Abstract
Properties of Segal’s entropy for semifinite and finite von Neumann algebras are investigated. In particular, its invariance with respect to a trace-preserving normal *-homomorphism is studied, as well as norm-continuity in the trace norm on the set of bounded in the operator norm density matrices.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In the paper, we want to address some questions concerning Segal’s entropy for normal states, or, more generally, positive operators, in semifinite von Neumann algebras. A particular instance of this entropy is the celebrated von Neumann entropy defined for density matrices in the algebra \(\mathbb {B}(\mathcal {H})\) of all bounded linear operators on a Hilbert space. However, in the case of an arbitrary semifinite von Neumann algebra, where instead of the canonical trace we have a normal semifinite faithful trace, substantial differences between these two entropies show up. These differences become still larger when we deal with a finite trace. The problems investigated are as follows: subinvariance and invariance of Segal’s entropy with respect to a unital normal positive linear map, and continuity properties of this entropy with respect to the trace norm.
2 Preliminaries and Notation
Let \(\mathcal {M}\) be a semifinite von Neumann algebra of operators acting on a Hilbert spaced \(\mathcal {H}\) with a normal semifinite faithful trace τ, identity \(\mathbb {1}\), and predual \(\mathcal {M}_{*}\). By \(\mathcal {M}^{+}\) we shall denote the set of positive operators in \(\mathcal {M}\), and by \(\mathcal {M}_{*}^{+}\) — the set of positive functionals in \(\mathcal {M}_{*}\). These functionals will be sometimes referred to as (non-normalised) states. The set of normalised states, i.e. the elements \(\rho \in \mathcal {M}_{*}^{+}\) such that \(\rho (\mathbb {1})=\parallel \rho \parallel =1\) will be denoted by \(\mathfrak {S}\).
The algebra of measurable operators \(\tilde {\mathcal {M}}\) is defined as a topological ∗-algebra of densely defined closed operators on \(\mathcal {H}\) affiliated with \(\mathcal {M}\) with strong addition + and strong multiplication ⋅, i.e.
where \(\overline {a+b}\) and \(\overline {ab}\) are the closures of the corresponding operators defined by addition and composition respectively on the natural domains given by the intersections of the domains of the a and b and of the range of b and the domain of a respectively. The translation-invariant measure topology is defined by a fundamental system of neighbourhoods of 0, {N(ε, δ):ε, δ > 0}, given by
Thus for operators \(a_{n}, a\in \tilde {\mathcal {M}}\), the sequence (a n ) converges to a in measure if for any ε, δ > 0 there exists n 0 such that for each \(n\geqslant n_{0}\) there exists a projection \(p\in \mathcal {M}\) such that
The following “technical” form of convergence in measure proved in [12, Proposition 2.7] is useful. Let
be the spectral decomposition of |a n −a| with spectral measure e n taking values in \(\mathcal {M}\) since a n −a, and thus |a n −a|, are affiliated with \(\mathcal {M}\). Then a n →a in measure if and only if for each ε > 0
For each \(\rho \in \mathcal {M}_{*}\), there is a measurable operator h such that
The space of all such operators is denoted by \(L^{1}(\mathcal {M},\tau )\), and the correspondence above is one-to-one and isometric, where the norm on \(L^{1}(\mathcal {M},\tau )\), denoted by ∥⋅∥1, is defined as
The space of all measurable operators h such that \(\tau (|h|^{p})<+\infty \), \(p\geqslant 1\), constitutes a Banach space \(L^{p}(\mathcal {M},\tau )\) with the norm
(In the theory of noncommutative L p-spaces for semifinite von Neumann algebras, it it shown that τ can be extended to the h’s as above; see e.g. [4, 10, 12] for a detailed account of this theory.) Moreover, to hermitian functionals in \(\mathcal {M}_{*}\) correspond selfadjoint operators in \(L^{1}(\mathcal {M},\tau )\), and to states in \(\mathcal {M}_{*}\) — positive operators in \(L^{1}(\mathcal {M},\tau )\). For a state ρ the corresponding element in \(L^{1}(\mathcal {M},\tau )\) will be denoted by h ρ and called the density matrix of ρ, thus
In particular,
so for normalised states, we have for their density matrices the equality τ(h ρ )=1.
