Reflected Entropy and Canonical Purifications

Reflected entropy is the entanglement entropy of a particular doubled purification of a mixed bipartite state. “Canonical” means canonical after the algebra, representation, and conjugation are fixed; it does not mean that the auxiliary copies are physically prepared. The construction measures total correlations and differs from both mutual information and negativity.

Required background. Mutual Information and Regulator-Independent Correlations supplies relative entropy, monotonicity, and the continuum distinction between type-III algebras and split type-I factors. Helpful background. Entanglement Negativity in QFT supplies a contrasting mixed-state measure based on a partial transformation rather than a purification.

The shared structure map places these constructions side by side, while the canonical comparison table records their different domains. Statements below about holographic identification are limited to primary literature available through 10 August 2026 and to the state classes stated explicitly.

Canonical vectorization and normalization

Choose a factorized orthonormal basis $|i\rangle_{AB}$ and define vectorization by

$$|X\rangle\!\rangle =\sum_{i,j}X_{ij} |i\rangle_{AB}|j\rangle_{A^*B^*}.$$

For a normalized density operator $\rho_{AB}$, its positive square root gives

$$|\sqrt{\rho_{AB}}\rangle\!\rangle \in{\cal H}_{A}\otimes{\cal H}_{B} \otimes{\cal H}_{A^*}\otimes{\cal H}_{B^*}.$$

The Hilbert–Schmidt inner product supplies both essential checks:

$$\langle\!\langle\sqrt\rho|\sqrt\rho\rangle\!\rangle =\operatorname{Tr}\rho=1, \qquad \operatorname{Tr}_{A^*B^*} |\sqrt\rho\rangle\!\rangle \langle\!\langle\sqrt\rho|=\rho_{AB}.$$

The reflected entropy is the entropy across the $AA^*:BB^*$ cut,

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

These definitions and recovery identities are Dutta and Faulkner 2021, §1, eqs. (1)–(3). A factorized basis change acts by local unitaries on the doubled factors, so the entropy is basis independent even though a coordinate vectorization was used to calculate it.

The diagram shows which copies are auxiliary and which partial trace defines the measure. Inspect the $AA^*:BB^*$ cut rather than interpreting $A^*$ and $B^*$ as new laboratory systems.

Vectorization turns $\sqrt{\rho_{AB}}$ into a normalized pure state on $ABA^*B^*$. Tracing $BB^*$ produces $\rho_{AA^*}$, and its entropy equals that across the complementary $AA^*:BB^*$ cut. The construction assumes a fixed type-I tensor product or split realization. Schematic; not to scale.

Continuum algebras and the split factor

Standard form supplies a canonical positive-cone vector for a normal state on a von Neumann algebra, but it does not by itself turn a sharp QFT region into a tensor factor with a finite entropy. For spacelike-separated regions with

$${\cal A}(A)\subset{\cal N}\subset{\cal A}(B)',$$

the split property provides an intermediate type-I factor ${\cal N}$. If the state vector $|\psi\rangle$ is cyclic and separating for ${\cal A}(A)$, ${\cal A}(B)$, and ${\cal A}(A)\vee{\cal A}(B)$, modular conjugation selects a state-dependent canonical factor ${\cal N}_\psi$ Doplicher and Longo 1984, §§1–2; Dutta and Faulkner 2021, §7, eqs. (99) and (104)–(108).

A lattice regulator or another split factor can also define a type-I reflected entropy, but its collar width, center, and limiting prescription remain part of the result. Calling the purification canonical does not erase those choices.

