Choosing an Entanglement Measure for the Physical Question

There is no universally best entanglement number. Entropy coefficients, mutual information, negativity, canonical-purification diagnostics, entropy-cone tests, and channel distances act on different inputs and support different conclusions. Selection begins with a task, an algebra, a state or channel class, an allowed operation set, and a continuum prescription.

Required background. Use the Information-Measure Domain and Comparison Atlas to type the inputs and strongest permissible conclusion before comparing values. Helpful background. Entanglement Negativity in QFT supplies partial-transpose conventions, Reflected Entropy and Canonical Purifications supplies the doubled construction, and Multipartite Information and Entropy Cones supplies state-class-dependent inequalities.

The chapter’s structure map shows why these branches do not imply one another. Its canonical comparison table records their common domain boundaries, and its validity map gives the variations that every reported claim should survive.

Type the question before choosing the quantity

Five declarations prevent most category errors:

1. Task. Is the goal geometric universality, RG ordering, total correlation, NPT certification, symmetry-accessible entanglement, multipartite compatibility, information flow, or channel discrimination?
2. Input. Is the object a state, a pair of commuting algebras, a charge-block decomposition, an entropy vector, or a channel?
3. Operations. Which measurements, local operations, reference frames, ancillas, and energy budgets are available?
4. Continuum realization. Is the quantity intrinsic to local algebras, finite only at positive separation, or defined first in a lattice or split type-I factor?
5. Failure criterion. What counterexample would show that the number does not answer the task?

The answer may legitimately be “no single scalar.” For example, classifying a phase generally needs response and excitation data in addition to an entanglement diagnostic.

Decision matrix

Information quantities selected by a typed physical question.

Physical question and quantity	Typed input domain	Operational conclusion	Continuum realization	Failure mode
Is a geometric coefficient universal? Matched log, corner, sphere, or topological term	Declared dimension, state, shape, counterterm class, and limiting family	Invariance under that regulator and counterterm class	Balanced subtraction, response coefficient, or controlled fit from matched representatives	A cutoff-independent-looking constant can retain shape, center, or scheme dependence
Does an RG flow obey an irreversible ordering? Dimension-specific entropic $c$- or $F$-quantity	Vacuum QFT, Lorentz symmetry, nested regions, and theorem-specific dimension	Monotonicity only under the theorem’s hypotheses	Derivative or mutual-information prescription that removes local terms	Raw $S_A$ can grow with region size and need not be an RG monotone
Are two regions correlated? Mutual information $I(A{:}B)$	One state on two commuting algebras; preferably positive separation	Total classical and quantum correlation and bounds on connected observables	Algebraic relative entropy, or a matched type-I limit in which boundary terms cancel	$I>0$ does not certify entanglement
Is a bipartite mixed state non-PPT? Logarithmic negativity $\mathcal E$	Type-I bipartition, state, bosonic or fermionic transpose, and grading	$\mathcal E>0$ certifies NPT entanglement	Lattice or split factor, or a separately defined algebraic replacement	$\mathcal E=0$ means PPT, not separable, in general dimensions
What does the canonical doubling encode? Reflected entropy $S_R$	Density operator, its support, and a chosen type-I realization of $AB$	Entropy of the $AA^*$ reduction of the canonical purification	Controlled split-factor or cutoff limit; the auxiliary copies are not a physical preparation	It is not monotone under all partial traces and is not a general operational correlation measure
How is entanglement distributed across charge sectors? $p_q$ and $S(\rho_{A,q})$	State, additive charge, sector projectors, and superselection/reference-frame resources; a pure global state if sector entropy is called entanglement	Number uncertainty and within-sector entropy; for a pure state, the sector-average accessible entanglement	Charge-resolved regulator or split with the same symmetry action at every cutoff	Adding the Shannon sector term can overstate distillable entanglement under a superselection rule
Which linear entropy constraints hold? Entropy vector and a declared cone	All nonempty party unions, purifier convention, and quantum, stabilizer, holographic, or other state class	Membership or violation for that cone	Every entropy uses one common algebra and regulator prescription; homogeneous combinations must be finite	Satisfying a special-class inequality does not prove membership in that state class
How distinguishable are two evolutions? Energy-constrained diamond distance	Channels $\Phi,\Psi$, input Hamiltonian $H$, mean-energy bound $E$, and arbitrary reference ancilla	Optimal equal-prior discrimination bias on the constrained input set	Intrinsic channel optimization with a physical energy restriction; regulator convergence must preserve $H$ and $E$	Changing the Hamiltonian, energy convention, ancilla, or input domain changes the task

The state class in an entropy-cone claim is equally consequential: holographic entropy vectors obey constraints beyond those of general quantum states Bao et al. 2015, §§2–4. Passing one specialized-cone inequality is necessary but not sufficient for class membership.

