Week 7 — Connes Cocycle and Araki-Uhlmann Relative Entropy

A single faithful normal state gives a modular flow. Physics rarely gives us just one state. We perturb the vacuum, compare thermal states, restrict states to subalgebras, and ask whether one state can be distinguished from another. Week 7 supplies the comparison calculus: the relative modular operator, the Connes cocycle, and Araki-Uhlmann relative entropy.

How to use this chapter

• In class: calculate the relative modular operator for two faithful matrices, derive the cocycle, and recover Umegaki relative entropy before stating the representation-independent theory.
• For self-study: keep the three objects separate on paper: ${\Delta }_{\omega |\varphi }$ compares states, $(D\omega :D\varphi {)}_t$ compares flows, and $S(\omega \Vert \varphi )$ is a number in $[0,+\infty ]$. Work one noncommuting $2\times 2$ example.
• Instructor checkpoint: test cocycle ordering and support conditions. Relative entropy is intrinsic and monotone, but it need not be finite.

0. Reading

Primary:

• Takesaki, Theory of Operator Algebras II, chs. VIII–IX, for relative modular theory, spatial derivatives, and Connes cocycles.
• Ohya & Petz, Quantum Entropy and Its Use, ch. 5, for finite-dimensional relative entropy and monotonicity.

Secondary:

• Bratteli & Robinson, Vol. I, §2.5, for the single-vector Tomita–Takesaki background. That section does not supply the relative modular/Radon–Nikodym theory used here.
• Witten, “Notes on some entanglement properties of QFT,” arXiv:1803.04993, §3–4.

Optional research reading:

• Araki, “Relative entropy of states of von Neumann algebras,” Publ. RIMS 11 (1976).
• Uhlmann, “Relative entropy and the Wigner–Yanase–Dyson–Lieb concavity in an interpolation theory,” Commun. Math. Phys. 54 (1977) 21–32, for monotonicity.
• Connes, “Une classification des facteurs de type III,” Ann. Sci. ENS 6 (1973).

1. Why compare modular flows?

Let $\omega$ and $\varphi$ be two faithful normal states on the same von Neumann algebra $\mathcal{M}$. Week 6 gives two flows:

$${\sigma }_t^{\omega },{\sigma }_t^{\varphi }.$$

The basic question is:

What operator converts the $\varphi$-clock into the $\omega$-clock?

In classical probability, comparing two measures uses a Radon-Nikodym derivative $d\omega /d\varphi$. In noncommutative probability, the derivative is no longer a single positive function. It becomes a one-parameter family of unitaries:

$$(D\omega /D\varphi {)}_t.$$

This is the Connes cocycle.

The same comparison object produces relative entropy. In type III, where neither state has a density matrix with respect to a trace, this is the entropy that remains.

2. Relative Tomita operators

For the cleanest statement, put $\mathcal{M}$ in a standard representation and choose cyclic-separating vector representatives ${\Omega }_{\omega }$ and ${\Omega }_{\varphi }$ for $\omega$ and $\varphi$.

Definition 2.1. The relative Tomita operator $S_{\omega ,\varphi }$ is initially defined on $\mathcal{M}{\Omega }_{\varphi }$ by

$$S_{\omega ,\varphi }(a{\Omega }_{\varphi })=a^{\ast }{\Omega }_{\omega },a\in \mathcal{M}.$$

It is closable. Its polar data define the relative modular operator

$${\Delta }_{\omega ,\varphi }:=S_{\omega ,\varphi }^{\ast }{S}_{\omega ,\varphi }.$$

Proof status. [Stated only — refs: Araki 1976; Takesaki Vol. II, chs. VIII–IX.] Closability and representation-independent construction parallel Week 5 but require the standard form of a von Neumann algebra or spatial derivatives to avoid representation artifacts.