Observe that for a finite (normalised) τ, we have \(\mathcal {M}\subset L^{1}(\mathcal {M},\tau )\) while this is not the case for τ infinite since then \(\tau (\mathbb {1})=+\infty \).
Below we present some simple facts which can be regarded as part of the folklore of noncommutative probability theory.
For an arbitrary \(a\in L^{1}(\mathcal {M},\tau )\) we have the spectral decomposition
Thus for any ε > 0, we get
Consequently, we obtain the Chebyschev inequality (cf. [3, Lemma 1.1]
Taking into account the above-mentioned “technical” form of convergence in measure we have
Lemma 1
If a sequence (a n ) of operators in \(L^{1}(\mathcal {M},\tau )\) converges in ∥⋅∥ 1 -norm, then it converges in measure.
For \(x\in L^{1}(\mathcal {M},\tau )\), define a functional x τ on \(\mathcal {M}\) by the formula
The Segal entropy of ρ, denoted by H(ρ), is defined as
i.e. for the spectral representation of h ρ
we have
Following Segal [8], we shall be interested only in the case when \(h_{\rho }\in \mathcal {M}^{+}\).
Accordingly, we define Segal’s entropy for \(h\in \mathcal {M}^{+}\) by the formula
where h has the spectral representation as in (1). Let us note that the existence of Segal’s entropy is by no means guaranteed, however, for finite τ and normalised state ρ, we have, on account of the inequality
the relation
showing that, at least in this case, Segal’s entropy is well defined and nonnegative.
Remark 1
It should be noted that the original Segal definition of entropy differs from ours by a minus sign before the trace. However, for the sake of having nonnegative entropy for states on a finite von Neumann algebra we have adopted the definition as above.
3 Subinvariance and Invariance of Segal’s Entropy
Let \(\alpha \colon \mathcal {M}\to \mathcal {M}\) be a normal positive unital linear map such that
The (pre-)dual map α ∗ is defined on \(\mathcal {M}_{*}\) by the formula
Our aim is to establish the form of the density matrix of α ∗(ρ). To this end the following construction is employed.
Fix an element \(z\in \mathcal {M}^{h}\), and consider a linear map
defined by
(Since \(\overline {\tau (\alpha (x)z)}=\tau ((\alpha (x)z)^{*})=\tau (z\alpha (x))=\tau (\alpha (x)z)\) the values of f lie in \(\mathbb {R}\).) The function \(\mathbb {R}\ni t\mapsto |t|\) is convex, thus the Jensen inequality for positive unital maps yields
for each \(x\in \mathcal {M}^{h}\). Consequently, we get from the properties of trace
which means that f is bounded and \(\parallel {f}\parallel \leqslant \parallel {z}\parallel _{\infty }\). Hence f can be extended to a bounded linear functional on \(\overline {\{x\tau :x\in \mathcal {M}^{h}\cap L^{1}(\mathcal {M},\tau )\}}=\mathcal {M}_{*}^{h}\). This, in turn, yields that \(f\in \left (\mathcal {M}_{*}^{h}\right )^{*}\simeq \mathcal {M}^{h}\). Denote the element in \(\mathcal {M}^{h}\) corresponding to f by \(\tilde {\alpha }(z)\). We then have
We have thus obtained a map \(\mathcal {M}^{h}\ni z\mapsto \tilde {\alpha }(z)\in \mathcal {M}^{h}\) which is clearly linear. For \(\tilde {\alpha }(z)\) regarded as a linear functional on \(\mathcal {M}_{*}^{h}\), we have on account of relation (2)
which means that
so \(\parallel \tilde {\alpha }\parallel \leqslant 1\). Further, for each \(x\in \mathcal {M}^{h}\cap L^{1}(\mathcal {M},\tau )\), we have
hence
