute the basis as a set of Wannier functionsFang et al. (2018b).

Here we choose an operator $A$ whose eigenstates $\left\{\left|\phi_{i}\right\rangle\right\}$ form a complete basis. For a quantum state $\ket{\varphi}$, we use its probability distribution over the basis $\left\{\left|\phi_{i}\right\rangle\right\}$ to define an entropy

[TABLE]

Entropy of this form has appeared in many different contexts and had different names Mirbach and Korsch (1998); Polkovnikov (2011); Santos et al. (2011); Han and Wu (2015); Anzà and Vedral (2017); Safranek et al. (2018); Alterman et al. (2018). In particular, without the minus sign, it was defined as the information of operator $A$ in Ref. Everett III (1973). To our judgement, it is fair to regard Eq. (1) as the generalization of the quantum entropy proposed by Wigner and von Neumann in 1929 v. Neumann (1929); von Neumann (2010). We are to show in this work that this GWvN entropy is independent of choice of $A$ in the sense of typicality.

Due to the normalization rule, all quantum states in a Hilbert space of $N$ dimensions lie on the $\left(2N-1\right)$-dimensional hypersphere $\mathbb{S}^{2N-1}$,

[TABLE]

We are interested in the statistical average, variance, and distribution of the GWvN entropy when the quantum states are sampled on this hypersphere uniformly at random. Let us look at the numerical results first. Shown in Fig. 1(a) are the distributions of the entropies for $N=110,510,5210$. The figure shows that the distribution $P(S)$ becomes narrower and the average gets closer to $\ln N$ quickly as $N$ increases. These two trends are further demonstrated in Fig. 1(b,c). These results indicate that when $N$ is large, which is the usual case for a quantum many-body system, the GWvN entropy is the same for overwhelming majority of the quantum states with a value that is very close to $\ln N$. It is clear that these results are independent of the choice of operator $A$.

We find that the distribution $P(S)$ of the GWvN entropies obtained numerically in Fig. 1 can be fit very well with a function ${\cal P}(s)=d\Pr(s)/ds$, where $\Pr(s)$ is a Fermi-Dirac-like function

[TABLE]

with $s=S/\ln N$ being the normalized GWvN entropy and the parameters

[TABLE]

Here $\Gamma^{\prime}(z)$ is the derivative of gamma function with respect to $z$. As shown in Fig. 1, all the three essential features of the entropy distribution, shape, average value, and variance, are captured very well by ${\cal P}(s)$. The empirical distribution in Eq.(3) is discussed and compared with rigorous results of concentration of measure and Levy’s lemma in Appendix A.

We turn to analytical results, which can be obtained for the average value and the variance of the GWvN entropy over all quantum states on $\mathbb{S}^{2N-1}$. This is achieved with the introduction of an auxiliary function $H_{\lambda}=-\sum_{j}\left|z_{j}\right|^{2\lambda}$ Jones (1990), which is related to the GWvN entropy as

[TABLE]

We average $\left|z_{j}\right|^{2\lambda}$ over all points on $\mathbb{S}^{2N-1}$ and obtain (see Appendix B for details)

[TABLE]

Therefore, the average GWvN entropy reads

[TABLE]

Asymptotically, we have $\mu_{N}=\ln N+\gamma-1+O(1/N)$, where $\gamma\approx 0.577$ is the Euler-Mascheroni constant. This confirms our numerical and fitting results (see Fig. 1(b)). In fact, when $N$ is of order $10^{19}$ (about 63 spins in spin-1/2 model, still a small system), $\langle S\rangle$ is already $0.99\ln N$, less than 1% off the maximal value.

Similarly, with the auxiliary function $H_{\lambda}$, we find the variance of the GWvN entropy according to (7)(38)(39)

[TABLE]

where $\Psi_{1}\left(z\right)=d^{2}\ln\Gamma\left(z\right)/dz^{2}$ is trigamma function. Asymptotically, $\Psi_{1}\left(N+1\right)=1/(N+1)+O(1/N^{2})$. So, the leading term of the variance of GWvN entropy is $\sim 0.3/N$ Mirbach and Korsch (1998). It agrees with the numerical result and the fitting result(see Fig. 1(c)).

