1. Introduction
Let
be the algebra of all bounded linear operators on a Hilbert space
, and let tr be the canonical trace on
. For a state
ρ on
represented by a density matrix
D, its von Neumann entropy is defined by
Now, let
H be the Hamiltonian of a physical system whose (bounded) observables are represented by
. The expected value of the energy in the state
ρ is given by
Let E, belonging to the spectrum of H, be a fixed energy level. We are interested in the states for which the expected value of the energy equals E; i.e., in the states ρ such that . The classical result says that the maximal value of the entropy for such states is attained for a so-called Gibbs state; that is, a state with the density matrix for some . In this note, we aim to show a similar result in the situation where is replaced by a finite von Neumann algebra, and the von Neumann entropy is replaced by Segal’s entropy.
2. Preliminaries and Notation
Let be a finite von Neumann algebra with a normal finite faithful trace τ, identity , and predual . As usual, we assume that τ is normalised (i.e., ), so τ is itself a normal faithful state. By , we shall denote the set of positive operators in , and by , the set of normal states of (i.e., ). For , the spectrum of x will be denoted by .
The trace τ can be extended to the space consisting of densely defined closed operators affiliated with such that for each , is finite; in particular, .
Let
be a von Neumann subalgebra of
. There exists a normal faithful conditional expectation
such that
. This expectation can be extended to a map from
onto
, denoted by the same symbol, which retains the basic properties of the original conditional expectation; in particular,
and
For each
, there is a selfadjoint positive operator
such that
By analogy with the case , the as above will be called the density matrix of ρ.
The Segal entropy of
ρ—denoted by
—is defined as
where
is the density matrix of
ρ; i.e., for the spectral representation of
we have
Let us note that Segal’s entropy is well-defined (though it may be minus infinity) and nonpositive for the states, since, on account of the inequality
we have
and the right-hand side of the inequality above is always finite, and equals zero for
ρ being a state.
The Segal entropy in connection with quantum measurement theory was employed in [
1], where it was shown that a weakly repeatable measurement is a maximal state entropy one, and that under some natural conditions, the converse is true, even in a slightly stronger form; i.e., a maximal state entropy measurement satisfying these conditions is repeatable (The reader should be warned that the definition of Segal’s entropy adopted in [
1] differs from ours by a minus sign, so the conclusions there are about minimal entropy instead of maximal). So far, Segal’s entropy has not found many other applications in physics or information theory, and we hope that this paper may arouse some interest among physicists or information theorists for this notion.
Some further properties of Segal’s entropy were investigated in [
2,
3,
4,
5,
6,
7,
8].
3. Maximum Segal’s Entropy States
The function introduced below plays a crucial role in our further considerations.
Lemma 1. Let h be a selfadjoint element in which is not a multiple of the identity, and let f be defined asThe function f is strictly increasing. Proof. Define on
a scalar product
by the formula
For
and
, we obtain by the Schwarz inequality
Hence, for all with the equality if and only if for some λ which contradicts the assumption. Thus, for all , and we arrive at the conclusion of the lemma. ☐
In what follows, we fix an arbitrary Hamiltonian
h (i.e.,
h is a selfadjoint element of
) and put
For each , we define the Gibbs states as the states with density matrices . Let us begin with the following simple observation.
Lemma 2. The following inclusion holdsMoreover, if for some Gibbs state we have or , then, respectively, or . Proof. We have
consequently, for each
we get—applying
ρ to the elements of the inequality above—
which shows the inclusion.
Furthermore, assume for example that for some
, we have
. Then,
and since
and
τ is faithful, we obtain
which yields
. In the same way, we show that
if
. ☐
The result above shows that for the Gibbs states we cannot have or unless h is a multiple of the identity. However, the values between and can be attained as shown in the following proposition.
Proposition 1. Let . Then, there is a unique such that for the Gibbs state , we have .