The geometric and RG rows need especially careful wording. In $1+1$ dimensions, Lorentz symmetry and strong subadditivity make $C(\ell)=3\ell S'(\ell)$ monotone for the vacuum interval Casini and Huerta 2007, pp. 7032–7035, Eqs. (2)–(7). This does not license the same formula in every dimension. In $2+1$ dimensions, the concentric-circle mutual-information prescription supplies a regulator-controlled $F$-quantity and avoids treating a fitted raw sphere constant as self-evidently universal Casini et al. 2015, §§2–3.

For natural logarithms, type-I mutual information is a relative entropy,

$$I(A{:}B)=D(\rho_{AB}\Vert\rho_A\otimes\rho_B).$$

Pinsker’s inequality and trace-norm duality imply, for bounded $X\in\mathcal A_A$ and $Y\in\mathcal A_B$,

$$\left\lvert\langle XY\rangle-\langle X\rangle\langle Y\rangle\right\rvert \leq \lVert X\rVert\,\lVert Y\rVert\sqrt{2I(A{:}B)}.$$

This is a bound on total connected correlation, not an entanglement witness Wolf et al. 2008, pp. 070502-1–070502-2.

Logarithmic negativity is

$$\mathcal E(A{:}B)=\log\left\lVert\rho_{AB}^{T_B}\right\rVert_1.$$

Its monotonicity and trace-norm definition are developed in Vidal and Werner 2002, §§II–III. A positive partial transpose is only a necessary separability condition in general: explicit $3\times3$ and $2\times4$ PPT-entangled states exist Horodecki 1997, pp. 333–338.

For a type-I state $\rho_{AB}$, vectorize $\sqrt{\rho_{AB}}$ to obtain its canonical purification on $ABA^*B^*$ and define

$$S_R(A{:}B)=S(AA^*)_{\lvert\sqrt\rho\rangle}.$$

This construction and its split-property continuum interpretation are given in Dutta and Faulkner 2021, §§2 and 7. It is safest to call $S_R$ a canonical-purification diagnostic: counterexamples show that it can increase under discarding a subsystem, so it is not a correlation measure in the axiomatic monotonicity sense Hayden, Lemm, and Sorce 2023, Theorem 1.

If $\rho_A=\bigoplus_q p_q\rho_{A,q}$ is block diagonal in a conserved charge, then

$$S(\rho_A)=H(\{p_q\})+\sum_q p_q S(\rho_{A,q}).$$

The sector probabilities and within-sector entropies answer different questions Goldstein and Sela 2018, pp. 200602-1–200602-3. Under a particle-number superselection rule without a shared reference frame, the operationally accessible pure-state entanglement is the sector average, not automatically the full mode entropy Wiseman and Vaccaro 2003, pp. 097902-1–097902-4.

Finally, for channels from input $A$ with Hamiltonian $H_A$, define

$$\lVert\Phi-\Psi\rVert_{\diamond,E} =\sup_{\substack{\rho_{AR}\geq0,\ \operatorname{Tr}\rho_{AR}=1\\ \operatorname{Tr}(H_A\rho_A)\leq E}} \left\lVert[(\Phi-\Psi)\otimes\operatorname{id}_R](\rho_{AR})\right\rVert_1.$$

With equal priors, the best success probability is

$$p_{\mathrm{succ}} =\frac12+\frac14\lVert\Phi-\Psi\rVert_{\diamond,E}.$$

The energy constraint is essential in infinite-dimensional systems Shirokov 2018, §4. A channel distance is not an entanglement measure merely because an ancilla appears in its optimization.

Four applications and their adversarial replacements

Entanglement harvesting

After specifying detector worldlines, switching functions, smearings, field state, perturbative order, and causal relation, the protocol produces a two-detector density matrix $\rho_{D_1D_2}$. If the question is whether the detectors harvested NPT entanglement, use detector negativity and report the transpose convention; see Entanglement Harvesting from Quantum Fields.

Tempting replacement: mutual information. The separable state

$$\rho_{\mathrm{cc}} =\frac12\left(\lvert00\rangle\langle00\rvert +\lvert11\rangle\langle11\rvert\right)$$

has $I(D_1{:}D_2)=\log2$ but $\mathcal E=0$. Mutual information can show harvested total correlation; it cannot certify that the harvested resource is entanglement.

