Relative Modular Operators and Connes Cocycles

Relative modular theory compares two states on the same algebra without introducing a trace or pretending that their logarithms commute. The relative modular operator supplies Araki relative entropy, while the Connes cocycle is an algebra-valued unitary that converts one state’s modular flow into the other’s. Both constructions depend on the order of the states and on their supports.

Required background. Tomita–Takesaki flow supplies the standard representation and modular automorphisms.

Helpful background. Araki relative entropy supplies the information-theoretic application and support conventions.

The chapter’s structure map locates relative modular data within a standard algebra–state representation. Its comparison table and validity guide keep support, analyticity, and geometric claims separate.

Relative Tomita data and label order

Let $\phi$ and $\psi$ be faithful normal states represented by vectors $\Phi$ and $\Psi$ in the natural cone of a standard form of $\mathcal A$. On the dense set $\mathcal A\Psi$, define

$$S_{\Phi\mid\Psi,0}(A\Psi)=A^*\Phi.$$

This antilinear operator is closable. Its closure has positive square

$$\Delta_{\Phi\mid\Psi} =S_{\Phi\mid\Psi}^*S_{\Phi\mid\Psi},$$

the relative modular operator. With this ordering convention,

$$S(\psi\Vert\phi) =-\langle\Psi,\log\Delta_{\Phi\mid\Psi}\,\Psi\rangle.$$

Araki defines the extended-valued quantity and its relative modular operator in Araki 1976, § 2, pp. 810–817, especially eqs. (2.1)–(2.3). The expectation formula, rather than the symbol alone, fixes which state appears first.

In the finite standard representation on Hilbert–Schmidt matrices,

$$\Delta_{\Phi\mid\Psi}(X)=\rho_\phi X\rho_\psi^{-1}.$$

Left and right multiplication commute as superoperators, so spectral calculus gives

$$\log\Delta_{\Phi\mid\Psi} =L_{\log\rho_\phi}-R_{\log\rho_\psi}.$$

Evaluating on $\Psi=\rho_\psi^{1/2}$ yields

$$-\langle\rho_\psi^{1/2}, \log\Delta_{\Phi\mid\Psi}\,\rho_\psi^{1/2}\rangle =\operatorname{Tr}\rho_\psi (\log\rho_\psi-\log\rho_\phi).$$

This does not identify $\log\Delta$ with a difference of two operators acting on one copy of the original Hilbert space. The left/right labels are essential.

Connes cocycles change the modular frame

For faithful normal states, choose any auxiliary faithful normal state or weight $\chi$ and set

$$u_s=[D\psi:D\phi]_s :=\Delta_{\psi\mid\chi}^{is} \Delta_{\phi\mid\chi}^{-is}.$$

The auxiliary object cancels, $u_s$ belongs to $\mathcal A$, and

$$u_{s+t}=u_s\,\sigma_s^\phi(u_t).$$

It is a cocycle, rather than an ordinary representation, because the second factor is transported by the reference flow. It intertwines the two modular groups:

$$\sigma_s^\psi(A) =u_s\,\sigma_s^\phi(A)\,u_s^*.$$

For a third faithful state $\omega$, consistent label conventions give the chain rule

$$[D\psi:D\phi]_s[D\phi:D\omega]_s =[D\psi:D\omega]_s.$$

Connes’ original Radon–Nikodym cocycle construction and modular-group comparison appear in Connes 1973, § 1.2. The chain-rule form for state representatives is developed in Araki 1974, §§ 3–4, pp. 319–334.

In a finite standard representation, for any faithful density matrices—commuting or not—

$$u_s=\rho_\psi^{is}\rho_\phi^{-is}.$$

If the matrices commute, this becomes

$$u_s=e^{is(\log\rho_\psi-\log\rho_\phi)}.$$

For noncommuting states the ordered product remains correct, but the single exponential does not: the Baker–Campbell–Hausdorff series contains commutators.

