Sufficiency, Conditional Expectations, and Petz Recovery

Equality in data processing certifies exact recoverability only after the channel, the reference state, its support, and the family of states to be recovered have been fixed. For an inclusion $\mathcal N\subset\mathcal M$, a faithful normal reference state admits a state-preserving conditional expectation precisely when $\mathcal N$ is invariant under its modular flow. Under those hypotheses the expectation and the Petz transpose channel express the same sufficiency mechanism.

Required background. Araki relative entropy and data processing provide the equality to be analyzed; normal completely positive maps fix the channel category. Helpful background. Split inclusions give the worked type-I factor, and modular automorphisms state the invariance criterion. Connections include conditional mutual information, approximate Markov recovery, Petz and rotated recovery, covariant recovery, and information–disturbance bounds.

Conditional expectations and modular invariance

A normal conditional expectation $E:\mathcal M\to\mathcal N$ is a normal unital completely positive projection satisfying the bimodule identity

$$E(N_1AN_2)=N_1E(A)N_2, \qquad N_1,N_2\in\mathcal N.$$

Let $\phi$ be a faithful normal state on $\mathcal M$. Takesaki’s theorem states that a $\phi$-preserving normal conditional expectation exists if and only if

$$\sigma_t^\phi(\mathcal N)=\mathcal N \quad\text{for every }t\in\mathbb R.$$

When it exists, it is unique and commutes with the modular flow Takesaki 1972, Theorems 1–2, pp. 309–315. Thus “project onto the subalgebra” is not a construction: positivity, the bimodule property, normality, and modular invariance are substantive requirements.

The proof mechanism uses the GNS vector $\Omega_\phi$. Modular invariance makes the closed subspace $\overline{\mathcal N\Omega_\phi}$ stable under the modular operator and conjugation. The orthogonal projection $e_\mathcal N$ then satisfies

$$e_\mathcal N A\Omega_\phi=E(A)\Omega_\phi,$$

and modular theory proves that the right-hand side defines a normal positive bimodule projection. Without invariance, the orthogonal projection need not send left multiplication by $\mathcal M$ to an element of $\mathcal N$.

Sufficiency and the Petz map

Let $\Phi:\mathcal A\to\mathcal B$ be a normal unital completely positive map in the Heisenberg direction, and let $\rho,\sigma$ be normal states on $\mathcal B$, with $\sigma$ faithful or with all objects reduced to its support. Data processing gives

$$S_{\mathcal A}(\rho\circ\Phi\Vert\sigma\circ\Phi) \leq S_{\mathcal B}(\rho\Vert\sigma).$$

The channel is sufficient for the pair $\{\rho,\sigma\}$ if there is a normal unital completely positive $\mathcal R:\mathcal B\to\mathcal A$ such that

$$\rho\circ\Phi\circ\mathcal R=\rho, \qquad \sigma\circ\Phi\circ\mathcal R=\sigma.$$

Under the stated faithfulness and normality hypotheses, equality in data processing is equivalent to sufficiency; the recovery is the transpose map determined by $\sigma$. For modular-analytic elements define the KMS inner product

$$\langle X,Y\rangle_\sigma^{\mathrm{KMS}} =\left\langle \Delta_\sigma^{1/4}X\xi_\sigma, \Delta_\sigma^{1/4}Y\xi_\sigma \right\rangle.$$

The transpose map is the KMS adjoint,

$$\langle\Phi(A),B\rangle_\sigma^{\mathrm{KMS}} =\langle A,\mathcal R_\sigma(B)\rangle_{\sigma\circ\Phi}^{\mathrm{KMS}},$$

first on an analytic core and then by normal extension. In matrix algebras its Schrödinger predual becomes the familiar formula

$$\mathcal R_{\sigma,*}(x) =\sigma^{1/2}\Phi\!\left(\Phi_*(\sigma)^{-1/2}x \Phi_*(\sigma)^{-1/2}\right)\sigma^{1/2},$$

with generalized inverses on supports and with picture conventions made explicit. Petz proved the von Neumann-algebra sufficiency/equality theorem using relative modular operators Petz 1986, Theorems 4–5, pp. 126–129.

