Entropy Inequalities
Some inequalities and relations among entropies of reduced quantum mechanical density matrices are discussed and proved. While these are not as strong as those available for classical systems they are nonetheless powerful enough to establish the existence
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© by Springer-Verlag 1970
Entropy Inequalities HUZIHIRO ARAKI
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan ELLIOTT
H. LIEB*
Department of Mathematics Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA Received March 2, 1970
Abstract. Some inequalities and relations among entropies of reduced quantum mechanical density matrices are discussed and proved. While these are not as strong as those available for classical systems they are nonetheless powerful enough to establish the existence of the limiting mean entropy for translationally invariant states of quantum continuous systems.
I. Introduction
In this note we shall be concerned with inequalities satisfied by the entropies of reduced density matrices. We begin with some definitions and a statement of our main Theorem 1. Section II contains the proof of the main theorem when the dimension is finite. Section III contains some other inequalities that can be derived from Theorem 1 by application of certain transformations. Section IV contains the proof of the main theorem when the dimension is infinite. Section V deals with the application of our theorem to the existence of the mean entropy for trans lationally invariant states of a quantum continuous system. Definition 1. A density matrix, g, on a Hilbert space, H, is a self adjoint non-negative trace class operator on H whose trace is unity. Definition 2. If (} is a density matrix,
S(g) = - Trg lng
(1.1)
is the entropy associated with g.
Since 0 ~ g ~ 1, we have - e- 1 ~ g lng ~ 0 and ('Pj' (g In(}) 'Pj) ~ 0 for any 'Pj. Hence
S= - I('Pj'(} In(} 'P)
(1.2)
j
* Work supported by National Science Foundation Grant G P-9414.
M. Loss et al. (eds.), Inequalities © Springer-Verlag Berlin Heidelberg 2002
47
With H. Araki in Commun. Math. Phys. 18, 160--170 (1970) Entropy Inequalities
161
exists for any orthonormal basis {1p j} and 0 ~ S ~ + 00. If S < 00 for one basis {1p j}, then the nonnegative operator - (} In (} is in the trace class and hence S is independent of the basis {1pj} and is finite for all {1pJ. Otherwise S must be + 00. Therefore (1.2) does not depend on the orthonormal basis {1pj} and defines the right hand side of(1.1). Definition 3. If (}12 is a density matrix on H1 ® H2 then (}1 , the reduced density matrix, is a density matrix on H1 defined by
(}1
=
Tr2 (}12 .
(1.3)
Here Tr2 means the partial trace defined by (x, (}ly)= I(x®e j ,(}12[y®eJ), j
where {eJ is any complete orthonormal basis in H2 and x, y E H1 . Notation. If (}12 is a density matrix on H1®H2 then we will denote S((}12) by S12 and S((}l) by Sl. A theorem that is true classically [lJ (meaning that all relevant density matrices commute) is the following: (1.4)
We believe that (1.4) is true quantum mechanically as has been conjectured by Lanford and Robinson [2J, but have been unable to prove it. We can, however, prove the following which is as good for some applications. Theorem 1. Let (}123 be a density matrix on H1 ®H2®H3. Then
S123 Furthermore,
~
S12
+ S23
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