Inequalities for quantum thermodynamics in infinite space

Document Type : Full length research Paper


1 Department of mathematics, University of Zanjan, Iran

2 Department of physics, University of Zanjan


In statistical mechanics, the upper limit of entropy is very important for determining the final state of the system based on the Helmholtz variational principle. Hence many attempts have been made to calculate the entropy of the system, and a thermodynamic theory based on Renyi entropy has recently been presented that can describe new states in the thermodynamic system. The exact determination of entropy can not be done in many cases, and therefore approximate methods are used. An approximate solution often involves obtaining a high limit for entropy, which determines the final state of the system. This paper presents an upper limit for quantum entropy. For this purpose, the calculations are carried out in a separable Hilbert space with orthogonal bases. Using the Shanon definition of entropy, the upper limit is calculated for the relative entropy of two commutative operators.


[1] D.A. Edwards, The Mathematical Foundations of Quantum Mechanics, Princeton university press, (1979). ISBN: 9780691178561
##[2] A. Rényi, On measures of entropy and information, Hungarian Academy of Sciences, Budapest Hungary, (1961).
##[3] N. Bebiano, J. da Providência, J.P. da Providência, Renyi's quantum thermodynamical Inequalities, The Electronic Journal of Linear Algebra, 33 (2017) 63-73.  
##[4] A. Misra, U. Singh, M.N. Bera, A. Rajagopal, Quantum Rényi relative entropies affirm universality of thermodynamics, Physical Review E, 92(4) (2015) 042161.
##[5] N. Bebiano, R. Lemos, J. Da Providência, Inequalities for quantum relative entropy, Linear Algebra and its Applications, 401 (2005) 159-72.
##[6] F. Hiai, Equality cases in matrix norm inequalities of Golden-Thompson type, Linear and Multilinear Algebra, 36(4) (1994) 239-49.
##[7] F. Hiai, D. Petz, The Golden-Thompson trace inequality is complemented. Linear Algebra and its Applications, 181 (1993) 153-85.
##[8] T. Ando, F. Hiai, Log majorization and complementary Golden-Thompson type inequalities, Linear Algebra and its Applications, 197 (1994) 113-31.
##[9] G.J. Murphy, C*-algebras and operator theory, Academic press (2014).
##[10] S. Power, Another proof of Lidskii's theorem on the trace, Bulletin of the London Mathematical Society, 15(2) (1983) 146-8.
##[11] C. Shannon, W. Weaver, The mathematical theory of communication, University of illinois, Urbana, (1949).
##[12] H.F. Trotter, On the product of semi-groups of operators, Proceedings of the American Mathematical Society, 10 (1959) 545-551.