which by virtue of [1, Corollary 3.2.6] shows the positivity of \(\tilde {\alpha }\) (the reasoning in [1, Corollary 3.2.6] is performed for a map between C*-algebras while in our case we deal with a map from the hermitian part of \(\mathcal {M}\) into itself but the proof of positivity remains exactly the same also in this restricted situation). Now \(\tilde {\alpha }\), being a bounded map, can be extended to a bounded linear map on the whole of \(\mathcal {M}\), denoted by the same symbol, and this extended map is obviously positive and satisfies relations (4) and (5). Moreover, since each element in \(\mathcal {M}\) is a linear combination of two elements from \(\mathcal {M}^{h}\), we have
For the dual map \(\tilde {\alpha }^{*}\) defined on \(\mathcal {M}^{*}\) by the formula
we have, for the functionals x τ with \(x\in \mathcal {M}\cap L^{1}(\mathcal {M},\tau )\),
Since \(|\tau (\alpha (x))|=|\tau (x)|<+\infty \), it follows that \(\alpha (x)\in \mathcal {M}\cap L^{1}(\mathcal {M},\tau )\), consequently, \((x\tau )\circ \tilde {\alpha }=\alpha (x)\tau \in \mathcal {M}_{*}\). This means that
and since \(\{x\tau :x\in \mathcal {M}\cap L^{1}(\mathcal {M},\tau )\}\) is dense in \(\mathcal {M}_{*}\), and \(\parallel \tilde {\alpha }^{*}\parallel =1\), the relation above yields
showing that \(\tilde {\alpha }\) is normal.
Now take arbitrary \(z\in \mathcal {M}^{+}\), and let \(x_{i}\in \mathcal {M}^{+}\cap L^{1}(\mathcal {M},\tau )\) be such that \(x_{i}\uparrow \mathbb {1}\). Then
which implies, on account of the normality of τ,
On the other hand, we have, since \(\alpha (x_{i})\uparrow \alpha (\mathbb {1})=\mathbb {1}\),
showing that \(\tau (\tilde {\alpha }(z))=\tau (z)\), and thus \(\tau \circ \tilde {\alpha }=\tau \).
The results of our considerations may by summarised as follows. Let \(\mathcal {M}\) be a von Neumann algebra with a normal faithful semifinite trace τ, and let α be a normal positive unital linear map on \(\mathcal {M}\) such that τ∘α = τ. Then there is a normal positive unital linear map \(\tilde {\alpha }\) on \(\mathcal {M}\) such that \(\tau \circ \tilde {\alpha }=\tau \), and
The map \(\tilde {\alpha }\) will be called conjugate to α.
Lemma 2
Let \(\rho \in \mathcal {M}_{*}\) be such that \(h_{\rho }\in \mathcal {M}\) . Then
Proof
For each \(x\in \mathcal {M}\), we have
and the conclusion follows. □
Remark 2
The same result can be obtained in a stronger form without the assumption \(h_{\rho }\in \mathcal {M}\). However, this would require extension of \(\tilde {\alpha }\) from \(\mathcal {M}\) to \(L^{1}(\mathcal {M},\tau )\), which is not needed for our present purposes.
Theorem 3
Let \(\mathcal {M}\) , τ and α be as before. Then for every \(\rho \in \mathcal {M}_{*}^{+}\) , the following relation holds
Proof
Taking into account Lemma 2, we obtain
The function \([0,+\infty )\ni t\mapsto t\log t\) is operator convex, and the map \(\tilde {\alpha }\) is positive and unital, so from Jensen’s inequality we get
consequently,
showing the claim. □
Remark 3
It should be noted that the result above, in a slightly stronger form, was obtained in [6, Proposition 7.3]. However, the existence of the conjugate map \(\tilde {\alpha }\) satisfying the basic relation (6) was simply taken for granted without showing its properties. Since these properties will be important in our further considerations, we have decided to present in detail the whole construction of this map.
