Petz Recovery Map in Quantum Information

• Petz recovery map is a canonical quantum operation that reverses channels using the equality condition of the data‐processing inequality, thereby enabling state recoverability.
• It provides necessary and sufficient conditions for quantum Markovianity while ensuring stability and robustness in entropy monotonicity through quantitative bounds.
• Its explicit construction in both finite-dimensional matrices and operator-algebraic frameworks underpins applications in quantum error correction, entanglement wedge reconstruction, and channel reversibility.

The Petz recovery map is a canonical quantum operation that reverses a quantum channel on a specified reference state and plays an essential role at the interface of quantum information theory, operator algebras, quantum error correction, and mathematical physics. It is structurally determined by the equality case of the data-processing inequality for quantum relative entropy and admits a closed form in terms of the channel and reference state. The Petz map provides necessary and sufficient recoverability conditions for quantum Markovianity, underpins stability and robustness bounds for entropy monotonicity, and admits quantitative guarantees in both exact and approximate scenarios.

1. Definitions and Mathematical Structure

Let $$\mathcal{N}:\mathcal{B}(\mathcal{H})\to\mathcal{B}(\mathcal{K})$$ be a completely positive, trace-preserving (CPTP) map (quantum channel), and let $\sigma > 0$ be a full-rank reference state (density operator) on $\mathcal{H}$. The Petz recovery map $$\mathcal{R}_{\sigma,\mathcal{N}}:\mathcal{B}(\mathcal{K})\to\mathcal{B}(\mathcal{H})$$ is defined by: $$\boxed{ \mathcal{R}_{\sigma,\mathcal{N}}(X) = \sigma^{1/2} \, \mathcal{N}^\dagger \left[\mathcal{N}(\sigma)^{-1/2} \, X \, \mathcal{N}(\sigma)^{-1/2}\right] \sigma^{1/2} }$$ where $\mathcal{N}^\dagger$ is the Hilbert–Schmidt adjoint (trace-dual) of $\mathcal{N}$, and all inverses are defined on the support of $\mathcal{N}(\sigma)$. In operator-algebraic settings, the same construction generalizes using GNS and modular conjugation structures.

Key properties:

• $\mathcal{R}_{\sigma,\mathcal{N}}$ is CPTP on the support of $\mathcal{N}(\sigma)$.
• $$\mathcal{R}_{\sigma,\mathcal{N}}(\mathcal{N}(\sigma)) = \sigma$$.
• If $$S(\rho\|\sigma) = S(\mathcal{N}(\rho)\|\mathcal{N}(\sigma))$$ for some $\rho$, then $$(\mathcal{R}_{\sigma,\mathcal{N}}\circ\mathcal{N})(\rho) = \rho$$.

The definition remains structurally identical in infinite-dimensional von Neumann algebra settings, with the modular theory (Tomita–Takesaki) replacing explicit matrix-inverse operations (Furuya et al., 2020).

2. Role in Data-Processing Inequality and Equality Characterizations

The Umegaki relative entropy

$$S(\rho\|\sigma) := \operatorname{Tr}[\rho(\log\rho - \log\sigma)]$$

obeys the monotonicity (data-processing) inequality under all CPTP maps $\mathcal{N}$: $$S(\rho\|\sigma) \geq S\left(\mathcal{N}(\rho) \| \mathcal{N}(\sigma)\right).$$ Petz’s theorem asserts the equivalence: $$S(\rho\|\sigma) = S(\mathcal{N}(\rho)\|\mathcal{N}(\sigma)) \iff \rho = \mathcal{R}_{\sigma,\mathcal{N}}\circ\mathcal{N}(\rho),\; \sigma = \mathcal{R}_{\sigma,\mathcal{N}}\circ\mathcal{N}(\sigma).$$ Quantitatively, for $\rho,\sigma$ faithful and suitable $\mathcal{N}$, for any $\rho$,

$$S(\rho\|\sigma) - S(\mathcal{N}(\rho)\|\mathcal{N}(\sigma)) \geq -2\log F\left(\rho, (\mathcal{R}_{\sigma,\mathcal{N}}\circ\mathcal{N})(\rho)\right)$$

where $F$ is the Uhlmann fidelity.

This property places the Petz map at the core of equality-saturation characterizations for quantum Markov chains, quantum channels, and sufficiency of subalgebras (Carlen et al., 2017, Furuya et al., 2020).

3. Explicit Constructions and Algebraic Frameworks

3.1. Finite-Dimensional Matrix Algebras

In the finite-dimensional setting, for von Neumann algebras $\mathcal{M}$, $\mathcal{N}$ with conditional expectation $\mathcal{E}_\tau$, the Petz recovery map for $\rho\in\mathcal{M}$ reads (Carlen et al., 2017): $$\mathcal{R}_{\rho}(X) = \rho^{1/2} \, \rho_{\mathcal{N}}^{-1/2} \, X \, \rho_{\mathcal{N}}^{-1/2} \, \rho^{1/2}, \quad X\in\mathcal{N}$$ here $\rho_{\mathcal{N}} := \mathcal{E}_\tau(\rho)$, and $\mathcal{R}_\rho$ is the adjoint of the Accardi–Cecchini coarse-graining.

