Entanglement-Dynamics Diagnostic Comparison

Different dynamical diagnostics answer different operational questions. Subregion entropy can measure bipartite entanglement for a globally pure state; mutual information measures total regional correlation; connected correlators test selected observables; operator entanglement characterizes a regulated evolution map; densities and currents measure transport; and recovery tests whether dispersed information remains usable. Agreement strengthens a model. Disagreement is often the scientific result rather than a defect to be averaged away.

Required background. The information-measure domain atlas supplies definition domains, and continuum finite-time windows supplies the common comparison interval.

Helpful background. Front extraction, operator entanglement, and open and monitored dynamics provide the principal specialized diagnostics.

Begin from the physical question

Choose the task before calculating the convenient quantity.

How entangled is a bipartition of a pure state? Use

$$S_A=-\operatorname{tr}\rho_A\log\rho_A.$$

For a globally mixed or open-system state, $S_A$ also contains classical and environmental mixedness. Use a mixed-state entanglement measure, an entanglement witness, or a task-specific resource instead.

How much total correlation connects two regions? In a type-I regulator,

$$I(A:B)=S_A+S_B-S_{AB} =D(\rho_{AB}\Vert\rho_A\otimes\rho_B).$$

Mutual information includes classical and quantum correlation. It need not certify entanglement or signaling.

Which observable carries a detectable disturbance? Use a connected correlator or, for a signaling task, compare receiver statistics under two local encodings. For bounded $M_A,M_B$ and natural logarithms,

$$I(A:B)\geq \frac{|\langle M_AM_B\rangle -\langle M_A\rangle\langle M_B\rangle|^2} {2\|M_A\|^2\|M_B\|^2}.$$

Small mutual information controls every bounded connected correlator, but one vanishing correlator does not establish small mutual information Wolf et al. 2008, Eq. (5).

How nonfactorizable is an evolution map? In finite dimension, vectorize $U$ with normalization $1/\sqrt{d_Ad_B}$ and compute entropy across $(AA'):(BB')$. Product unitaries have $S_{\rm op}=0$ and equal-$d$ SWAP has $S_{\rm op}=2\log d$ Zanardi 2001, Eqs. (2)–(9). A bosonic or QFT version additionally needs a finite-energy reference, regulator, and cutoff order.

How does a conserved quantity move? Use density, current, structure factor, or full counting statistics. A diffusion constant is not an entanglement velocity.

Can an encoded state be reconstructed? Specify encoder, accessible output, decoder class, metric, energy budget, and error. Small local correlations can coexist with excellent joint recovery.

Does an environment return distinguishability? Use a channel-level non-Markovianity criterion with its optimization and divisibility assumptions. A dip in one entropy or a revival for one estimator is not a universal backflow measure.

OTOCs and circuit complexity are neighboring diagnostics, but their primary tasks are operator growth and computational structure. A correlation with entropy growth in one model does not make them substitutes for subregion entanglement.

Bell and classical correlations

The Bell state and its dephased version have the same subsystem entropy:

$$|\Phi^+\rangle=\frac{|00\rangle+|11\rangle}{\sqrt2}, \qquad \rho_{\rm cl}=\frac12|00\rangle\langle00| +\frac12|11\rangle\langle11|.$$

For both states,

$$S_A=S_B=\log2.$$

For the Bell state, $S_{AB}=0$ and $I(A:B)=2\log2$. For the separable classical state, $S_{AB}=\log2$ and $I(A:B)=\log2$. Thus:

• unchanged $S_A$ does not imply unchanged entanglement in a mixed-state process;
• nonzero mutual information does not imply entanglement;
• a mixed-state entanglement measure distinguishes the two;
• a record that reveals the dephasing phase can restore a conditioned Bell trajectory.

This benchmark should accompany every open-system comparison because it exposes several category errors with exact arithmetic.

Recovery can survive local decorrelation

Let $R$ and $X$ be independent uniform classical bits and define

$$Y=R\oplus X.$$

Each single share is independent of $R$:

$$I(R:X)=I(R:Y)=0.$$

Together they recover $R$ exactly because $R=X\oplus Y$, so

$$I(R:XY)=\log2.$$

This is a classical secret-sharing benchmark for information delocalization. Small mutual information between a reference and either small output region does not mean the encoded information was destroyed; it can be stored in joint correlations.

Quantum conditional mutual information makes the recovery question quantitative. For a tripartite state, there exists a recovery map $\mathcal R_{B\to BC}$ such that, using unsquared fidelity $F=\|\sqrt\rho\sqrt\sigma\|_1$,

$$I(A:C|B)_\rho \geq-2\log F\!\left( \rho_{ABC}, (\operatorname{id}_A\otimes\mathcal R)(\rho_{AB}) \right).$$

Fawzi and Renner 2015, main result and Theorem 5.1 establish this approximate-recovery relation. If $I(A:C|B)=0.02$ nats, the bound guarantees some recovery with

$$F\geq e^{-0.01}\approx0.99005.$$

Existence of a channel with high fidelity does not guarantee that the decoder is spatially local, computationally efficient, causal within the desired time, symmetry respecting, or feasible at the available energy. Those are additional tests.

Backflow needs a channel definition

The Breuer–Laine–Piilo distinguishability measure Breuer, Laine, and Piilo 2009, Eqs. (1)–(4) uses trace distance

$$D(t;\rho_1,\rho_2)=\frac12 \|\Lambda_t(\rho_1)-\Lambda_t(\rho_2)\|_1$$

and

$$\mathcal N_{\rm BLP} =\max_{\rho_1,\rho_2} \int_{\dot D>0}\dot D(t)\,dt.$$

An increase for one fixed pair is a sufficient witness and a lower bound on the optimized quantity. No increase for one chosen pair does not prove Markovianity. BLP backflow is also not equivalent to failure of CP divisibility; Rivas, Huelga, and Plenio 2010 formulate a distinct divisibility-based criterion.