When $\omega =\varphi$, this reduces to the ordinary Tomita operator and ordinary modular operator:

$${\Delta }_{\omega ,\omega }={\Delta }_{\omega }.$$

Physical picture. The ordinary modular operator measured the thermal weights of one state against itself; the relative modular operator measures one state’s excitations weighted by another state’s thermal structure. In the finite-dimensional formula of §3, ${\Delta }_{\omega ,\varphi }=L_{\rho }{R}_{{\sigma }^{-1}}$: the “numerator” state supplies the weights, the “denominator” state supplies the reference orbit. The physics use-case is exactly the situations where two states coexist: a perturbed vacuum compared to the vacuum, a black-hole microstate compared to the Hartle–Hawking state, an excited wedge state compared to Rindler. Everything this week — cocycle, relative entropy — is a different scalar or unitary extracted from this single object. It is worth internalizing that ${\Delta }_{\omega ,\varphi }$ exists for any pair of faithful normal states on any von Neumann algebra: the comparison calculus never needs a trace, which is why it is the toolset that survives in QFT.

3. Finite-dimensional model

Let $\mathcal{M}=M_n(\mathbb{C})$ and let $\rho ,\sigma >0$ be density matrices. Work in the Hilbert-Schmidt space

$${\mathcal{H}}_{HS}=M_n(\mathbb{C}),⟨X,Y⟩=\mathrm{Tr}(X^{\ast }Y).$$

The algebra acts by left multiplication:

$$L_{a}X=aX.$$

The vector representing ${\omega }_{\rho }$ is ${\Omega }_{\rho }={\rho }^{1/2}$, since

$$⟨{\rho }^{1/2},L_a{\rho }^{1/2}⟩=\mathrm{Tr}(\rho a).$$

3.1 Relative modular operator

For $X=a{\sigma }^{1/2}$,

$$S_{\rho ,\sigma }X=a^{\ast }{\rho }^{1/2}.$$

One checks (Problem 1) that

$$\boxed{{\Delta }_{\rho ,\sigma }(X)=\rho X{\sigma }^{-1}.}$$

Equivalently,

$${\Delta }_{\rho ,\sigma }=L_{\rho }{R}_{{\sigma }^{-1}},$$

where $R_{b}X=Xb$ is right multiplication.

This formula is the finite-dimensional template. The left density matrix belongs to the numerator state; the right density matrix belongs to the reference state.

3.2 Ordinary modular operator

Putting $\rho =\sigma$ gives

$${\Delta }_{\sigma }(X)=\sigma X{\sigma }^{-1}.$$

Therefore the modular flow on the left algebra is

$${\sigma }_t^{\sigma }(a)={\sigma }^{-it}a{\sigma }^{it},$$

as in Week 6 (in our upper-strip $\beta =+1$ convention; note the minus sign on the exponent of ${\Delta }_{\sigma }^{-it}$).

4. The Connes cocycle

Definition 4.1. The Connes cocycle derivative of $\omega$ with respect to $\varphi$ is

$$\boxed{(D\omega /D\varphi {)}_t:={\Delta }_{\omega ,\varphi }^{-it}{\Delta }_{\varphi }^{it}.}$$

(Sign convention matches our Week 5 §4: ${\sigma }_t^{\omega }=\mathrm{Ad}({\Delta }_{\omega }^{-it})$.)

Theorem 4.2 (Connes Radon-Nikodym theorem). [Stated only — refs: Connes 1973; Takesaki Vol. II Ch. VIII.] For faithful normal states $\omega ,\varphi$ on $\mathcal{M}$,

$$(D\omega /D\varphi {)}_t\in \mathcal{M}$$

is a strongly continuous family of unitaries satisfying the cocycle identity

$$(D\omega /D\varphi {)}_{t+s}=(D\omega /D\varphi {)}_t{\sigma }_t^{\varphi }\left((D\omega /D\varphi {)}_s\right).$$

It intertwines the modular flows:

$${\sigma }_t^{\omega }(a)=(D\omega /D\varphi {)}_t{\sigma }_t^{\varphi }(a)(D\omega /D\varphi {)}_t^{\ast }.$$

Reversing the two states gives the same-time adjoint identity

$$\boxed{(D\omega /D\varphi {)}_t^{\ast }=(D\varphi /D\omega {)}_t.}$$

This identity is a useful guardrail: reversing the indices takes an adjoint, not a replacement $t\mapsto -t$.

This is the noncommutative Radon-Nikodym theorem in the form we need later.