All the above results, analytical or numerical, demonstrate clearly that there is typicality for the GWvN entropy when the dimension of the occupied Hilbert space $N$ is large enough. In other words, for a closed many-body quantum system, the GWvN entropy of almost any of its quantum state is very close to $\ln N$ and the exception is very rare. These results are independent of choice of operator $A$ as long as the eigenstates of $A$ forms a complete basis. When $N$ can be regarded as the number of microstates (eg. energy eigenstates) of a quantum system, GWvN entropy is consistent with the well known Boltzmann entropy. We will discuss more the physical implication of these results.

III GWvN entropy of subsystems

The GWvN entropy can be extended to quantum subsystems which are described by mixed states. We consider a closed quantum system that consists of two subsystems. We focus on one of the subsystems and call the other subsystem environment. The whole system is in a pure quantum state $\left|\varphi\right\rangle=\sum_{i,\alpha}c_{i\alpha}\left|\phi_{i},\psi_{\alpha}\right\rangle$ with normalization $\sum_{i,\alpha}\left|c_{i\alpha}\right|^{2}=1$. Here the Roman indices represent the subsystem and the Greek indices the environment. By convention, $\{\phi_{i},i=1,2,\cdots,n\}$ is a complete basis for the subsystem and $\{\psi_{\alpha},\alpha=1,2,\cdots,m\}$ is a complete basis for the environment. Tracing out the environment in the density matrix of the system, $\rho=\sum_{ij,\alpha\beta}c_{i\alpha}c_{j\beta}^{*}\left|\phi_{i},\psi_{\alpha}\right\rangle\left\langle\phi_{j},\psi_{\beta}\right|$ , we obtain the reduced density matrix for the subsystem

[TABLE]

where $p_{ij}=\sum_{\alpha=1}^{m}c_{i\alpha}c_{j\alpha}^{*}$. The definition of GWvN entropy for the subsystem is

[TABLE]

We are interested in the average value and variance of $S(\rho_{s})$ when the quantum state $\left|\varphi\right\rangle$ of the whole system is sampled uniformly over the hypersphere $\mathbb{S}^{2N-1}$. Here $N=nm$.

We introduce another auxiliary function $K_{\lambda}=-\sum_{i=1}^{n}p_{ii}^{\lambda}$, which is related to the entropy as

[TABLE]

Direct computation (see the Appendix B for details) shows that

[TABLE]

When $\lambda=1$, we have $\langle p_{ii}\rangle=1/n$ as expected. This leads to

[TABLE]

With Eq.(11) we have the average of the GWvN entropy

[TABLE]

where the approximation is asymptotic and it holds when both $n$ and $m$ are very large. The variance of $S(\rho_{s})$ can also be derived analytically. With Eqs. (14,38,39), we have

[TABLE]

When $N>m\gg 1$, we have

[TABLE]

Here we have used the condition $m>n$, which is usually the case. This shows that the variance of a subsystem’s entropy is effectively controlled by the environment and the whole system. The reason is that we are averaging over the Hilbert space of the whole system, where the overwhelming majority of the quantum states are almost maximally entangled Page (1993). If the subsystem and the environment are not entangled, then the subsystem can be regarded as an isolated system, which is already discussed in the last section. It is clear from these results that the GWvN entropy of a subsystem has typicality.

Similarly, the GWvN entropy of the environment is typical with the average being $\left\langle S\left(\rho_{e}\right)\right\rangle\approx\ln m$ and the variance of order $1/(nN)$.

It is interesting to compare these results on mixed states with the familiar von Neumann entropy. For this purpose, we assume that $m\gg n$ since the environment should usually be much larger than the subsystem. Page computed the average value of the von Neumann entropy $S_{v}(\rho_{s})=-\text{Tr}\rho_{s}\ln\rho_{s}$ when the quantum state $\ket{\varphi}$ is sampled randomly Page (1993). He found that the average value is approximately $\ln n-n/(2m)$, which is consistent with the GWvN entropy asymptotically. Page did not compute the variance of $S_{v}(\rho_{s})$. This shows that the GWvN entropy for a system with a large environment is consistent with the von Neumann entropy, which was already noticed in Ref. Han and Wu (2015). However, for the environment, its von Neumann entropy $S_{v}(\rho_{e})=S_{v}(\rho_{s})\approx\ln n$, very different from its GWvN entropy $\ln m$.