Now we want to investigate the situation where the entropy does not change under a transformation of states.
Theorem 4
Let α be a *-homomorphism such that τ∘α = τ. Put \(\mathcal {N}=\alpha (\mathcal {M})\) . The relation
holds if and only if \(h_{\rho }\in \mathcal {N}\).
Proof
Let us start with some preliminary analysis. For arbitrary \(x,y\in \mathcal {M}\), we have
which shows that
Let α(x i )→y σ-weakly. Then, as \(\tilde {\alpha }\) is normal,
and the normality of α yields
which shows that \(\mathcal {N}\) is σ-weakly closed, consequently, \(\mathcal {N}\) is a von Neumann algebra. The map \(\mathbb {E}=\alpha \circ \tilde {\alpha }\) is a projection onto \(\mathcal {N}\), so it is a conditional expectation, moreover, \(\mathbb {E}\) is normal and \(\tau \circ \mathbb {E}=\tau \).
For arbitrary \(x,y,z\in \mathcal {M}\), we have, since \((\mathbb {E} y)\alpha (z)\in \mathcal {N}\),
and
which shows that
Since \(\tilde {\alpha }\circ \mathbb {E}=\tilde {\alpha }\), we obtain for arbitrary \(x,y\in \mathcal {M}\)
and consequently,
It follows that for arbitrary polynomial W
and thus
for an arbitrary function f which is continuous on \(sp\tilde {\alpha }(x)\cup {sp}\mathbb {E} x\). Observe that since α is a *-homomorphism, we get
and thus
for an arbitrary function f which is continuous on \(sp\tilde {\alpha }(x)\cup {sp} x\). In particular, for the function \(f(t)=t\log t\) we have, for every \(x\in \mathcal {M}^{+}\), the equalities
Assume first that \(h_{\rho }\in \mathcal {N}\). Then h ρ = α(h) for some \(h\in \mathcal {M}\), and thus \(\tilde {\alpha }(h_{\rho })=h\) which shows that h is a density matrix. Taking into account the relations (7) and (8), we obtain
Now assume that H(α ∗(ρ)) = H(ρ). Since the function f as above is operator convex, and \(\tilde {\alpha }\) is positive and unital, it follows from Jensen’s inequality that
Consequently, taking into account the relations (8) and (9), we obtain
which shows, in particular, that
Consequently, we get
and the inequality (9) and the faithfulness of τ yield
i.e. we have equality in Jensen’s inequality. From [5, Appendix B.5], it follows that
which means that \(h_{\rho }\in \mathcal {N}\). □
As a corollary we obtain
Corollary 5
Let α be a *-automorphism. Then for every \(\rho \in \mathcal {M}_{*}^{+}\) we have
4 Continuity Properties of Segal’s Entropy
In this section we want to investigate the continuity of Segal’s entropy.
Lemma 6
Let \(\rho \in \mathfrak {S}\) , and let \(0<\alpha \leqslant \beta \) . Then for each \(h\in \mathcal {M}^{+}\) , we have
Proof
Set \(\gamma =\frac {\beta }{\alpha }\). Since \(\gamma \geqslant 1\), and the function \(\mathbb {R}_{+}\ni t\mapsto t^{\gamma }\) is convex, Jensen’s inequality yields
for each \(z\in \mathcal {M}^{+}\). Putting z = h α, we obtain
which shows the claim. □
Define the quantum mechanical Rényi entropy (≡α-entropy) S α , α ≠ 1, by the formula (see e.g. [6, Part II, Chapter 7])
We want to investigate some properties of this entropy.
Proposition 7
Fix \(\rho \in \mathfrak {S}\) . The function α↦S α (ρ) is decreasing on the interval \((1,+\infty )\).