A faithful two-mode benchmark

Use two regulated fermionic modes with occupation basis

$$\lvert00\rangle,\quad\lvert01\rangle,\quad \lvert10\rangle,\quad\lvert11\rangle,$$

and full algebra $M_4(\mathbb C)$. At a fixed lattice spacing $a$, choose the faithful states

$$\rho_\phi=\frac{\mathbf1_4}{4}, \qquad \rho_\psi=\frac{1}{10} \operatorname{diag}(1,2,3,4).$$

Write $q=(1,2,3,4)/10$. On the matrix units,

$$\Delta_{\Phi\mid\Psi}(E_{ij}) =\frac{1}{4q_j}E_{ij},$$

so the complete relative modular spectrum is read from the four column labels. The cocycle is

$$u_s =\operatorname{diag}\!\left( (4q_1)^{is},(4q_2)^{is},(4q_3)^{is},(4q_4)^{is} \right),$$

and direct substitution verifies both the cocycle identity and the intertwining relation. The relative entropy is

$$S(\psi\Vert\phi) =\sum_{i=1}^4q_i\log(4q_i) \simeq0.106440.$$

This is simultaneously a bipartite two-qubit calculation and a two-mode regulated field reduction. The structural identities match exactly at fixed $a$ because the regulated regional algebra is type I. No continuum claim follows until a family of lattices, embeddings of the regional algebras, and convergence of the compared observables have been specified.

The smallest eigenvalue is $\delta=0.1$. If every $q_i$ is reconstructed within $\eta<\delta$, then

$$\lVert\rho_\psi^{-1}\rVert \leq\frac{1}{\delta-\eta}, \qquad \lvert\delta\log q_i\rvert \leq\frac{\eta}{\delta-\eta}.$$

Thus $(a,\eta,\delta)$ are the regulator, numerical uncertainty, and support-gap controls. The benchmark has no hidden statistical error when the displayed matrices are used exactly.

Support restrictions and extended relative entropy

For nonfaithful states, support inclusion is directional. If

$$s(\psi)\nleq s(\phi),$$

then $S(\psi\Vert\phi)=+\infty$. If $s(\psi)\leq s(\phi)$, the logarithmic spectral expectation may still diverge; support inclusion is necessary but not sufficient for finiteness in infinite dimension. Araki’s nonfaithful extension and support reduction are given in Araki 1977, §§ 2–3, pp. 176–184.

For a concrete unequal-support test, take

$$\rho_\psi=\lvert0\rangle\langle0\rvert, \qquad \rho_\phi=\operatorname{diag}(3/4,1/4).$$

Then $S(\psi\Vert\phi)=\log(4/3)$, but $\rho_\psi^{is}$ is not a unitary on the full two-dimensional space. On the support $p_\psi=\lvert0\rangle\langle0\rvert$, the reduced cocycle is

$$u_s=(4/3)^{is}p_\psi,$$

a partial isometry in the unreduced algebra and a unitary in the support corner. Reversing the comparison gives $s(\phi)\nleq s(\psi)$ and $S(\phi\Vert\psi)=+\infty$. Full-algebra inverses, full unitarity, and unrestricted analytic continuation are precisely the claims that fail; real-time support-reduced identities survive.

Support inclusion also does not guarantee a finite answer. On the diagonal algebra $\ell^\infty(\mathbb N)$, let

$$p_n=\frac{6}{\pi^2n^2}, \qquad q_n=(e-1)e^{-n}.$$

Both distributions have full support, but

$$\sum_{n=1}^\infty p_n\log\frac{p_n}{q_n}=+\infty$$

because the summand has a positive harmonic-order tail. Faithfulness therefore removes kernels; it does not supply logarithmic integrability.

Perturbations and modular response

For a differentiable faithful family $\psi_\lambda$ with $\psi_0=\phi$, the derivative of $[D\psi_\lambda:D\phi]_s$ is transported by the reference modular flow. Resolvent or modular-frequency representations require their own analytic domains. The state-susceptibility page distinguishes the vanishing first variation of relative entropy from its positive quadratic term.

