Proof of the strong subadditivity of quantum-mechanical entropy

Abstract

We prove several theorems about quantum-mechanical entropy, in particular, that it is strongly subadditive.with A = P123 and B = exp(-1np2 + Inp12 + InP23)' One finds F(P123) '" 8 123 + 8 2 -8 12 -8 23 "" Tr123 [exp(lnpI2 -lnP2 + Inp23) -P123]'

Inequalities for the ranks of multipartite quantum states

“…Although many proofs of strong subadditivity are known (e.g. [13,[19][20][21]), none provides any intuition regarding the ranks we consider. Furthermore, the argument we have employed is limited by the fact that by a local invertible filtering one can make only one reduced state equal to the maximally mixed state; indeed, if it were possible to find, for given |ψ ABCD , another state |ψ ′ ABCD with the same Schmidt ranks for all bipartite partitions, but such that ψ ′ A and ψ ′ D are maximally mixed on their respective subsystems, then strong subadditivity (3) and weak monotonicity (4) would hold for S 0 -and we know that already to be false.…”

“…Given a multipartite system, how do the entropies of the different subsystems relate to each other? For the case of the von Neumann entropy, S(ρ) = − Tr ρ log ρ, they must satisfy the well-known strong subadditivity and weak monotonicity relations [13] …”

A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy

“…(That is, the subscripts indicate the spaces "remaining" after the traces.) Then differentiating in p at p = q = 1, we obtain the strong subadditivity of the quantum entropy [14], [12]:…”

“…That is, when computing Ψ p,q of a positive operator on H 1 ⊗ (H 2 ⊗ C 2 ), H 1 is the first factor, and H 2 ⊗ C 2 is the second factor. We then define 14) with A and A related as in (1.13) Notice that if A happens to be self adjoint, and A = B − C where B and C are positive, then…”

“…Another was a certain subadditivity property, and this has only very recently been shown to be false by Hansen [7]. The hope that subadditivity would hold under additional assumptions was shown to be false by Seiringer [21] The general case Tr(A p KA q K * ) with p, q > 0, p + q ≤ 1 (and K not necessarily self-adjoint) was done in [12] in connection with a proof of strong subadditivity of entropy [14]. This concavity result was the impetus to several papers and several different proofs.…”