Bounds and exact limiting cases

For finite-dimensional or otherwise finite type-I entropies,

$$I(A{:}B) \leq S_R(A{:}B) \leq2\min\{S(A),S(B)\}.$$

The lower bound follows from strong subadditivity, and the upper bound from positivity of $I(A{:}A^*)$ and $I(B{:}B^*)$ in the canonical purification Dutta and Faulkner 2021, §2, eqs. (20)–(25). Two normalization checks are especially useful:

$$\rho_{AB}=\rho_A\otimes\rho_B \quad\Longrightarrow\quad I=S_R=0,$$

and

$$\rho_{AB}=|\psi\rangle\langle\psi| \quad\Longrightarrow\quad S_R=2S(A)=I(A{:}B).$$

The canonical purification is well defined when $\rho$ is nonfaithful: take its positive square root on its support. What can become singular are formulas that insert $\rho^{-1}$, a modular inverse, or inverse covariance blocks. Those formulas must be restricted to the support or defined by a controlled faithful limit.

Two-mode Gaussian application

Use quadratures $R=(q_A,p_A,q_B,p_B)$ with $[q,p]=i$ and vacuum covariance $V_{\rm vac}=\mathbb I/2$. If a two-mode Gaussian covariance matrix has Williamson form

$$V=S\left(\nu_1\mathbb I_2\oplus\nu_2\mathbb I_2\right)S^T, \qquad \nu_k\geq\frac12,$$

purify each thermal normal mode with a star mode using the block

$$V_{\rm pur}(\nu_k)= \begin{pmatrix} \nu_k\mathbb I_2 & \sqrt{\nu_k^2-1/4}\,Z\\ \sqrt{\nu_k^2-1/4}\,Z & \nu_k\mathbb I_2 \end{pmatrix}, \qquad Z=\operatorname{diag}(1,-1).$$

Apply the Gaussian unitary associated with $S$ to the physical normal modes and its conjugate to the star modes. Explicitly, if $\Theta=\bigoplus_k\operatorname{diag}(1,-1)$ flips every momentum in the position-basis convention, the star phase-space map is $\Theta S\Theta$. Reorder the result as $AA^*BB^*$ and restrict the doubled covariance matrix to $AA^*$. If its symplectic eigenvalues are $\mu_j$, then

$$S_R=\sum_j h(\mu_j), \qquad h(\nu)= \left(\nu+\frac12\right)\log\left(\nu+\frac12\right) -\left(\nu-\frac12\right)\log\left(\nu-\frac12\right).$$

This correlation-matrix construction is developed for free scalar QFT in Bueno and Casini 2020, §2.1, eqs. (16)–(27). It makes the regulator and numerical controls explicit: require every input and output symplectic eigenvalue to be at least $1/2$ within tolerance, vary lattice spacing at fixed physical geometry, and compare $S_R$ with both bounds.

Two exact two-mode reductions benchmark the algorithm.

Product thermal modes. For

$$V=\nu_A\mathbb I_2\oplus\nu_B\mathbb I_2, \qquad \nu_A,\nu_B>\frac12,$$

the canonical purification factorizes between $AA^*$ and $BB^*$. Hence $S_R=0=I$, saturating the lower bound even though each individual mode is mixed.

Pure two-mode squeezed state. Let

$$V(r)=\frac12 \begin{pmatrix} \cosh2r\,\mathbb I_2 & \sinh2r\,Z\\ \sinh2r\,Z & \cosh2r\,\mathbb I_2 \end{pmatrix}, \qquad n=\sinh^2r.$$

With $g(n)=(n+1)\log(n+1)-n\log n$, each one-mode reduction has entropy $g(n)$. Since the physical state is pure,

$$I(A{:}B)=S_R(A{:}B)=2g(n),$$

which saturates both bounds. Although this state is nonfaithful, the support definition is regular; only an inverse-based implementation needs a limiting prescription.

Purification and holographic scope

Altering the purifier by an arbitrary unitary generally changes $S(AA^*)$. It still purifies $\rho_{AB}$, but it no longer defines reflected entropy unless it implements the same canonical standard-form identification. The shared validity map records this failure separately from support and regulator failures.

For Euclidean-prepared holographic states on a time-reflection-symmetric slice, the leading semiclassical proposal is

$$S_R(A{:}B)=2E_W(A{:}B)+O(G_N^0), \qquad E_W=\frac{\operatorname{Area}(\Sigma_{AB})}{4G_N}.$$

The factor of two and normalization are Dutta and Faulkner 2021, §1, eqs. (4) and (8). The original derivation imposes the time-reflection setup in §1; its dynamical extension is proposed there, not established as a generic QFT theorem. Neither this geometric relation nor an arbitrary purification defines $S_R$ outside its stated domain.

