Facchi Paolo, Gramegna Giovanni, Konderak Arturo
Dipartimento di Fisica and MECENAS, Università di Bari, I-70126 Bari, Italy.
Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy.
Entropy (Basel). 2021 May 21;23(6):645. doi: 10.3390/e23060645.
Given the algebra of observables of a quantum system subject to selection rules, a state can be represented by different density matrices. As a result, different von Neumann entropies can be associated with the same state. Motivated by a minimality property of the von Neumann entropy of a density matrix with respect to its possible decompositions into pure states, we give a purely algebraic definition of entropy for states of an algebra of observables, thus solving the above ambiguity. The entropy so-defined satisfies all the desirable thermodynamic properties and reduces to the von Neumann entropy in the quantum mechanical case. Moreover, it can be shown to be equal to the von Neumann entropy of the unique representative density matrix belonging to the operator algebra of a multiplicity-free Hilbert-space representation.
对于受选择规则约束的量子系统的可观测量代数，一个态可以由不同的密度矩阵表示。因此，不同的冯·诺依曼熵可以与同一个态相关联。受密度矩阵的冯·诺依曼熵相对于其可能分解为纯态的极小性性质的启发，我们给出了可观测量代数态的熵的纯代数定义，从而解决了上述模糊性。如此定义的熵满足所有理想的热力学性质，并且在量子力学情形下简化为冯·诺依曼熵。此外，可以证明它等于属于无重数希尔伯特空间表示的算子代数的唯一代表性密度矩阵的冯·诺依曼熵。