3.2. Operator Algebras and Modular Theory

For normal unital CP maps $\Phi:\mathcal{A}\to\mathcal{B}$ between von Neumann algebras and faithful normal states, the Petz dual is

$$\Phi^P_{\rho} (b) = \rho_A^{-1/2}\; \Phi^* \left( \rho_B^{1/2} \, b \, \rho_B^{1/2} \right)\; \rho_A^{-1/2}$$

with $\Phi^*$ the Schrödinger-picture adjoint (Furuya et al., 2020). Exact recoverability is characterized by the existence of a $\rho$-preserving conditional expectation onto the correctable subalgebra.

3.3. Gaussian and Fermionic Systems

The Petz map preserves the Gaussian structure for both bosonic and fermionic systems (Chen et al., 8 Nov 2025, Swingle et al., 2018). For single-mode Gaussian loss channels, the Petz map is again a single-mode Gaussian map (either a beam-splitter or a phase-insensitive amplifier), with explicit parameters determined by the transmissivity, environment noise, and reference state.

For fermionic Gaussian channels, the Petz map is structurally Gaussian; its covariance-matrix action is explicitly constructed via composition of sandwich, adjoint, and normalization kernels in the Grassmann path integral formalism.

4. Solution Set Structure and Characterizations

Describing all solutions to the Petz equation $\sigma = \mathcal{R}_{\rho}(\sigma_{\mathcal{N}})$ reduces to analyzing the fixed-point algebra $$C = \{X \in \mathcal{N} : \mathcal{A}_\rho(X) = X\}$$ of the Accardi–Cecchini map. $C$ is a von Neumann subalgebra of $\mathcal{N}$, and its center $Z$ consists of a finite family of orthogonal projections. States $\sigma$ solving the Petz equation are classified as block-diagonal: $$\sigma = \bigoplus_{j=1}^J \beta_j\,(\tau_j \otimes \omega_j),\quad \beta_j > 0$$ with $\tau_j$ determined by blocks of $\rho$ and $\omega_j$ arbitrary states; this includes all conditionally expectation–invariant states onto $C$ under $\rho$ (Carlen et al., 2017).

5. Stability, Robustness, and Approximate Recovery Bounds

Carlen–Vershynina establish a quantitative stability bound for the data-processing inequality in terms of the trace-norm deviation from perfect Petz recovery (Carlen et al., 2017): $$S(\rho\|\sigma) - S(\rho_{\mathcal{N}}\|\sigma_{\mathcal{N}}) \geq \left(\frac{1}{8\pi}\right)^{4} \|\Delta_{\sigma,\rho}\|^{-2} \| \mathcal{R}_{\rho}(\sigma_{\mathcal{N}}) - \sigma \|_1^4$$ with $\Delta_{\sigma,\rho}$ the relative modular operator.

The same structural result extends to operator-convex quasi-entropies: $$S_{f}(\rho\|\sigma) := \operatorname{Tr}[f(\Delta_{\sigma,\rho})\rho]$$ Equality is achieved only when $\sigma$ is a fixed point of the Petz map, with analogous stability bounds for small deviations.

Pinched and rotated Petz maps, as in the measured relative entropy strengthened bounds (Sutter et al., 2015), yield

$$D(\rho\|\sigma) - D(\mathcal{N}(\rho)\|\mathcal{N}(\sigma)) \geq D_{\mathrm{meas}}\left(\rho \Big\| (\mathcal{R}_{\sigma,\mathcal{N},\rho} \circ \mathcal{N})(\rho)\right)$$

with $\mathcal{R}_{\sigma,\mathcal{N},\rho}$ a convex combination of rotated Petz maps.

Approximate recoverability for the Petz map has been refined using the second sandwiched Rényi entropy $$\tilde D_2(\rho\|\sigma) = \log \operatorname{Tr}\left[(\sigma^{-1/4} \rho \sigma^{-1/4})^2\right]$$. Order-$\epsilon$ recovery is guaranteed when the data-processing inequality for $\tilde D_2$ is saturated up to $O(\epsilon^2 / d)$ for Hilbert space dimension $d$, with Petz-map reconstruction error scaling as $O(\sqrt{d\,\Delta \tilde D_2})$ (Cree et al., 2021).