Proof
We have
and the conclusion follows from Lemma 6. □
Assume that the state \(\rho \in \mathfrak {S}\), with the density matrix \(h_{\rho }\in \mathcal {M}^{+}\), has finite Segal’s entropy, and take a closer look at the function \(\alpha \mapsto \tau (h_{\rho }^{\alpha })\). Its difference quotient at point 1 equals
where
is the spectral representation of h ρ . For the net of functions {f α :α > 1} defined as
a little of calculus shows that for each fixed λ we have \(f_{\alpha }(\lambda )\leqslant f_{\beta }(\lambda )\), for α < β, so this net is increasing. Moreover, we have
and the Lebesgue Monotone Convergence Theorem yields
(To be more precise, for passing to the limit we should divide the integral into two parts \({{\int }_{0}^{1}}\) and \({\int }_{1}^{\infty }\), and for the integral \({{\int }_{0}^{1}}\) consider the positive functions (−f α ).)
Hence, we have shown that the function \(\alpha \mapsto \tau (h_{\rho }^{\alpha })\) is differentiable at α = 1, and
Consequently, for the function \(\alpha \mapsto \log \tau (h_{\rho }^{\alpha })\) we have
Observe now that for the Rényi entropy we have
consequently, Segal’s entropy H is a limit of a decreasing net of functions {−S α :α > 1} (note that we consider the limit at 1, so for α < β we have \(-S_{\alpha }\leqslant -S_{\beta }\)).
Now fix α > 1, and consider the function \(\mathfrak {S}\ni \rho \mapsto S_{\alpha }(\rho )\). For \(\rho ,\varphi \in \mathfrak {S}\) we have the following estimate
From this estimate we obtain
Proposition 8
The Rényi entropy is continuous in the ∥⋅∥ 1 -norm on the set \(\{\rho \in \mathfrak {S}:\parallel {h}_{\rho }\parallel _{\infty }\leqslant c\}\) for every \(c\geqslant 0\) . Moreover, for \(\mathcal {M}=\mathbb {B}(\mathcal {H})\) and the canonical trace τ= tr, the Rényi entropy is continuous on the whole of \(\mathfrak {S}\).
Proof
For an arbitrary semifinite trace, we have on the set \(\{\rho \in \mathfrak {S}:\parallel {h}_{\rho }\parallel _{\infty }\leqslant c\}\)
so if \(\parallel \rho _{n}-\rho \parallel =\parallel {h}_{\rho _{n}}-h_{\rho }\parallel _{1}\to 0\), then \(\tau (h_{\rho _{n}}^{\alpha })\to \tau (h_{\rho }^{\alpha })\), and thus S α (ρ n )→S α (ρ), which proves the continuity of the Rényi entropy on the set \(\{\rho \in \mathfrak {S}:\parallel {h}_{\rho }\parallel _{\infty }\leqslant c\}\) in the general case.
In the case \(\mathcal {M}=\mathbb {B}(\mathcal H)\) and the canonical trace, we have for every \(h\in \mathcal {M}^{+}\), \(\parallel {h}\parallel _{\infty }=\max \{\lambda :\lambda \in {sp} h\}\leqslant \operatorname {tr} h=\parallel {h}\parallel _{1}\), and thus
Consequently, if \(\parallel \rho _{n}-\rho \parallel =\parallel {h}_{\rho _{n}}-h_{\rho }\parallel _{1}\to 0\), then for the density matrices we have \(\parallel {h}_{\rho _{n}}\parallel _{1}\leqslant c\), and hence \(\parallel {h}_{\rho }\parallel _{1}\leqslant c\) for some c, so the estimate (10) holds, proving, as before, the continuity of the Rényi entropy on \(\mathfrak {S}\). □
The results obtained so far lead to the following theorem on semicontinuity of Segal’s entropy.
Theorem 9
Segal’s entropy is norm-upper semicontinuous on the set \(\{\rho \in \mathfrak {S}:\parallel {h}_{\rho }\parallel _{\infty }\leqslant c\}\) for every \(c\geqslant 0\) . In the case \(\mathcal {M}=\mathbb {B}(\mathcal H)\) and the canonical trace, Segal’s entropy is norm-upper semicontinuous on \(\mathfrak {S}\) (consequently, von Neumann’s entropy is norm-lower semicontinuous on \(\mathfrak {S}\) ).