Strictly, the GWvN entropy is not additive. However, in the sense of typicality, it is additive. This is evident in the following result

[TABLE]

When $N$, $m$, $n$ are large, we can safely ignore the constant $1-\gamma$ and therefore have $\braket{S(\rho_{s})}+\braket{S(\rho_{e})}=\braket{S(\rho)}$. The result can be easily generalized to multipartite systems where each subsystem has a Hilbert space of large dimension.

The dimension $N$ of the Hilbert space in the above discussion should be regarded as the dimension of a sub-space that is physically relevant. As an example and also as an important application, let us consider quantum microcanonical ensemble, which is characterized by an energy shell $[E,E+\Delta E]$ with large but finite number of energy eigenstates  Landau and Lifshitz (1980); Huang (1987). Suppose that $N$ is the number of energy eigenstates in the shell. According to the above results, the GWvN entropy of any quantum state (pure or mixed) in the microcanonical ensemble is typically $\ln N$ with a very small variance. This is consistent with the well known Boltzmann entropy.

In the above discussion, we have fixed the basis $\left\{\left|\phi_{i}\right\rangle\right\}$ (or operator $A$) while sampling the quantum states in the Hilbert space. It is clear that we can equivalently fix the quantum state while sampling all possible bases. In fact, this is exactly what von Neumann did in his 1929 paperv. Neumann (1929); von Neumann (2010).

IV Dynamic evolution of GWvN entropy

Our results for the GWvN entropy so far are kinematic and have nothing to do with the Hamiltonian of a quantum system. In this section, we investigate how the GWvN entropy evolves dynamically and find that the GWvN entropy has dynamical typicality, similar to the dynamical behavior of observables found in Ref.  Bartsch and Gemmer (2009); Reimann (2018). For an isolated quantum system with a set of energy eigenstates $\ket{E_{j}}$, its dynamical evolution is given by

[TABLE]

where $a_{j}=\braket{E_{j}}{\Psi(0)}$ is determined by the initial quantum state $\ket{\varphi(0)}$. In general, because $a_{j}$ approaches quickly to zero as $j\rightarrow\infty$, there are only a finite number of energy eigenstates occupied. During the dynamical evolution, this number of occupied states does not change as $a_{j}$ is independent of time. This means that when a quantum system evolves dynamically, it dynamical path in the Hilbert space will lie entirely in this sub-Hilbert space of occupied states.

Let us consider a quantum system with a macroscopic number of particles. In this case, the dimension $N$ of the sub-Hilbert space of occupied states is in general enormously large. It is natural to expect that almost all the quantum states on the dynamical path are typical and their GWvN entropies are very close to $\ln N$ with small fluctuations. This is exactly what is implied in the quantum H theorems proved in Ref.v. Neumann (1929); von Neumann (2010); Han and Wu (2015). However, according to these proofs, there are exceptions that occur when the system’s Hamiltonian has a great deal of degeneracy in its eigen-energies and eigen-energy differences. These degeneracies are shown to be closely connected to the integrability of the Hamiltonians Zhang et al. (2016). In other words, when the system is integrable, its quantum dynamics will be restricted by various good quantum numbers and can not adequately sample the sub-Hilbert space. When the system is non-integrable, its quantum dynamics can adequately sample the sub-Hilbert space so that the quantum states involved in the dynamics are typical.

We now show how the GWvN entropy relaxes dynamically in a quantum chaotic system. We choose a complete orthonormal basis, $\left\{\left|\psi_{0}\right\rangle,\left|\xi_{i}\right\rangle\right\}(i=1,\cdots,N-1)$, where $\left|\psi_{0}\right\rangle$ is the initial state. In this case, the GWvN entropy is zero initially. As the system evolves under unitary operator $U\left(t\right)=\exp\left(-iHt/\hbar\right)$, its GWvN entropy changes with time as

[TABLE]

where $p\left(t\right)=\left|\langle\psi\left|\psi\left(t\right)\right\rangle\right|^{2}$ is the surviving probability (also called fidelity) Flambaum and Izrailev (2001a); Torres-Herrera et al. (2014) and the overline denotes averaging over the other $(N-1)$ basis $\left\{\left|\xi_{i}\right\rangle\right\}$. The averaging is justified by that the system is quantum chaotic and its dynamics can sample adequately in the subspace spanned by $\left\{\left|\xi_{i}\right\rangle\right\}$.

