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Strong subadditivity of quantum entropy

In quantum information theory, strong subadditivity of quantum entropy (SSA) is the relation among the von Neumann entropies of various quantum subsystems of a larger quantum system consisting of three subsystems (or of one quantum system with three degrees of freedom). It is a basic theorem in modern quantum information theory. It was conjectured by D. W. Robinson and D. Ruelle[1] in 1966 and O. E. Lanford III and D. W. Robinson[2] in 1968 and proved in 1973 by E.H. Lieb and M.B. Ruskai,[3] building on results obtained by Lieb in his proof of the Wigner-Yanase-Dyson conjecture.[4]

The classical version of SSA was long known and appreciated in classical probability theory and information theory. The proof of this relation in the classical case is quite easy, but the quantum case is difficult because of the non-commutativity of the reduced density matrices describing the quantum subsystems.

Some useful references here include:

  • "Quantum Computation and Quantum Information"[5]
  • "Quantum Entropy and Its Use"[6]
  • Trace Inequalities and Quantum Entropy: An Introductory Course[7]
  1. ^ Robinson, Derek W.; Ruelle, David (1967). "Mean entropy of states in classical statistical mechanics". Communications in Mathematical Physics. 5 (4). Springer Science and Business Media LLC: 288–300. Bibcode:1967CMaPh...5..288R. doi:10.1007/bf01646480. ISSN 0010-3616. S2CID 115134613.
  2. ^ Lanford, Oscar E.; Robinson, Derek W. (1968). "Mean Entropy of States in Quantum‐Statistical Mechanics". Journal of Mathematical Physics. 9 (7). AIP Publishing: 1120–1125. Bibcode:1968JMP.....9.1120L. doi:10.1063/1.1664685. ISSN 0022-2488.
  3. ^ Lieb, Elliott H.; Ruskai, Mary Beth (1973). "Proof of the strong subadditivity of quantum‐mechanical entropy" (PDF). Journal of Mathematical Physics. 14 (12). AIP Publishing: 1938–1941. Bibcode:1973JMP....14.1938L. doi:10.1063/1.1666274. ISSN 0022-2488.
  4. ^ Lieb, Elliott H (1973). "Convex trace functions and the Wigner-Yanase-Dyson conjecture". Advances in Mathematics. 11 (3): 267–288. doi:10.1016/0001-8708(73)90011-X. ISSN 0001-8708.
  5. ^ M. Nielsen, I. Chuang, Quantum Computation and Quantum Information, Cambr. U. Press, (2000)
  6. ^ M. Ohya, D. Petz, Quantum Entropy and Its Use, Springer (1993)
  7. ^ E. Carlen, Trace Inequalities and Quantum Entropy: An Introductory Course, Contemp. Math. 529 (2009).

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