Reflected Entropy and Canonical Purifications

Reflected entropy assigns an entropy to a canonical purification of a mixed bipartite state. It captures total correlations through a construction different from mutual information and negativity; the purification is canonical relative to the chosen algebraic realization, not a claim that the doubled state is physically prepared.

Required background. Use Mutual Information and Regulator-Independent Correlations. Helpful background. Entanglement Negativity in QFT provides a contrasting mixed-state measure.

Evidence cutoff. Research-sensitive comparisons on this page include primary literature available through 10 August 2026.

Canonical purification

For a density operator $\rho_{AB}$, vectorize its positive square root in a doubled Hilbert space,

$$\lvert\sqrt{\rho_{AB}}\rangle \in\mathcal H_A\otimes\mathcal H_B \otimes\mathcal H_{A^*}\otimes\mathcal H_{B^*}.$$

The reflected entropy is

$$S_R(A{:}B)_\rho =S\!\left(\rho_{AA^*}\right), \qquad \rho_{AA^*} =\operatorname{Tr}_{BB^*} \lvert\sqrt{\rho_{AB}}\rangle \langle\sqrt{\rho_{AB}}\rvert.$$

Standard form or the GNS construction makes this definition basis independent at the algebraic level. In the continuum, a split inclusion or regulator supplies the type-I entropy; the split distance and limiting prescription remain part of the result.

The chapter diagram locates reflected entropy beside negativity on the mixed-state branch, not on the ordinary mutual-information branch.

The canonical purification turns a mixed $AB$ state into a pure state on $ABA^*B^*$. Entropy across $AA^*:BB^*$ measures correlations of the chosen algebraic realization; it is not a physical preparation protocol. Schematic.

Properties and a Gaussian check

For finite systems, reflected entropy is symmetric, nonnegative, and obeys $S_R(A{:}B)\geq I(A{:}B)$ in its standard domain. For a pure $\rho_{AB}$, it reduces to twice the entanglement entropy. These properties and a replica construction were introduced by Dutta and Faulkner 2021, §§2–3.

For a faithful two-mode Gaussian state, construct $\sqrt{\rho}$ through its covariance data, form the doubled covariance matrix, and compute $S(\rho_{AA^*})$. Check the pure-state and product-state limits. If a symplectic eigenvalue reaches a nonfaithful boundary, define the result by a controlled limit rather than inserting an unbounded inverse.

Holographic proposals relate reflected entropy to an entanglement-wedge cross section in special large-N states. That relation is not the definition and is not a generic QFT theorem. Primary results available through 10 August 2026 remain limited to their stated holographic state classes.

Purification and continuum limits

The validity map shows which choices the word “canonical” does not remove.

Canonical purification is canonical after the algebra and representation are fixed. Nonfaithful limits, changed split factors, alternative purifications, and holographic geometric identifications require separate justification. Schematic.

Report the doubled algebra, purification convention, support treatment, regulator or split distance, and dimension. Do not interpret the auxiliary copies as laboratory systems unless a preparation protocol is supplied.

References

• Dutta, Souvik, and Thomas Faulkner. “A Canonical Purification for the Entanglement Wedge Cross-Section.” Journal of High Energy Physics 03 (2021): 178. DOI. Open preprint.

Further reading

• Louisia, Kyan, Takato Mori, and Herbie Warner. “Zoo of Correlation Inequalities in Holography and Beyond.” Journal of High Energy Physics 06 (2026): 265. DOI.