Corollary 4.3 (state-independence of the modular flow modulo inner automorphisms). [Proved, given Theorem 4.2.] Since the cocycle is a unitary in $\mathcal{M}$, the intertwining relation says ${\sigma }_t^{\omega }$ and ${\sigma }_t^{\varphi }$ differ by an inner automorphism at each $t$. Hence the image of the modular flow in the outer automorphism group,

$$\delta :\mathbb{R}\to \mathrm{Out}(\mathcal{M})=\mathrm{Aut}(\mathcal{M})/\mathrm{Inn}(\mathcal{M}),$$

is the same for every faithful normal state. A von Neumann algebra carries a canonical one-parameter subgroup of $\mathrm{Out}(\mathcal{M})$ — a dynamics intrinsic to the algebra itself.

Physical picture, with the classification boundary visible. Changing a faithful normal state changes its modular flow by unitaries inside $\mathcal{M}$, so the outer class ${\delta }_t$ is state-independent. For semifinite factors this outer flow is trivial. For type III it is nontrivial as a flow, although particular times can lie in its kernel. That kernel is Connes’ $T$-invariant: for type III${}_{\lambda }$, $0<\lambda <1$, it is $(2\pi /|\log \lambda |)\mathbb{Z}$; for type III${}_1$ it is $\{0\}$. These facts distinguish the periodic III${}_{\lambda }$ case from the aperiodic III${}_1$ case. They do not by themselves classify all type III${}_0$ factors, whose flow-of-weights data are essential. Thus “the algebra carries an outer clock” is a useful image, provided we do not promote the image into a complete classification theorem.

4.1 Finite-dimensional verification

Using the Hilbert-Schmidt formulas,

$${\Delta }_{\rho ,\sigma }^{-it}=L_{{\rho }^{-it}}{R}_{{\sigma }^{it}},{\Delta }_{\sigma }^{it}=L_{{\sigma }^{it}}{R}_{{\sigma }^{-it}}.$$

Thus

$${\Delta }_{\rho ,\sigma }^{-it}{\Delta }_{\sigma }^{it}=L_{{\rho }^{-it}{\sigma }^{it}}.$$

So the cocycle is represented inside $\mathcal{M}$ by

$$\boxed{u_t=(D{\omega }_{\rho }/D{\omega }_{\sigma }{)}_t={\rho }^{-it}{\sigma }^{it}.}$$

The adjoint/index identity is visible immediately:

$$({\rho }^{-it}{\sigma }^{it}{)}^{\ast }={\sigma }^{-it}{\rho }^{it}=(D{\omega }_{\sigma }/D{\omega }_{\rho }{)}_t.$$

The cocycle identity becomes

$$u_{t+s}=u_t{\sigma }_t^{\sigma }(u_s).$$

Indeed,

$$\begin{array}{rl}{u}_t{\sigma }_t^{\sigma }(u_s)& ={\rho }^{-it}{\sigma }^{it}{\sigma }^{-it}({\rho }^{-is}{\sigma }^{is}){\sigma }^{it}\\ & ={\rho }^{-it}{\rho }^{-is}{\sigma }^{is}{\sigma }^{it}\\ & ={\rho }^{-i(t+s)}{\sigma }^{i(t+s)}=u_{t+s}.\end{array}$$

The intertwining identity is just as direct:

$$\begin{array}{rl}{u}_t{\sigma }_t^{\sigma }(a)u_t^{\ast }& ={\rho }^{-it}{\sigma }^{it}{\sigma }^{-it}a{\sigma }^{it}{\sigma }^{-it}{\rho }^{it}\\ & ={\rho }^{-it}a{\rho }^{it}={\sigma }_t^{\rho }(a).\end{array}$$

This calculation is worth memorizing. It is the finite-dimensional version of the machinery used in cocycle-perturbation theory.

5. Araki-Uhlmann relative entropy

The same relative modular operator defines relative entropy.