Proof
For Segal’s entropy H we have
and Rényi’s entropies S α are continuous as functions of state on the set \(\{\rho \in \mathfrak {S}:\parallel {h}_{\rho }\parallel _{\infty }\leqslant c\}\) for an arbitrary algebra \(\mathcal {M}\), and the set \(\mathfrak {S}\) for the algebra \(\mathbb {B}(\mathcal {H})\), respectively. Since the net {−S α :α > 1} is decreasing as α→1+, H is upper semicontinuous as a limit of a decreasing net of continuous functions. (Since von Neumann’s entropy is minus Segal’s entropy the lower semicontinuity of von Neumann’s entropy follows.) □
Remark 4
Lower semicontinuity of von Neumann’s entropy was proved in [11], while the result as above for Segal’s entropy was obtained in [6, Proposition II.7.6] by using a technique of rearrangements. A simple proof above unifies these two results.
Our last result concerns continuity of Segal’s entropy in finite von Neumann algebras for not necessarily normalised states.
Theorem 10
Let \(\mathcal {M}\) be a finite von Neumann algebra with a normal finite faithful trace τ. Then Segal’s entropy is norm-continuous on the set \(\{\rho \in \mathcal {M}_{*}^{+}:\parallel {h}_{\rho }\parallel _{\infty }\leqslant c\}\) for every \(c\geqslant 0\).
Proof
Let \(\parallel \rho _{n}-\rho \parallel =\parallel {h}_{\rho _{n}}-h_{\rho }\parallel _{1}\to 0\). By virtue of Lemma 1, \(h_{\rho _{n}}\to h_{\rho }\) in measure. Let f be the function defined as
Since f is continuous, we get, on account of [7, Theorem 2.1],
Since for the operators \(h_{\rho _{n}}\) we have \(sp h_{\rho _{n}}\subset [0,c]\), it follows that
for some m, and consequently, there is a real number M such that
By virtue of [9, Theorem 4.8] (see also [2, Theorem 3.6]), we infer that
and the inequality
yields
which finishes the proof. □
References
Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics, vol. 1. Springer-Verlag, New York-Heidelberg-Berlin (1987)
Fack, T., Kosaki, H.: Generalized s-numbers of τ-measurable operators. Pac. J. Math. 123, 269–300 (1986)
Łuczak, A.: Laws of large numbers in von Neumann algebras and related results. Stud. Math. 81, 231–243 (1985)
Nelson, E.: Notes on non-commutative integration. J. Funct. Anal. 15, 103–116 (1974)
Neshveyev, S., Strmer, E.: Dynamical Entropy in Operator Algebras. Springer, Berlin-Heidelberg-New York (2006)
Ohya, M., Petz, D.: Quantum Entropy and Its Use. Springer, Berlin-Heidelberg-New York (2004)
Padmanabhan, A.R.: Probabilistic aspects of von Neumann algebras. J. Funct. Anal. 31, 139–149 (1979)
Segal, I.E.: A note on the concept of entropy. J. Math. Mech. 9(4), 623–629 (1960)
Stinespring, W.F.: Integration theorems for gages and duality for unimodular groups. Trans. Am. Math. Soc. 90, 15–56 (1959)
Takesaki, M.: Theory of Operator Algebras II Encyclopaedia of Mathematical Sciences, vol. 125. Springer, Berlin–Heidelberg–New York (2003)
Wehrl, A.: Three theorems about entropy and convergence of density matrices. Rep. Math Phys. 10(2), 159–163 (1976)
Yeadon, F.J.: Non-commutative l p -spaces. Math. Proc. Cambridge Philos. Soc. 77, 91–102 (1975)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Łuczak, A., Podsędkowska, H. Properties of Segal’s Entropy for Quantum Systems. Int J Theor Phys 56, 3783–3793 (2017). https://doi.org/10.1007/s10773-017-3310-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10773-017-3310-1