For a qubit dephasing channel with coherence factor $\kappa(t)$, the pair $|+\rangle,|-\rangle$ has

$$D(t)=|\kappa(t)|.$$

If $\kappa(t)=e^{-\gamma t}$, the trace distance decreases monotonically and this pair contributes no backflow. If $\kappa(t)=\cos(gt)$ on $0\leq t\leq\pi/g$, $D$ decreases from one to zero and then revives to one, contributing exactly one to the positive-derivative integral. Each fixed-time dephasing map is CPTP because $|\kappa|\leq1$, but the family is not CP divisible through the revival.

An entropy dip, mutual-information revival, or trajectory-conditioned increase may be physically interesting, yet it is not automatically $\mathcal N_{\rm BLP}$. Keep the state pair, channel, record, and optimization fixed.

A matched diagnostic suite

Run one regulated preparation through integrable, chaotic, localized, and open or monitored comparators. On the same accepted physical-time window, compute:

• $S_A$ for pure global states, and a declared mixed-state entanglement measure otherwise;
• $I(A:B)$ and several bounded connected correlators;
• density and current profiles for every conserved quantity;
• regulated operator entanglement with identity and SWAP controls;
• an explicit encoded-state recovery fidelity;
• a channel distinguishability or divisibility diagnostic in open evolution;
• average trajectory entanglement and entanglement of the unconditional state separately.

Keep raw physical units before any collapse. Use the same cutoff, geometry, uncertainty model, and time interpolation. Fit an effective description on one subset and test another. The comparison should permit disagreement rather than normalizing curves until they coincide.

Expected informative disagreements include:

• $S_1$ grows ballistically while a conserved density diffuses;
• a selected correlator vanishes while mutual information remains nonzero;
• local mutual information falls while a joint decoder succeeds;
• operator entanglement is maximal for SWAP although product inputs remain unentangled;
• every monitored trajectory remains entangled while the unconditional state becomes separable;
• an estimator revives for a fixed pair without establishing a universal non-Markovianity measure.

Direct observation, extraction, interpretation

Report every conclusion in three layers.

1. Direct observation: “The half-maximum contour of $\Delta I(A:B)$ follows this fitted curve over $t\in[t_1,t_2]$.”
2. Extracted parameter: “A broadened-ballistic fit gives $v_f$ and exponent $\alpha$ with these uncertainties and threshold sensitivities.”
3. Interpretation: “Within this regulated preparation, the result is consistent with this effective mechanism and inconsistent with these controls.”

The third layer cannot be stronger than the first two. “Information travels at $v$” additionally needs an encoding, receiver, causal comparison, and recovery or capacity statement.

The chapter orientation map supplies the shared logical layers. Its failure controls specify the independent variations, while the diagnostic comparison provides the chapter-wide decision table.

Common pitfalls

Calling $S_A$ entanglement for a mixed global state. Use negativity, entanglement of formation, a witness, or an operational task with stated limitations.

Treating one recovery bound as an implementable decoder. Check locality, energy, symmetry, causality, and computational resources.

Calling one revival non-Markovianity. State the optimized or divisibility criterion and retain the same channel and conditioning.

Exercises

Compute $S_A$, $S_{AB}$, and $I(A:B)$ for the Bell state and the classically correlated bit.

Solution

Both have $S_A=S_B=\log2$. The Bell state is pure, so $S_{AB}=0$ and $I=2\log2$. The classical state has two equal joint eigenvalues, so $S_{AB}=\log2$ and $I=\log2$. Only the Bell state is entangled.

For independent uniform $R,X$ and $Y=R\oplus X$, verify the three mutual informations and give a decoder.

Solution

$X$ is independent of $R$. Because XOR with an independent uniform bit randomizes $R$, $Y$ is also independent of $R$. Hence $I(R:X)=I(R:Y)=0$. But $R=X\oplus Y$, so the pair determines $R$ exactly and $I(R:XY)=H(R)=\log2$. The decoder computes XOR.

Convert $I(A:C|B)=0.02$ nats into the fidelity guaranteed by the Fawzi–Renner inequality.

Solution

From $0.02\geq-2\log F$, one obtains $\log F\geq-0.01$ and

$$F\geq e^{-0.01}\approx0.99005.$$

This uses unsquared fidelity. If a source defines squared fidelity, the numerical exponent must be converted.

Evaluate the BLP contribution for $D(t)=|\cos gt|$ on $0\leq t\leq\pi/g$.

Solution

$D$ decreases from one to zero on $[0,\pi/(2g)]$, which contributes nothing to the positive-derivative integral. It increases from zero to one on $[\pi/(2g),\pi/g]$, contributing $1-0=1$. This is the contribution of the chosen pair; an optimized measure still requires the maximization contract.

References

• Breuer, Heinz-Peter, Elsi-Mari Laine, and Jyrki Piilo. “Measure for the Degree of Non-Markovian Behavior of Quantum Processes in Open Systems.” Physical Review Letters 103 (2009): 210401. DOI.
• Fawzi, Omar, and Renato Renner. “Quantum Conditional Mutual Information and Approximate Markov Chains.” Communications in Mathematical Physics 340 (2015): 575–611. DOI.
• Rivas, Ángel, Susana F. Huelga, and Martin B. Plenio. “Entanglement and Non-Markovianity of Quantum Evolutions.” Physical Review Letters 105 (2010): 050403. DOI.
• 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.
• Zanardi, Paolo. “Entanglement of Quantum Evolutions.” Physical Review A 63 (2001): 040304(R). DOI.