Definition 5.1. The Araki-Uhlmann relative entropy of $\omega$ with respect to $\varphi$ is

$$\boxed{S(\omega \Vert \varphi ):=-⟨{\Omega }_{\omega },\log {\Delta }_{\varphi ,\omega }{\Omega }_{\omega }⟩.}$$

This bracket is an extended spectral integral, not an ordinary operator-domain test. Let $E$ be the spectral measure of ${\Delta }_{\varphi ,\omega }$ and

$${\mu }_{\varphi ,\omega }(B):=⟨{\Omega }_{\omega },E(B){\Omega }_{\omega }⟩.$$

Then

$$S(\omega \Vert \varphi )=-{\int }_{[0,\infty )}\log \lambda d{\mu }_{\varphi ,\omega }(\lambda ).$$

The value is $+\infty$ when $\mathrm{supp}\omega \nleqq \mathrm{supp}\varphi$ (equivalently, the relative spectral measure sees the zero-support obstruction), or when

$${\int }_{(0,1)}(-\log \lambda )d{\mu }_{\varphi ,\omega }(\lambda )=+\infty .$$

There is no additional divergence hidden at large $\lambda$: ${\log }^{+}\lambda \le \lambda$, and $\int \lambda d{\mu }_{\varphi ,\omega }=\Vert {\Delta }_{\varphi ,\omega }^{1/2}{\Omega }_{\omega }{\Vert }^2$ is controlled (equal to $1$ for normalized faithful vector representatives in this setup). Thus membership in $D(\log \Delta )$, which demands a stronger square-log moment, is not the finiteness criterion for relative entropy.

The order of the indices matters. The definition uses ${\Delta }_{\varphi ,\omega }$, not ${\Delta }_{\omega ,\varphi }$.

Theorem 5.2. [Stated only — refs: Araki 1976; Uhlmann, Commun. Math. Phys. 54 (1977) 21–32.] For normal states on a von Neumann algebra:

$$S(\omega \Vert \varphi )\ge 0,$$

with equality iff $\omega =\varphi$ under the usual support assumptions. It is monotone under restriction to subalgebras:

$$S(\omega {|}_{\mathcal{N}}\Vert \varphi {|}_{\mathcal{N}})\le S(\omega {|}_{\mathcal{M}}\Vert \varphi {|}_{\mathcal{M}})(\mathcal{N}\subset \mathcal{M}).$$

Monotonicity is the algebraic form of data processing: throwing away observables cannot make two states easier to distinguish.

Physical picture (finite-dimensional i.i.d. version first). The quantum Stein lemma identifies $S(\rho \Vert \sigma )$ with the optimal asymptotic type-II error exponent when one discriminates ${\rho }^{\otimes n}$ from ${\sigma }^{\otimes n}$ while keeping the type-I error bounded. This gives a precise meaning to distinguishability and makes monotonicity natural: discarding observables cannot improve the optimal rate. Extensions exist for von Neumann algebras and more general asymptotic settings, but their hypotheses should be stated before importing the slogan wholesale. In finite dimensions, $S(\rho \Vert \sigma )=+\infty$ exactly when $\mathrm{supp}\rho \nleqq \mathrm{supp}\sigma$; then an event forbidden by $\sigma$ has positive probability under $\rho$. In infinite dimensions, relative entropy can also diverge because $\log \sigma$ is too singular even when the support inclusion holds, so $+\infty$ does not automatically mean perfect one-shot discrimination. For local QFT, the safe statement is that Araki relative entropy quantifies distinguishability using observables in the chosen region, and restriction to a smaller algebra cannot increase it.

6. Finite-dimensional entropy calculation

Let ${\omega }_{\rho }(a)=\mathrm{Tr}(\rho a)$ and ${\omega }_{\sigma }(a)=\mathrm{Tr}(\sigma a)$ with $\rho ,\sigma >0$. We compute $S({\omega }_{\rho }\Vert {\omega }_{\sigma })$ from Definition 5.1.