Common pitfalls

Subtracting logarithms inside an exponential. Preserve $\rho_\psi^{is}\rho_\phi^{-is}$ unless the logarithms commute. The first omitted Baker–Campbell–Hausdorff term is already quadratic in $s$.

Suppressing the reference algebra. Relative modular data compare states on one specified algebra. Changing the region is a different problem.

Treating support inclusion as an integrability estimate. It removes an immediate kernel obstruction but does not bound the logarithmic spectral expectation.

Exercises

1. Reproduce the benchmark’s relative modular eigenvalues, cocycle, and relative entropy. Verify the intertwining relation on one off-diagonal matrix unit $E_{ij}$.

Solution

Since $\rho_\phi=\mathbf1_4/4$,

$$\Delta_{\Phi\mid\Psi}(E_{ij}) =\rho_\phi E_{ij}\rho_\psi^{-1} =\frac{1}{4q_j}E_{ij}.$$

Also $\rho_\phi^{-is}=4^{is}\mathbf1$, giving

$$u_sE_{ij}u_s^* =\left(\frac{q_i}{q_j}\right)^{is}E_{ij}.$$

The reference flow $\sigma_s^\phi$ is trivial, so this equals $\sigma_s^\psi(E_{ij})$ and proves intertwining. Finally,

$$-\operatorname{Tr}\rho_\psi (\log\rho_\phi-\log\rho_\psi) =\sum_iq_i\log(4q_i),$$

whose numerical value is approximately $0.106440$.

2. Let

$$\rho_\phi=\operatorname{diag}(2/3,1/3), \qquad \rho_\psi=H\operatorname{diag}(3/4,1/4)H,$$

where $H$ is the Hadamard matrix. Show that the ordered cocycle cannot equal $e^{is(\log\rho_\psi-\log\rho_\phi)}$ for generic $s$.

Solution

Up to scalar multiples of the identity,

$$\log\rho_\psi=\frac{\log3}{2}\sigma_x+\text{scalar}, \qquad \log\rho_\phi=\frac{\log2}{2}\sigma_z+\text{scalar}.$$

Their commutator is

$$[\log\rho_\psi,\log\rho_\phi] =-\frac{i\log3\log2}{2}\sigma_y\neq0.$$

The Baker–Campbell–Hausdorff expansion gives

$$\rho_\psi^{is}\rho_\phi^{-is} =\exp\!\left( is(\log\rho_\psi-\log\rho_\phi) +\frac{s^2}{2} [\log\rho_\psi,\log\rho_\phi] +O(s^3) \right).$$

The commutator term is absent from the proposed single exponential, so the two expressions disagree for generic nonzero $s$.

3. Analyze both orders of the unequal-support qubit example. Identify the inverse, entropy, and cocycle statement that survives in each order.

Solution

For $\psi=\lvert0\rangle\langle0\rvert$ and faithful $\phi$,

$$S(\psi\Vert\phi) =-\langle0\rvert\log\rho_\phi\lvert0\rangle =\log(4/3).$$

The inverse of $\rho_\psi$ does not exist on the full space, so the faithful full-algebra formula is unavailable. Compression to $p_\psi$ gives the unitary scalar cocycle $(4/3)^{is}$ in the corner, represented by the partial isometry $(4/3)^{is}p_\psi$ before compression. In the reverse order, $s(\phi)$ is not contained in $s(\psi)$, so $S(\phi\Vert\psi)=+\infty$ and no normalized support-reduced comparison contains the full state $\phi$.

References

• Araki, Huzihiro. “Some Properties of Modular Conjugation Operator of von Neumann Algebras and a Non-Commutative Radon–Nikodym Theorem with a Chain Rule.” Pacific Journal of Mathematics 50 (1974): 309–354. DOI; Open PDF.
• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI; Open PDF.
• Araki, Huzihiro. “Relative Entropy for States of von Neumann Algebras II.” Publications of the Research Institute for Mathematical Sciences 13 (1977): 173–192. DOI; Open article.
• Connes, Alain. “Une classification des facteurs de type III.” Annales Scientifiques de l’École Normale Supérieure 6 (1973): 133–252. DOI; Numdam.