Proof. Let
f be the function as in Lemma 1; in particular, we have
Take an arbitrary
, and observe that
Let
be the spectral representation of
h. We have
Since
for
, we obtain for such
λ
and the Lebesgue Dominated Convergence Theorem yields
To estimate the remaining integral, take an arbitrary fixed
to obtain
Since
and
, we get
consequently,
The estimates obtained yield (after passing to the limit in Formula (
1)),
and since
α was arbitrary in
, it follows that
On the other hand, since
, Lemma 2 yields the inequality
; hence,
An analogous reasoning leads first to the inequality
and hence
Now, since
f is continuous and increasing, the Darboux property yields that for each
there is a unique
such that
☐
Now we are in a position to prove the main result of the paper.
Theorem 1. Let h be a Hamiltonian in , and let be arbitrary, where and are as before. Then, there exists a unique such thatthat is, the maximal value of Segal’s entropy for the states in which the energy level is fixed is attained for a Gibbs state. Moreover, this Gibbs state is the only one for which the maximal value of entropy is attained. Proof. Let
be such that for the Gibbs state
according to Proposition 1, the
β as above exists and is unique. For
being the density matrix of
, we have
Let
, with the density matrix
, be such that
. We have
On the other hand,
and thus
Now the basic inequality
obtained in ([
7], Theorem 1) for density matrices
a and
b of states
and
, respectively, with finite entropy, yields that for all
ρ with finite entropy, we have
and since
ρ was arbitrary, the claim follows.
Now assume that for some state
ρ with density matrix
such that
, we have
On account of Equality (
2), this yields
Let
be the von Neumann algebra generated by
(more precisely,
is the (abelian) algebra generated by the spectral projections of
), and let
be a faithful normal conditional expectation from
onto
such that
. The operator
—as a product of an operator from
and an operator from
—is in
, so we may apply conditional expectation
to it, obtaining
The function
is operator concave; hence, from Jensen’s inequality, we obtain (keeping in mind that
is bounded)
consequently,
which yields the inequality
Relations (
5) and (
6) now yield
i.e.,
Since
is a density matrix, we get (by virtue of Inequality 3),
Now
(and consequently,
) are in
, so
commutes with
, and from the equality above we infer—taking into account ([
7], Theorem 2)—that
in particular,
is bounded.
Now let
be the (abelian) von Neumann algebra generated by the Hamiltonian
h (in particular, we have
and
), and let
be a normal faithful conditional expectation from
onto
such that
. The function
is operator convex, so from Jensen’s inequality we obtain
By virtue of Inequality (
3) and the fact that
is a density matrix, from Equality (
4) we get
Since
and
the faithfulness of
τ yields
From ([
3], Appendix B.5), it follows that
i.e., Equality (
7) becomes
or
Now, since
, it follows that
and
commute, and from Equality (
8) we obtain—referring once more to ([
7], Theorem 2)—that
which ends the proof. ☐
Remark 1. It should be noted that the equalitymeans thatwhere is the relative entropy of the states ρ, . This relative entropy is defined in ([4],Chapter 5) by means of the relative modular operator, but it can be shown that for finite von Neumann algebras with a normal faithful finite trace τ, we have Now, referring to ([4], Corollary 5.6) gives the equality In our approach, we have chosen a simpler and more straightforward way without reference to any advanced theory of von Neumann algebras; in particular, to a rather sophisticated definition of the relative entropy based on the notion of the relative modular operator.
4. Conclusions
We have shown that for an arbitrary finite von Neumann algebra
with a normal finite trace
τ, a state that maximises Segal’s entropy with a given energy level is a unique Gibbs state, i.e., a state
defined for
by the formula
with some fixed selfadjoint
. This result is analogous to the classical one concerning von Neumann’s entropy defined by means of the canonical trace on the full algebra
. Since the definition of Segal’s entropy applied to the full algebra and the canonical trace leads to von Neumann’s entropy, it would be interesting to obtain such a result for Segal’s entropy in the case of semifinite von Neumann algebras. Another interesting question would be extending Segal’s definition to the states with their density matrices not necessarily in the algebra, and proving the maximisation condition in this case. However, in both the cases a serious difficulty arises, namely, an appropriate definition of Gibbs state is not clear due to the fact that the operator
need not be of trace class. Overcoming these difficulties seems to be an interesting challenge.