Since

$${\Delta }_{\sigma ,\rho }=L_{\sigma }{R}_{{\rho }^{-1}},$$

we have

$$\log {\Delta }_{\sigma ,\rho }=L_{\log \sigma }-R_{\log \rho }.$$

Acting on ${\Omega }_{\rho }={\rho }^{1/2}$,

$$\log {\Delta }_{\sigma ,\rho }{\rho }^{1/2}=\log \sigma {\rho }^{1/2}-{\rho }^{1/2}\log \rho .$$

Therefore

$$\begin{array}{rl}S({\omega }_{\rho }\Vert {\omega }_{\sigma })& =-⟨{\rho }^{1/2},\log {\Delta }_{\sigma ,\rho }{\rho }^{1/2}⟩\\ & =-\mathrm{Tr}\left({\rho }^{1/2}\log \sigma {\rho }^{1/2}\right)+\mathrm{Tr}\left(\rho \log \rho \right)\\ & =\boxed{\mathrm{Tr}\rho (\log \rho -\log \sigma )}.\end{array}$$

So Araki-Uhlmann relative entropy reduces exactly to the usual Umegaki relative entropy in finite dimensions. (The convention sign-flip that appears in the cocycle does not affect the entropy formula — entropy is built from $\log \Delta$, which is convention-independent in the operator-theoretic sense.)

7. Worked example: two thermal states

Let

$${\rho }_{\beta }=\frac{e^{-\beta H}}{Z(\beta )},Z(\beta )=\mathrm{Tr}(e^{-\beta H}),$$

and compare ${\rho }_{{\beta }_1}$ with ${\rho }_{{\beta }_2}$ for the same Hamiltonian.

Since

$$\log {\rho }_{\beta }=-\beta H-\log Z(\beta ),$$

we get

$$\begin{array}{rl}S({\rho }_{{\beta }_1}\Vert {\rho }_{{\beta }_2})& =\mathrm{Tr}{\rho }_{{\beta }_1}\left[(-{\beta }_{1}H-\log Z({\beta }_1))-(-{\beta }_{2}H-\log Z({\beta }_2))\right]\\ & =({\beta }_2-{\beta }_1)⟨H{⟩}_{{\beta }_1}+\log Z({\beta }_2)-\log Z({\beta }_1).\end{array}$$

Equivalently, using $F(\beta )=-(1/\beta )\log Z(\beta )$,

$$S({\rho }_{{\beta }_1}\Vert {\rho }_{{\beta }_2})=({\beta }_2-{\beta }_1)⟨H{⟩}_{{\beta }_1}-{\beta }_{2}F({\beta }_2)+{\beta }_{1}F({\beta }_1).$$

This is not just a formula. It says relative entropy between thermal states is a free-energy comparison measured in the state ${\rho }_{{\beta }_1}$.

Second-order expansion: relative entropy as heat capacity. Set ${\beta }_2={\beta }_1+\delta \beta$ and expand. The first-order term vanishes — differentiate the formula in ${\beta }_2$ at ${\beta }_2={\beta }_1$ using ${\partial }_{\beta }\log Z=-⟨H{⟩}_{\beta }$, and the two contributions cancel. This is positivity at work: $S\ge 0$ with equality at $\delta \beta =0$ forces the expansion to start at quadratic order. The quadratic term is

$$S({\rho }_{{\beta }_1}\Vert {\rho }_{{\beta }_1+\delta \beta })=\frac{1}{2}(\delta \beta {)}^2(⟨H^2{⟩}_{{\beta }_1}-⟨H{⟩}_{{\beta }_1}^2)+O(\delta {\beta }^3),$$

the energy variance — i.e. $C/{\beta }^2$ in terms of the heat capacity $C$. So for nearby temperatures, relative entropy is the thermodynamic metric: distinguishability of neighboring thermal states is controlled by the system’s energy fluctuations, the same quantity that controls how sharply temperature can be measured. This is the simplest instance of the general principle that the second variation of $S(\omega \Vert \varphi )$ defines an information metric (Kubo–Mori/Bogoliubov) on the state space — the structure underlying “first law of entanglement” calculations in QFT.

8. Why this survives in type III

Von Neumann entropy

$$S(\rho )=-\mathrm{Tr}(\rho \log \rho )$$

requires a density matrix and a trace. For the local QFT examples classified in Block C, phase-space/nuclearity input supplies injectivity or hyperfiniteness, while a separate scaling or modular-spectrum theorem supplies the III${}_1$ subtype; factoriality and separable predual then give the standard hyperfinite III${}_1$ factor. There is no canonical local trace, hence no local density matrix relative to such a trace and no regulator-independent sharp-region von Neumann entropy.

