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在量子统计力学中，以科学家约翰·冯·纽曼命名的冯纽曼熵是古典吉布士熵在量子力学领域的延伸。对于由密度矩阵ρ描述的量子力学系统，冯·诺依曼熵是

$S=-\mathrm {tr} (\rho \ln \rho ),$

当$\mathrm {tr}$ 代表 迹数 ， ln 代表一个自然对数矩阵。当ρ用代表自身的特征值和特征向量 |1〉, |2〉, |3〉, 可以写成以下形式：

$\rho =\sum _{j}\eta _{j}\left|j\right\rangle \left\langle j\right|~,$

进而可以将“冯纽曼熵”改写成以下形式：

$S=-\sum _{j}\eta _{j}\ln \eta _{j}.$

在这个形式中，S可视为信息论中的熵。

背景

约翰·冯·诺伊曼于1932年所出版的著作《量子力学的数学基础》中建立了一套严谨的数学架构来计算量子力学。他在书中提供一套量测的理论，波函数的塌缩在其中被视为一种不可逆的过程。

John von Neumann established a rigorous mathematical framework for quantum mechanics in his 1932 work Mathematical Foundations of Quantum Mechanics. In it, he provided a theory of measurement, where the usual notion of wave-function collapse is described as an irreversible process (the so-called von Neumann or projective measurement).

这边会使用密度矩阵来进行计算。约翰·冯·诺伊曼和朗道不约而同都使用了密度矩阵进行运算，但两人使用的动机并不相同。给予朗道灵感的是，不可能用状态向量来描述复合量子系统的子系统这件事。而约翰·冯·诺伊曼使用密度矩阵是为了发展量子统计力学和量子测量理论。

The density matrix was introduced, with different motivations, by von Neumann and by Lev Landau. The motivation that inspired Landau was the impossibility of describing a subsystem of a composite quantum system by a state vector. On the other hand, von Neumann introduced the density matrix in order to develop both quantum statistical mechanics and a theory of quantum measurements.

发展密度矩阵是为了将古典统计力学的计算工具拓展到量子的领域。在古典物理架构之下，为了计算一个系统中与热力学有关的物理量，必须先计算一个系统的配分函数。冯纽曼熵以希尔伯特空间作为基础，使用密度函数来描述状态，并在希尔伯特空间中进行计算。统计密度矩阵计算的知识可以产生一种，与原先的计算方式不同，但是概念上相同的计算方法。假设有一个波函数|Ψ〉 ，这个波函数借由一组量子数n₁, n₂, ..., n_N来决定。

The density matrix formalism was developed to extend the tools of classical statistical mechanics to the quantum domain. In the classical framework we compute the partition function of the system in order to evaluate all possible thermodynamic quantities. Von Neumann introduced the density matrix in the context of states and operators in a Hilbert space. The knowledge of the statistical density matrix operator would allow us to compute all average quantities in a conceptually similar, but mathematically different way. Let us suppose we have a set of wave functions |Ψ〉 that depend parametrically on a set of quantum numbers n₁, n₂, ..., n_N. The natural variable which we have is the amplitude with which a particular wavefunction of the basic set participates in the actual wavefunction of the system. Let us denote the square of this amplitude by p(n₁, n₂, ..., n_N). The goal is to turn this quantity p into the classical density function in phase space. We have to verify that p goes over into the density function in the classical limit, and that it has ergodic properties. After checking that p(n₁, n₂, ..., n_N) is a constant of motion, an ergodic assumption for the probabilities p(n₁, n₂, ..., n_N) makes p a function of the energy only.

After this procedure, one finally arrives at the density matrix formalism when seeking a form where p(n₁, n₂, ..., n_N) is invariant with respect to the representation used. In the form it is written, it will only yield the correct expectation values for quantities which are diagonal with respect to the quantum numbers n₁, n₂, ..., n_N.

Expectation values of operators which are not diagonal involve the phases of the quantum amplitudes. Suppose we encode the quantum numbers n₁, n₂, ..., n_N into the single index i or j. Then our wave function has the form

$\left|\Psi \right\rangle \,=\,\sum _{i}a_{i}\,\left|\psi _{i}\right\rangle .$

The expectation value of an operator B which is not diagonal in these wave functions, so

$\left\langle B\right\rangle \,=\,\sum _{i,j}a_{i}^{*}a_{j}\,\left\langle i\right|B\left|j\right\rangle .$

The role which was originally reserved for the quantities $\left|a_{i}\right|^{2}$ is thus taken over by the density matrix of the system S.