One direction needs no formula. If a recovery $\mathcal R$ fixes both states, apply data processing first to $\Phi$ and then to $\mathcal R$:

$$S(\rho\Vert\sigma) \geq S(\rho\circ\Phi\Vert\sigma\circ\Phi) \geq S(\rho\circ\Phi\circ\mathcal R \Vert\sigma\circ\Phi\circ\mathcal R) =S(\rho\Vert\sigma).$$

Both inequalities must be equalities. The converse—from equality to construction of a normal completely positive recovery—is the genuinely modular part of Petz’s theorem and is where faithfulness and support reduction enter.

Exact equality for one pair does not recover every state. Nor does small entropy loss by itself imply exact recovery; approximate conclusions require an additional quantitative theorem and often a rotated or universal recovery map.

Split inclusion as an exact model

Suppose two separated local algebras admit a split representation

$$\mathcal N_1\vee\mathcal N_2 \simeq \mathcal N_1\,\overline\otimes\,\mathcal N_2 \subset B(\mathcal H_1\otimes\mathcal H_2).$$

Choose faithful normal states $\phi_1,\chi_2$ and the product reference $\phi=\phi_1\otimes\chi_2$. The map

$$E_{\chi_2}(A_1\otimes A_2)=\chi_2(A_2)A_1\otimes1$$

extends normally to a $\phi$-preserving conditional expectation onto $\mathcal N_1\otimes1$. Its modular flow factorizes, $\sigma_t^\phi=\sigma_t^{\phi_1}\otimes\sigma_t^{\chi_2}$, so the target algebra is invariant, independently verifying Takesaki’s criterion.

For every state $\rho_1\otimes\chi_2$ in the declared family, discarding the second factor and restoring $\chi_2$ is exact recovery. Relative entropy factorization gives

$$S(\rho_1\otimes\chi_2\Vert\phi_1\otimes\chi_2) =S(\rho_1\Vert\phi_1),$$

so data processing is saturated. This realizes the Petz recovery construction without assuming that the original sharp local algebras are type I.

Now perturb the reference to a faithful correlated state whose modular group does not preserve $\mathcal N_1\otimes1$. The same slice formula may still be a normal conditional expectation for a chosen $\chi_2$, but it is not reference-state preserving, and it is not the Petz map for the perturbed reference. This adversarial test separates algebraic projection from statistical sufficiency.

Exercises

1. Verify the bimodule law. Prove that $E_{\chi_2}$ is an $\mathcal N_1\otimes1$ bimodule map and preserves $\phi_1\otimes\chi_2$.

Solution

For elementary tensors, multiply on the left and right by $B_1\otimes1$ and $C_1\otimes1$. The slice gives $B_1\chi_2(A_2)A_1C_1\otimes1$, which is exactly $(B_1\otimes1)E_{\chi_2}(A_1\otimes A_2)(C_1\otimes1)$. Moreover $(\phi_1\otimes\chi_2)(E_{\chi_2}(A_1\otimes A_2))=\phi_1(A_1)\chi_2(A_2)$; normality extends both identities.

2. Equality is family-dependent. Let $\rho_{12}$ be correlated but have marginal $\rho_1$. Explain why discarding factor 2 cannot generally be recovered by adjoining the fixed state $\chi_2$.

Solution

The recovered state is $\rho_1\otimes\chi_2$, which has no correlations. It equals $\rho_{12}$ only when the latter already has that product form. Hence exact recovery for the product family does not imply sufficiency for all normal states.

References

• Petz, Dénes. “Sufficient Subalgebras and the Relative Entropy of States of a von Neumann Algebra.” Communications in Mathematical Physics 105 (1986): 123–131. DOI.
• Takesaki, Masamichi. “Conditional Expectations in von Neumann Algebras.” Journal of Functional Analysis 9 (1972): 306–321. DOI.