Note that $\left|\langle\psi\left(t\right)\left|\xi_{i}\right\rangle\right|^{2}=\left(1-p\left(t\right)\right)\cdot\left|\langle\psi^{\prime}\left(t\right)\left|\xi_{i}\right\rangle\right|^{2}$, where $\left|\psi^{\prime}\left(t\right)\right\rangle$ is the normalized state projected by $\left|\psi\left(t\right)\right\rangle$ onto the subspace spanned by $\left\{\left|\xi_{i}\right\rangle\right\}$. Similar to Eq. (6), we have

[TABLE]

Eventually we find

[TABLE]

where $f(p)=-p\log p-(1-p)\log(1-p)$. The deviation is negligible with the order $O(1/N)$ by typicality arguments. More precisely, it is a direct result from (8). A very interesting fact is that this expression of $S\left(t\right)$ explicitly verifies the validity of the conjectured form proposed by Flambaum and Izrailev Flambaum and Izrailev (2001b).

V connection to thermodynamics

The GWvN entropy can also be used to derive the Gibbs ensemble with the maximal entropy principle Jaynes (1957); Neumann (1927). We choose the operator $A=H$ and its eigenstates $\ket{E_{i}}$ form a complete basis. For a typical quantum state $\rho$, its GWvN entropy is $S=-\sum_{i}\text{Tr}\left(\rho P_{E_{i}}\right)\ln\text{Tr}\left(\rho P_{E_{i}}\right)$, where $P_{E_{i}}=\ket{E_{i}}\bra{E_{i}}$. We want to maximize it under the condition that \left\langle H\right\rangle=\text{Tr\left(\rho H\right)} is constant. Mathematically, this can be done as

[TABLE]

with the restriction $\text{Tr}\left(\Omega H\right)=0$ and $\text{Tr}\left(\Omega\right)=0$. This leads to

[TABLE]

It can be inferred that $\sum_{i}\ln\left(\text{Tr}\left(\rho P_{E_{i}}\right)\right)P_{E_{i}}$ is linear function of $H$. Therefore the density matrix has the general form

[TABLE]

where $\alpha$ is normailzation factor and $\beta$ is inverse temperature. Explicitly we have

[TABLE]

From standard textbook in statistical physics Landau and Lifshitz (1980); Huang (1987), Eq.(24) represents the well known Gibbs ensemble. And from (25) we have

[TABLE]

where the temperature $T=1/\beta$. This is exactly the first law of thermodynamics. The similar derivations can be done for general operator $A$ and its eigenstates $\ket{\phi_{i}}$ with the restriction of fixed $\left\langle A\right\rangle$, then we have the generalized Gibbs ensemble $\rho=\alpha\exp\left(-\lambda_{A}A\right)$, where $\alpha$ is normailzation factor and $\lambda_{A}$ is the generalized inverse temperature. With the Gibbs ensemble, other thermodynamical relations can also be easily derived. Our result here is consistent with previous results in Ref. Goldstein et al. (2006); Anzà and Vedral (2017); Anza et al. (2018).

VI Conclusion

We have generalized the quantum entropy proposed by Wigner and von Neumann in 1929. Although the definition uses a specific complete basis, the generalized Wigner-von Neumann entropy becomes typical when the dimension $N$ of the Hilbert space is large. We have shown analytically that when we sample a quantum state uniformly at random in the Hilbert space, the average of its GWvN entropy is asymptotic to $\ln N$ and its variance is of order $1/N$. As a result, when the GWvN entropy is applied to microcanonical ensemble, it is equivalent to the Boltzmann entropy. When it is extended to a subsystem with large environment, it is consistent with the von Neumann entropy. In the end, with the maximal entropy principle we have shown that the GWvN entropy can be used to obtain the Gibbs ensemble from which all thermodynamic relations can be derived.

Acknowledgements.

We thank stimulating discussion with Chushun Tian. This work was supported by the The National Key Research and Development Program of China (Grants No. 2017YFA0303302, No. 2018YFA0305602) . Z. H. and Z. W. contributed equally to this work.