Relative entropy does not suffer this failure. Araki’s definition uses the relative modular operator, which is available for normal states on von Neumann algebras of any type.

The course will use this distinction repeatedly:

• Not available in type III: local entanglement entropy as $-\mathrm{Tr}(\rho \log \rho )$ without a regulator or split inclusion.
• Available in type III: $S(\omega \Vert \varphi )$ for two normal states on the same local algebra.
• Bridge to crossed products: after adjoining a modular clock, one can recover a semifinite trace on a dressed algebra; Block D explains this carefully.

9. Relation to perturbation theory

Suppose ${\omega }_s$ is a differentiable family of faithful normal states with ${\omega }_0=\varphi$. The cocycle

$$u_t(s)=(D{\omega }_s/D\varphi {)}_t$$

measures how the modular flow changes under the perturbation:

$${\sigma }_t^{{\omega }_s}=\mathrm{Ad}(u_t(s))\circ {\sigma }_t^{\varphi }.$$

Differentiating in $s$ gives an abstract modular-response operator and is the seed of Connes-cocycle perturbation theory. We will not develop that theory here. In particular, this construction should not be conflated with every later calculation called a perturbation: the AAJ calculation in Semester II instead uses unitary covariance, spectral reweighting/Jacobians, and BCH commutators. Week 7 supplies the comparison theory for faithful weights and their modular flows; it is not a universal template for perturbative dressed dynamics.

10. What to take away

• Stated only: the relative Tomita operator gives a relative modular operator ${\Delta }_{\omega ,\varphi }$ for two faithful normal states.
• Stated only: the Connes cocycle $(D\omega /D\varphi {)}_t$ lies in $\mathcal{M}$ and intertwines the two modular flows.
• Model proof: in finite dimensions, $(D{\omega }_{\rho }/D{\omega }_{\sigma }{)}_t={\rho }^{-it}{\sigma }^{it}$ (our sign convention).
• Computed here: Araki-Uhlmann relative entropy reduces to $\mathrm{Tr}\rho (\log \rho -\log \sigma )$ for matrices.
• Research arc: relative entropy and cocycles are the type-III-safe tools that replace density matrices and von Neumann entropy in local QFT.

11. Looking ahead

Block C applies the machinery to free QFT. Week 8 introduces Weyl algebras and the free scalar field; Week 9 brings in Reeh-Schlieder, which supplies cyclic-separating vectors for local algebras; Week 10 identifies a modular flow geometrically through Bisognano-Wichmann. The abstract objects $S$, $J$, $\Delta$, and $(D\omega /D\varphi {)}_t$ will stop looking abstract.

12. Problem set

Core problems.

1. Relative modular operator in Hilbert-Schmidt form. Let $\rho ,\sigma >0$ on $M_n(\mathbb{C})$. Starting from

$$S_{\rho ,\sigma }(a{\sigma }^{1/2})=a^{\ast }{\rho }^{1/2},$$

verify that

$${\Delta }_{\rho ,\sigma }(X)=\rho X{\sigma }^{-1}.$$

2. Cocycle identity. With $u_t={\rho }^{-it}{\sigma }^{it}$ and ${\sigma }_t^{\sigma }(a)={\sigma }^{-it}a{\sigma }^{it}$, prove

$$u_{t+s}=u_t{\sigma }_t^{\sigma }(u_s).$$

3. Intertwining. Prove directly that

$${\sigma }_t^{\rho }(a)=u_t{\sigma }_t^{\sigma }(a)u_t^{\ast }.$$

4. Relative entropy. Derive

$$S({\omega }_{\rho }\Vert {\omega }_{\sigma })=\mathrm{Tr}\rho (\log \rho -\log \sigma )$$

from the Araki-Uhlmann definition.

5. Two thermal states. For $H=\mathrm{diag}(0,E)$, compute $S({\rho }_{{\beta }_1}\Vert {\rho }_{{\beta }_2})$ explicitly as a function of ${\beta }_1,{\beta }_2,E$.

Starred problems.