Exercises

1. Verify the canonical purification

Using $|X\rangle\!\rangle=\sum_{ij}X_{ij}|i\rangle|j\rangle$, prove its norm identity and physical partial-trace identity for $X=\sqrt\rho$.

Solution

Orthogonality gives

$$\langle\!\langle X|X\rangle\!\rangle =\sum_{ij}|X_{ij}|^2 =\operatorname{Tr}(XX^\dagger).$$

For $X=\sqrt\rho=\sqrt\rho^{\,\dagger}$, this is $\operatorname{Tr}\rho=1$. Tracing the second copy gives matrix elements

$$\sum_jX_{ij}X^*_{kj}=(XX^\dagger)_{ik}=\rho_{ik}.$$

Thus the doubled vector is normalized and reduces to the original state on $AB$.

2. Derive the upper bound

Use the canonical purification to show $S_R(A{:}B)\leq2S(A)$ and then obtain the symmetric upper bound.

Solution

The star reduction has the same spectrum as the corresponding physical reduction, so $S(A^*)=S(A)$. Positivity of mutual information between $A$ and $A^*$ gives

$$0\leq I(A{:}A^*) =S(A)+S(A^*)-S(AA^*) =2S(A)-S_R(A{:}B).$$

Hence $S_R\leq2S(A)$. Repeating the argument for $B,B^*$ and using purity of the doubled state, $S(BB^*)=S(AA^*)$, yields $S_R\leq2S(B)$. Combining them gives $S_R\leq2\min\{S(A),S(B)\}$.

3. Gaussian limiting cases

Compute $I$ and $S_R$ for the product thermal covariance and the pure two-mode squeezed covariance above. Identify which bounds are saturated.

Solution

For the product covariance, $\rho_{AB}=\rho_A\otimes\rho_B$. Its canonical vector is $|\sqrt{\rho_A}\rangle\!\rangle\otimes|\sqrt{\rho_B}\rangle\!\rangle$, a product across $AA^*:BB^*$. Therefore $S_R=0$, while additivity gives $I=S(A)+S(B)-S(AB)=0$.

For the two-mode squeezed state, the physical state is pure and each one-mode reduction has mean occupation $n=\sinh^2r$ and entropy $g(n)$. Thus

$$I=2g(n), \qquad S_R=2S(A)=2g(n).$$

The product state saturates the lower bound at zero. The pure state saturates both the lower bound and $2\min\{S(A),S(B)\}$.

4. Alter the purification

Let $\rho_{AB}=\mathbb I_A/2\otimes\mathbb I_B/2$ for two qubits. Compare the canonical purification with the state obtained by swapping $A^*$ and $B^*$. What does this failure injection show?

Solution

The canonical purification is

$$|\Phi^+\rangle_{AA^*}\otimes|\Phi^+\rangle_{BB^*},$$

which is a product across $AA^*:BB^*$, so $S_R=0$. A unitary swap of $A^*$ and $B^*$ leaves the physical marginal $\rho_{AB}$ unchanged but produces

$$|\Phi^+\rangle_{AB^*}\otimes|\Phi^+\rangle_{BA^*}.$$

Both Bell pairs now cross the $AA^*:BB^*$ cut, so $S(AA^*)=2\log2$. The new state is a valid purification but not the canonical one. Therefore “entropy of a purification” is not a definition of reflected entropy; the canonical standard-form identification is essential.

References

• Bueno, Pablo, and Horacio Casini. “Reflected Entropy for Free Scalars.” Journal of High Energy Physics 11 (2020): 148. DOI. Open preprint.
• Doplicher, Sergio, and Roberto Longo. “Standard and Split Inclusions of von Neumann Algebras.” Inventiones Mathematicae 75 (1984): 493–536. DOI.
• 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.