Appendix A Concentration of measure and Levy’s lemma

Concentration of measure states that a function that depends in a Lipschitz way on many independent variables is almost constant. As a special form of concentration of measure, Levy’s lemma is commonly used in the typicality related literature Hayden et al. (2006); Singh et al. (2016); Fang et al. (2018a). Here we compare the distribution of the GWvN entropy $S$ Eq.(3) to Levy’s lemma. According to Levy’s lemma, for entropy $S(\varphi)$, where $\varphi$ is a random point drawn uniformly from hypersphere $\mathbb{S}^{2N-1}$, the upper bound of the deviations from expected value $\langle S\rangle$ is given by

[TABLE]

where $\eta=\sup\left|\nabla_{\varphi}S\right|$ is Lipschitz constant. Straightforward computation shows that

[TABLE]

The last inequality is true for dimension $N>e^{2}$, and the equality holds only when those $|z_{j}|^{2}$ are all the same. This means that \Pr\big{(}|S-\langle S\rangle|\geq\delta\big{)} has a sub-gaussian tail, which presents stronger convergence than a Poisson-like tail in the distribution of Eq.(3). However, the GWvN entropy is far away from zero only in a small region given by \big{|}S-\mu_{N}\big{|}\sim\ln N/c_{N}\sim N^{-1/2}. In this region our distribution is very accurate as indicated in Fig. 1(a).

Moreover, the variance of our distribution Eq.(3) is consistent with Levy’s lemma. Among all the distributions that satisfy Levy’s lemma, the one that has the the maximal variance should be given by

[TABLE]

where $f_{N}=2N/9\pi^{3}\eta^{2}$ is the factor in exponent of Eq.(27). It leads to the maximal variance that Levy’s lemma allows

[TABLE]

The variance (of order $1/N$) of our distribution gives a tighter bound than the above variance bound (of order $\eta^{2}/N$, namely $\ln^{2}N/N$).

Appendix B Averaging in Hilbert space

In this Appendix, we give out the derivation details that are needed in the main text, in particular, the derivations related to Eqs.(6,8,12,15).

Any quantum state $\ket{\varphi}$ in an $N$-dimensional Hilbert space can be expanded in an orthonormal basis with coefficients $z_{1},\dotsm,z_{N}$. $z_{i}$ is usually a complex number and we denote the real part and imaginary part as $x_{2i-1}$ and $x_{2i}$. Due to the normalization, the quantum state $\ket{\varphi}$ corresponds to point $\left(x_{1},x_{2},\dotsm,x_{2N}\right)$ on the $\left(2N-1\right)$-dimensional hypersphere $\mathbb{S}^{2N-1}$.

The average of a function in the $N$-dimensional Hilbert space is to integrate it on the hypersphere $\mathbb{S}^{2N-1}$. We can connect the integration on the sphere with an integration in the ballBaker (1997); Folland (2001). More precisely, let $B^{n}=\left\{x\in R^{n}\mathrel{\mathop{\ordinarycolon}}\left|x\right|\leq 1\right\}$ , suppose that $f\mathrel{\mathop{\ordinarycolon}}B^{n}\rightarrow R$ is continuous, then

[TABLE]

where $d\sigma_{n-1}(s)$ denotes an element on hypersphere $S^{n-1}$. If $f(rx)=r^{\alpha}f(x)$ is a homogeneous function of degree $\alpha$, we have

[TABLE]

Consider a function

[TABLE]

where $N=nm$ and

[TABLE]

Then the average $\left\langle g\right\rangle$ of $g\left(x_{1},\cdots,x_{2N}\right)$ on sphere $\mathbb{S}^{2N-1}$ reads

[TABLE]

where $\varOmega_{2N-1}=\frac{2\pi^{N}}{\Gamma(N)}$ is the surface area of $\mathbb{S}^{2N-1}$.

It is useful to calculate some derivatives of special forms of $g\left(x_{1},\cdots,x_{2N}\right)=\prod_{k=1}^{n}T_{k}^{\alpha_{k}}$.

[TABLE]

[TABLE]

[TABLE]

Here $\Psi(z)=\Gamma^{\prime}(z)/\Gamma(z)$ and $\Psi_{1}(z)=d^{2}\ln\Gamma\left(z\right)/dz^{2}$ are digamma function and trigamma function, respectively.