6. Applications in Quantum Error Correction, Markov Chains, and Quantum Channels

• Quantum Error Correction: The Petz map achieves universal recovery for codes and channels and realizes explicit decoders achieving the coherent information rate for quantum channel coding, including finite blocklength and one-shot regimes. For a code subspace $C$ and noise channel $\mathcal{E}$, the adapted Petz map is:

$$\mathcal{R}_{P,\mathcal{E}}(\cdot) = P\,\mathcal{E}^\dagger\left[\mathcal{E}(P)^{-1/2} (\cdot) \mathcal{E}(P)^{-1/2}\right] P$$

with $P$ the codespace projector (Biswas et al., 2023, Beigi et al., 2015). Stabilized and continuous-time protocols for Lindbladian dynamics leverage the Petz structure for engineered robust error suppression (Kwon et al., 2021).

• Quantum Markov Chains and Conditional Mutual Information: The saturation of strong subadditivity and conditional mutual information $I(A:C|B)=0$ is characterized by the existence of a perfect Petz recovery mapping, with the conditional mutual information serving as a measure of departure from perfect recoverability.
• Entanglement Wedge Reconstruction: In holographic duality (AdS/CFT), explicit Petz recovery realizes the entanglement wedge reconstruction map, reducing to the HKLL formula in symmetric cases and to modular-flow representations generally. The universal error bound is non-perturbatively small—$\le d_{\rm code}\sqrt{8\delta}$ for code dimension $d_{\rm code}$ and error $\delta$ (Chen et al., 2019, Bahiru et al., 2022).
• Quantum Channel Reversibility and Lindbladian Dynamics: Tabletop reversibility conditions—when the Petz map is physically identical to reversing the system-environment coupling—are realized under specific algebraic and dynamical constraints on the channel, reference state, and ancilla control (Song et al., 30 Oct 2025).
• Physical Platforms and Implementations: Resource-optimal circuit decompositions of the Petz map for specific channels (dephasing, amplitude-damping, depolarizing) on trapped-ion and NISQ devices have been constructed, with performance bottlenecks determined by ancilla count and CNOT depth. In the single-qubit rank-2 case, Petz recovery requires only one ancilla and three CNOT equivalences (Png et al., 29 Apr 2025).

7. Broader Contexts and Generalizations

• The structure and recoverability properties of the Petz map persist in infinite-dimensional algebras, fermionic and bosonic Gaussian systems, and for arbitrary reference states.
• For channels exhibiting strong scrambling, such as Haar-random evolutions in black hole models, the Petz map simplifies to the normalized channel adjoint (“Petz-lite”), matching previously proposed decoders in quantum gravity (Nakayama et al., 2023).
• In quantum many-body settings, Petz map recovery error and fidelity provide sharp diagnostics of phase structure and order, distinguishing, for example, phases by the scaling of fidelity loss with conditional mutual information and revealing operational meanings for topological entanglement entropy (Hu et al., 1 Aug 2024).

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Summary Table: Key Mathematical Objects

Construction	Formula / Property	Context/Condition
Petz recovery map (finite dim.)	$$\mathcal{R}_{\sigma,\mathcal{N}}(X) = \sigma^{1/2} \mathcal{N}^\dagger \left[\mathcal{N}(\sigma)^{-1/2} X \mathcal{N}(\sigma)^{-1/2}\right]\sigma^{1/2}$$	$\sigma$ full-rank, $\mathcal{N}$ CPTP
Equality case data-processing	$$S(\rho\|\sigma) = S(\mathcal{N}(\rho)\|\mathcal{N}(\sigma)) \Longleftrightarrow \rho = \mathcal{R}_{\sigma,\mathcal{N}}\circ\mathcal{N}(\rho)$$	All finite-dimensional/factor settings
Stability bound (trace norm)	$$S(\rho\|\sigma) - S(\mathcal{N}(\rho)\|\mathcal{N}(\sigma)) \geq C \| \mathcal{R}_{\sigma,\mathcal{N}}(\mathcal{N}(\rho)) - \rho\|_1^4$$	$C$ depends on modular, spectrum (Carlen et al., 2017)
Choi-matrix performance metric	$D\left(\mathcal{R}_{\sigma,\mathcal{N}} \circ \mathcal{N}, \mathds{1}\right) = \|J(\mathcal{R}\mathcal{N}) - J(\mathds{1})\|_1$	Channel-covariant, state-independent
Fermionic Gaussian Petz map	$$G_{\text{out}}^{\text{Petz}} = B_{\text{Petz}}(G_{\text{in}}- \Gamma) B_{\text{Petz}}^T$$, explicit $B_{\text{Petz}}$ in terms of covariance	Covariance-matrix formalism

The Petz recovery map thus forms a universal, algebraically, and physically grounded recovery channel indispensable in quantum information, error correction, mathematical physics, and practical device architectures. Its role as a necessary and sufficient witness for reversibility—both at the level of operator algebras and explicit quantum circuits—makes it a central object for the theoretical and experimental understanding of quantum noise and information flow.