RG monotonicity

For a Lorentz-invariant $1+1$-dimensional vacuum flow, use $C(\ell)=3\ell S'(\ell)$ with the theorem’s interval geometry. For a $2+1$-dimensional flow, use the appropriate renormalized or mutual-information $F$ prescription rather than transplanting the $1+1$ formula.

Tempting replacement: raw entropy. At a fixed point, $S(\ell)=(c/3)\log(\ell/\epsilon)+c_1$ increases with $\ell$ even though the RG central charge is constant. Its numerical decrease or increase along a regulator-dependent comparison is therefore not the claimed RG theorem.

Gauge-region accessibility

Choose the gauge-invariant region algebra, its center, and the allowed operations. If charge sectors are superselected, report both $H(\{p_q\})$ and $\sum_qp_qS(\rho_{A,q})$ and identify which term the protocol can distill. The detailed handoff is Centers, Edge Modes, and Distillable Entanglement.

Tempting replacement: raw mode entropy. A single particle in $(\lvert10\rangle+\lvert01\rangle)/\sqrt2$ has mode entropy $\log2$, but each fixed local-number sector is a product state. Without a shared phase reference, the sector-average accessible entanglement is zero. The Shannon uncertainty is real information, but it is not one distillable Bell pair.

Multipartite scrambling

Specify an input reference, a time-evolution channel, and output regions. A common operator-state diagnostic is

$$I_3(A{:}C{:}D) =I(A{:}C)+I(A{:}D)-I(A{:}CD),$$

with a negative value indicating that information about $A$ is available jointly in $CD$ more strongly than in either part separately for that declared channel state Hosur et al. 2016, §2. See Tripartite and Multipartite Scrambling Diagnostics.

Tempting replacement: $S_C(t)$ alone. A local output entropy can grow because of heating, dephasing, or entanglement with a nearby environment without showing that input information has delocalized across $C$ and $D$. A scrambling claim needs the reference correlations or channel recovery/decoupling task.

These four replacements are deliberately plausible. Their failures demonstrate why numerical similarity cannot substitute for domain matching.

Common pitfalls

Zero is not one universal verdict. Zero negativity means PPT; zero mutual information means a product state in its domain; zero topological subtraction means no detected long-range constant under its assumptions. These statements are not interchangeable.

A finite representative is not automatically intrinsic. Partial transpose, canonical purification, charge blocks, and raw entropy usually begin in a lattice or split factor. State what algebraic or balanced quantity is expected to survive cutoff removal.

A theorem does not travel without its domain. Dimension, Lorentz symmetry, region geometry, state class, energy constraint, and allowed operations are part of the conclusion.

Exercises

1. Derive the mutual-information correlation bound

For a finite-dimensional $\rho_{AB}$ and bounded $X,Y$, derive the bound on the connected correlator from Pinsker’s inequality.

Solution

Let $\Delta=\rho_{AB}-\rho_A\otimes\rho_B$. Then

$$\langle XY\rangle-\langle X\rangle\langle Y\rangle =\operatorname{Tr}[\Delta(X\otimes Y)].$$

Trace-norm duality gives

$$\left\lvert\operatorname{Tr}[\Delta(X\otimes Y)]\right\rvert \leq\lVert\Delta\rVert_1\lVert X\rVert\lVert Y\rVert.$$

For natural logarithms, quantum Pinsker says

$$D(\rho_{AB}\Vert\rho_A\otimes\rho_B) \geq\frac12\lVert\Delta\rVert_1^2.$$

The relative entropy on the left is $I(A{:}B)$, so $\lVert\Delta\rVert_1\leq\sqrt{2I(A{:}B)}$. Combining the inequalities proves the result. Nothing in the derivation separates classical from quantum correlation.

2. Compare classical correlation and NPT entanglement

Compute $I$ and $\mathcal E$ for $\rho_{\mathrm{cc}}$ above and for the Bell state $\lvert\Phi^+\rangle=(\lvert00\rangle+\lvert11\rangle)/\sqrt2$.

Solution

For $\rho_{\mathrm{cc}}$, each marginal is $I_2/2$, so $S_A=S_B=\log2$. The joint state has two probabilities $1/2$, hence $S_{AB}=\log2$ and

$$I(A{:}B)=\log2.$$

Partial transpose leaves this diagonal state positive, with trace norm one, so $\mathcal E=0$.