$\left\langle j\right|\,\rho \,\left|i\right\rangle \,=\,a_{j}\,a_{i}^{*}.$

Therefore, 〈B〉 reads

$\left\langle B\right\rangle \,=\,\mathrm {tr} (\rho \,B)~.$

The invariance of the above term is described by matrix theory. A mathematical framework was described where the expectation value of quantum operators, as described by matrices, is obtained by taking the trace of the product of the density operator ${\hat {\rho }}$ and an operator ${\hat {B}}$ (Hilbert scalar product between operators). The matrix formalism here is in the statistical mechanics framework, although it applies as well for finite quantum systems, which is usually the case, where the state of the system cannot be described by a pure state, but as a statistical operator ${\hat {\rho }}$ of the above form. Mathematically, ${\hat {\rho }}$ is a positive-semidefinite Hermitian matrix with unit trace.

定义

Given the density matrix ρ, von Neumann defined the entropy as

$S(\rho )\,=\,-\mathrm {tr} (\rho \,{\rm {\ln }}\rho ),$

which is a proper extension of the Gibbs entropy (up to a factor k_B) and the Shannon entropy to the quantum case. To compute S(ρ) it is convenient (see logarithm of a matrix) to compute the Eigendecomposition of $~\rho =\sum _{j}\eta _{j}\left|j\right\rangle \left\langle j\right|$. The von Neumann entropy is then given by

$S(\rho )\,=\,-\sum _{j}\eta _{j}\ln \eta _{j}~.$

Since, for a pure state, the density matrix is idempotent, ρ = ρ², the entropy S(ρ) for it vanishes. Thus, if the system is finite (finite-dimensional matrix representation), the entropy S(ρ) quantifies the departure of the system from a pure state. In other words, it codifies the degree of mixing of the state describing a given finite system. Measurement decoheres a quantum system into something noninterfering and ostensibly classical; so, e.g., the vanishing entropy of a pure state $\Psi =(\left|0\right\rangle +\left|1\right\rangle )/{\sqrt {2}}$, corresponding to a density matrix

$\rho ={1 \over 2}{\begin{pmatrix}1&1\\1&1\end{pmatrix}}$

increases to $S=\ln 2=0.69$ for the measurement outcome mixture

$\rho ={1 \over 2}{\begin{pmatrix}1&0\\0&1\end{pmatrix}}$

as the quantum interference information is erased.

性质

Some properties of the von Neumann entropy:

• S(ρ) is zero if and only if ρ represents a pure state.
• S(ρ) is maximal and equal to ln N for a maximally mixed state, N being the dimension of the Hilbert space.
• S(ρ) is invariant under changes in the basis of ρ, that is, S(ρ) = S(UρU^†), with U a unitary transformation.
• S(ρ) is concave, that is, given a collection of positive numbers λ_i which sum to unity ($\Sigma _{i}\lambda _{i}=1$) and density operators ρ_i, we have

$S{\bigg (}\sum _{i=1}^{k}\lambda _{i}\,\rho _{i}{\bigg )}\,\geq \,\sum _{i=1}^{k}\lambda _{i}\,S(\rho _{i}).$

• S(ρ) is additive for independent systems. Given two density matrices ρ_A , ρ_B describing independent systems A and B, we have

$S(\rho _{A}\otimes \rho _{B})=S(\rho _{A})+S(\rho _{B})$.

• S(ρ) is strongly subadditive for any three systems A, B, and C:

$S(\rho _{ABC})+S(\rho _{B})\leq S(\rho _{AB})+S(\rho _{BC}).$

This automatically means that S(ρ) is subadditive:

$S(\rho _{AC})\leq S(\rho _{A})+S(\rho _{C}).$

Below, the concept of subadditivity is discussed, followed by its generalization to strong subadditivity.