6*. Commuting states. Suppose $\rho$ and $\sigma$ commute. Show that the Connes cocycle is a one-parameter unitary group:

$$u_{t+s}=u_t{u}_s.$$

Explain why the cocycle identity is genuinely more general when $\rho$ and $\sigma$ do not commute.

7*. Positivity in finite dimensions. Prove Klein’s inequality

$$\mathrm{Tr}\rho (\log \rho -\log \sigma )\ge \mathrm{Tr}(\rho -\sigma ),$$

and use it to prove $S(\rho \Vert \sigma )\ge 0$ for density matrices.

8*. Monotonicity for partial trace. State the data-processing inequality

$$S({\rho }_{AB}\Vert {\sigma }_{AB})\ge S({\rho }_A\Vert {\sigma }_A).$$

Prove it in the special case where ${\rho }_{AB}$ and ${\sigma }_{AB}$ are block-diagonal classical-quantum states.

9*. Relative entropy for displaced Gaussian states (forward-looking). This problem uses the oscillator/Gaussian machinery developed in Week 8; it may be postponed until then. Let ${\rho }_{\mathrm{th}}=(1-q)q^N$ for one oscillator, with $0<q<1$ and mean occupation $\bar{n}=q/(1-q)$. Define ${\rho }_{\alpha }=D(\alpha ){\rho }_{\mathrm{th}}D(\alpha {)}^{\ast }$ and similarly ${\rho }_{\beta }$, using $D(\alpha {)}^{\ast }aD(\alpha )=a+\alpha$. You may use unitary invariance of relative entropy, $\log (U\rho U^{\ast })=U(\log \rho )U^{\ast }$, and

$$D(\gamma {)}^{\ast }ND(\gamma )=N+\gamma a^{\ast }+\overline{\gamma }a+|\gamma {|}^2.$$

Show that

$$S({\rho }_{\alpha }\Vert {\rho }_{\beta })=\log \left(\frac{\bar{n}+1}{\bar{n}}\right)|\alpha -\beta {|}^2=(-\log q)|\alpha -\beta {|}^2.$$

Then explain why the pure coherent-vector case on $\mathcal{B}(\mathcal{H})$ has a support singularity, and hence infinite relative entropy, when $\alpha \ne \beta$.

Self-study answer checkpoints

These checkpoints cover the core problems. Starred, project, and explicitly research-level problems remain source-led; they should be completed with the references and hypotheses named in the problem.

1. On Hilbert—Schmidt space the decisive identity is $${\Delta }_{\rho ,\sigma }=L_{\rho }{R}_{{\sigma }^{-1}},{\Delta }_{\rho ,\sigma }(X)=\rho X{\sigma }^{-1}.$$ In particular, on the defining core $X=a{\sigma }^{1/2}$ one obtains ${\Delta }_{\rho ,\sigma }X=\rho a{\sigma }^{-1/2}$.
2. Direct multiplication gives $$u_t{\sigma }_t^{\sigma }(u_s)={\rho }^{-it}{\rho }^{-is}{\sigma }^{is}{\sigma }^{it}=u_{t+s}.$$ No commutativity between $\rho$ and $\sigma$ is used.
3. The middle factors cancel: $$u_t{\sigma }_t^{\sigma }(a)u_t^{\ast }={\rho }^{-it}a{\rho }^{it}={\sigma }_t^{\rho }(a).$$
4. Since $\log {\Delta }_{\sigma ,\rho }=L_{\log \sigma }-R_{\log \rho }$, evaluation on ${\rho }^{1/2}$ gives $$S({\omega }_{\rho }\Vert {\omega }_{\sigma })=\mathrm{Tr}\rho (\log \rho -\log \sigma ).$$
5. With $Z_i=1+e^{-{\beta }_{i}E}$ and excited-state probability $q_i=e^{-{\beta }_{i}E}/Z_i$, the final result is $$S({\rho }_{{\beta }_1}\Vert {\rho }_{{\beta }_2})=({\beta }_2-{\beta }_1)E{q}_1+\log \frac{Z_2}{Z_1}.$$ For $E>0$ it is nonnegative and vanishes precisely when ${\beta }_1={\beta }_2$.

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Notes prepared for the algebraic-QFT course, Semester I Block B. Last revised 2026-08-24.