For the Bell state, the joint entropy vanishes and both marginal entropies equal $\log2$, giving $I=2\log2$. Its partial transpose has eigenvalues $(1/2,1/2,1/2,-1/2)$, so its trace norm is two and $\mathcal E=\log2$. Mutual information distinguishes the amounts of total correlation here, while negativity answers the NPT-certification question.

3. Separate number entropy from accessible entanglement

For $\lvert\psi\rangle=(\lvert10\rangle+\lvert01\rangle)/\sqrt2$, calculate the mode entropy and the sector-average entanglement when local particle number is superselected.

Solution

Tracing out $B$ gives

$$\rho_A=\frac12\lvert0\rangle\langle0\rvert +\frac12\lvert1\rangle\langle1\rvert,$$

so the mode entropy is $S(\rho_A)=\log2$. The local charge sectors have probabilities $p_0=p_1=1/2$, but the conditional state in each sector is pure. Therefore

$$\sum_qp_qS(\rho_{A,q})=0, \qquad H(\{p_q\})=\log2.$$

Without a resource that lifts the superselection restriction, the first quantity is the accessible entanglement. Replacing it with the full mode entropy incorrectly counts charge uncertainty as a distillable Bell pair.

4. Type a channel-discrimination claim

Two equally likely channels $\Phi$ and $\Psi$ act on an oscillator. State the optimization whose value determines the best success probability under mean input energy $E$, and list three changes that define a different task.

Solution

Choose the input Hamiltonian $H_A$ and optimize over input–reference states:

$$\lVert\Phi-\Psi\rVert_{\diamond,E} =\sup_{\operatorname{Tr}(H_A\rho_A)\leq E} \left\lVert[(\Phi-\Psi)\otimes\operatorname{id}_R](\rho_{AR})\right\rVert_1.$$

Helstrom discrimination of the two optimized output states gives

$$p_{\mathrm{succ}} =\frac12+\frac14\lVert\Phi-\Psi\rVert_{\diamond,E}.$$

Changing $H_A$ or its zero of energy without adjusting $E$, changing the admissible ancilla/reference class, or replacing the mean-energy bound by a hard spectral cutoff changes the feasible input set and hence the operational task. Removing the energy bound entirely can also make the infinite-dimensional norm physically uninformative.

References

• Bao, Ning, Sepehr Nezami, Hirosi Ooguri, Bogdan Stoica, James Sully, and Michael Walter. “The Holographic Entropy Cone.” Journal of High Energy Physics 09 (2015): 130. DOI. Open preprint.
• Casini, Horacio, and Marina Huerta. “A c-Theorem for the Entanglement Entropy.” Journal of Physics A: Mathematical and Theoretical 40 (2007): 7031–7036. DOI. Open preprint.
• Casini, Horacio, Marina Huerta, Robert C. Myers, and Alexandre Yale. “Mutual Information and the F-Theorem.” Journal of High Energy Physics 10 (2015): 003. DOI. Open preprint.
• 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.
• Goldstein, Moshe, and Eran Sela. “Symmetry-Resolved Entanglement in Many-Body Systems.” Physical Review Letters 120 (2018): 200602. DOI. Open preprint.
• Hayden, Patrick, Marius Lemm, and Jonathan Sorce. “Reflected Entropy: Not a Correlation Measure.” Physical Review A 107 (2023): L050401. DOI. Open preprint.
• Horodecki, Paweł. “Separability Criterion and Inseparable Mixed States with Positive Partial Transposition.” Physics Letters A 232 (1997): 333–339. DOI. Open preprint.
• Hosur, Pavan, Xiao-Liang Qi, Daniel A. Roberts, and Beni Yoshida. “Chaos in Quantum Channels.” Journal of High Energy Physics 02 (2016): 004. DOI. Open preprint.
• Shirokov, M. E. “On the Energy-Constrained Diamond Norm and Its Application in Quantum Information Theory.” Problems of Information Transmission 54 (2018): 20–33. DOI. Open preprint.
• Vidal, Guifré, and Reinhard F. Werner. “A Computable Measure of Entanglement.” Physical Review A 65 (2002): 032314. DOI. Open preprint.
• Wiseman, Howard M., and John A. Vaccaro. “Entanglement of Indistinguishable Particles Shared between Two Parties.” Physical Review Letters 91 (2003): 097902. DOI. Open preprint.
• Wolf, Michael M., Frank Verstraete, Matthew B. Hastings, and J. Ignacio Cirac. “Area Laws in Quantum Systems: Mutual Information and Correlations.” Physical Review Letters 100 (2008): 070502. DOI. Open preprint.