次可加性

If ρ_A, ρ_B are the reduced density matrices of the general state ρ_AB, then

$\left|S(\rho _{A})\,-\,S(\rho _{B})\right|\,\leq \,S(\rho _{AB})\,\leq \,S(\rho _{A})\,+\,S(\rho _{B})~.$

This right hand inequality is known as subadditivity. The two inequalities together are sometimes known as the triangle inequality. They were proved in 1970 by Huzihiro Araki and Elliott H. Lieb. While in Shannon's theory the entropy of a composite system can never be lower than the entropy of any of its parts, in quantum theory this is not the case, i.e., it is possible that S(ρ_AB) = 0, while S(ρ_A) = S(ρ_B) > 0.

Intuitively, this can be understood as follows: In quantum mechanics, the entropy of the joint system can be less than the sum of the entropy of its components because the components may be entangled. For instance, as seen explicitly, the Bell state of two spin-½s,

$\left|\psi \right\rangle =\left|\uparrow \downarrow \right\rangle +\left|\downarrow \uparrow \right\rangle ,$

is a pure state with zero entropy, but each spin has maximum entropy when considered individually in its reduced density matrix. The entropy in one spin can be "cancelled" by being correlated with the entropy of the other. The left-hand inequality can be roughly interpreted as saying that entropy can only be cancelled by an equal amount of entropy.

If system A and system B have different amounts of entropy, the smaller can only partially cancel the greater, and some entropy must be left over. Likewise, the right-hand inequality can be interpreted as saying that the entropy of a composite system is maximized when its components are uncorrelated, in which case the total entropy is just a sum of the sub-entropies. This may be more intuitive in the phase space formulation, instead of Hilbert space one, where the Von Neumann entropy amounts to minus the expected value of the ★-logarithm of the Wigner function, −∫ f ★ log_★ f  dx dp, up to an offset shift. Up to this normalization offset shift, the entropy is majorized by that of its classical limit.

强次可加性

主条目：Strong Subadditivity of Quantum Entropy

The von Neumann entropy is also strongly subadditive. Given three Hilbert spaces, A, B, C,

$S(\rho _{ABC})\,+\,S(\rho _{B})\,\leq \,S(\rho _{AB})\,+\,S(\rho _{BC}).$

This is a more difficult theorem and was proved in 1973 by Elliott H. Lieb and Mary Beth Ruskai, using a matrix inequality of Elliott H. Lieb proved in 1973. By using the proof technique that establishes the left side of the triangle inequality above, one can show that the strong subadditivity inequality is equivalent to the following inequality.

$S(\rho _{A})\,+\,S(\rho _{C})\,\leq \,S(\rho _{AB})\,+\,S(\rho _{BC})$

when ρ_AB, etc. are the reduced density matrices of a density matrix ρ_ABC. If we apply ordinary subadditivity to the left side of this inequality, and consider all permutations of A, B, C, we obtain the triangle inequality for ρ_ABC: Each of the three numbers S(ρ_AB), S(ρ_BC), S(ρ_AC) is less than or equal to the sum of the other two.

应用

The von Neumann entropy is being extensively used in different forms (conditional entropies, relative entropies, etc.) in the framework of quantum information theory. Entanglement measures are based upon some quantity directly related to the von Neumann entropy. However, there have appeared in the literature several papers dealing with the possible inadequacy of the Shannon information measure, and consequently of the von Neumann entropy as an appropriate quantum generalization of Shannon entropy. The main argument is that in classical measurement the Shannon information measure is a natural measure of our ignorance about the properties of a system, whose existence is independent of measurement.

Conversely, quantum measurement cannot be claimed to reveal the properties of a system that existed before the measurement was made. This controversy has encouraged some authors to introduce the non-additivity property of Tsallis entropy (a generalization of the standard Boltzmann–Gibbs entropy) as the main reason for recovering a true quantum information measure in the quantum context, claiming that non-local correlations ought to be described because of the particularity of Tsallis entropy.

相关条目

• Entropy (information theory)
• Linear entropy
• Partition function (mathematics)
• Quantum conditional entropy
• Quantum mutual information
• Quantum entanglement
• Strong subadditivity of quantum entropy
• Wehrl entropy

参考文献

Category:Quantum mechanical entropy Category:John von Neumann