Information Backflow

Information backflow is the idea that a reduced quantum system can temporarily regain information that had become inaccessible because of its interaction with an environment. A widely used operational diagnostic is an increase in trace distance between two evolving system states.

This page explains the trace-distance criterion, why it suggests memory, how it relates to divisibility, and where its limitations are. The broader comparison of non-Markovian terminology is What Non-Markovian Means; Non-Markovianity Measures compares the BLP measure with divisibility and entanglement-based measures.

Trace Distance

For two density operators $\rho$ and $\sigma$, the trace distance is

$$D(\rho,\sigma) = \frac12 \lVert\rho-\sigma\rVert_1,$$

where

$$\lVert X\rVert_1 = \operatorname{Tr}\sqrt{X^\dagger X}.$$

For density operators, $0\le D(\rho,\sigma)\le1$. The endpoints have clear meanings:

• $D(\rho,\sigma)=0$ means the states are identical;
• $D(\rho,\sigma)=1$ means the states can be perfectly distinguished by some measurement.

For equal prior probabilities, the optimal success probability for distinguishing $\rho$ from $\sigma$ is

$$P_{\mathrm{guess}} = \frac12 \left( 1+D(\rho,\sigma) \right).$$

Thus trace distance is not only a norm. It is an operational measure of distinguishability.

Contractivity Under Channels

Every completely positive trace-preserving map $\Phi$ contracts trace distance:

$$D(\Phi(\rho),\Phi(\sigma)) \le D(\rho,\sigma).$$

This expresses a simple principle: applying the same physical channel to both candidate states cannot make them easier to distinguish.

Now consider an open-system family

$$\rho_j(t)=\Phi_{t,0}(\rho_j(0)), \qquad j=1,2.$$

If the evolution from $s$ to $t$ is represented by a positive trace-preserving intermediate map $\Phi_{t,s}$, then

$$D(\rho_1(t),\rho_2(t)) \le D(\rho_1(s),\rho_2(s)).$$

Therefore, if the trace distance increases over an interval for some pair of initial states, the intermediate evolution cannot be positive and trace preserving over that interval. In particular, it cannot be CP-divisible.

Backflow Criterion

Define

$$\Delta(t) = D(\rho_1(t),\rho_2(t)).$$

A trace-distance backflow interval is an interval where

$$\frac{d\Delta(t)}{dt}>0.$$

During such an interval, the two reduced system states become more distinguishable. The usual interpretation is that information about the earlier system state, previously dispersed into environmental degrees of freedom or correlations, has become accessible again in the system.

The Breuer–Laine–Piilo diagnostic looks for whether such an increase exists for some pair of initial states. A corresponding measure integrates the positive parts and optimizes over pairs:

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

The optimization can be difficult in high dimension. For qubits and simple channels, symmetry often identifies good candidate pairs.

Relation to Divisibility

Trace-distance monotonicity is tied to positive divisibility. If a family is P-divisible, all trace distances between evolving system states are nonincreasing.

The implications are:

$$\text{CP divisible} \Longrightarrow \text{P divisible} \Longrightarrow \text{no trace-distance backflow}.$$

The reverse implications need not hold without extra assumptions. In particular:

• trace-distance backflow proves failure of P divisibility and hence failure of CP divisibility;
• failure of CP divisibility may occur without a trace-distance increase between system states;
• a diagnostic based on one pair of states can miss backflow visible to another pair.

This is why information backflow is a powerful operational witness, not a universal definition of non-Markovianity.

Example: Pure Dephasing

For a qubit dephasing map,

$$\rho_{01}(t) = \eta(t)\rho_{01}(0), \qquad \rho_{00}(t)=\rho_{00}(0).$$

Choose the pair

$$\rho_+(0)=\lvert+\rangle\langle+\rvert, \qquad \rho_-(0)=\lvert-\rangle\langle-\rvert,$$

where

$$\lvert\pm\rangle = \frac{\lvert0\rangle\pm\lvert1\rangle}{\sqrt2}.$$

The difference between these states is entirely coherence. Under dephasing,

$$D(\rho_+(t),\rho_-(t)) = |\eta(t)|.$$

Thus trace-distance backflow occurs exactly when $|\eta(t)|$ increases. For exponential Markovian dephasing,

$$\eta(t)=e^{-\Gamma_\phi t}, \qquad \Gamma_\phi\ge0,$$

there is no backflow.

Example: Amplitude Damping

For an amplitude-damping-like map with excited-state survival probability $q(t)$,

$$\rho_{ee}(t)=q(t)\rho_{ee}(0),$$

and the ground state remains fixed. Compare the initial states $\lvert e\rangle\langle e\rvert$ and $\lvert g\rangle\langle g\rvert$. In the basis $(\lvert g\rangle,\lvert e\rangle)$, their evolved difference has eigenvalues $\pm q(t)$, so

$$D(\rho_e(t),\rho_g(t)) = q(t).$$

If $q(t)$ increases, the excited and ground preparations become more distinguishable again. This is a direct population-return version of information backflow.

For ordinary Markovian zero-temperature damping,

$$q(t)=e^{-\Gamma_1t},$$

so the trace distance decreases monotonically.

What Is Flowing?

The word “information” should be read operationally. It does not mean a conserved substance moving along a wire. It means distinguishability of alternative system preparations as observed through measurements on the reduced system.

In a system-environment picture, loss of distinguishability can occur because:

• the environment carries away records of the initial state;
• the system becomes correlated with inaccessible degrees of freedom;
• different initial states relax toward a common attractor;
• environmental noise blurs phase or population information.

Backflow means some of that distinguishability becomes visible again in the system. This may happen through coherent exchange with a structured mode, finite-reservoir recurrence, delayed feedback, or a time-dependent effective generator.

What the Criterion Does Not Prove

Trace-distance backflow is a sufficient witness of non-Markovianity in the divisibility sense, but its absence is not a complete certificate of Markovianity in every sense.

Absence of observed backflow may mean:

• the tested state pair was not optimal;
• the dynamics is CP-indivisible but still P-divisible;
• memory is present in observables not captured by the chosen distinguishability task;
• the sampled time grid missed short backflow intervals;
• experimental noise or state-preparation errors masked a small increase.

Likewise, a trace-distance increase in reconstructed data should be compared with error bars. Small apparent increases can come from tomography noise, finite sampling, or inconsistent state reconstruction.

Practical Detection

A numerical or experimental workflow should:

1. choose candidate pairs of initial states;
2. reconstruct or compute $\rho_1(t)$ and $\rho_2(t)$ on a time grid;
3. compute $D(\rho_1(t),\rho_2(t))$ with a stable trace-norm routine;
4. estimate uncertainties or numerical tolerances;
5. identify statistically significant intervals of increase;
6. compare with CP-divisibility or Choi tests when the dynamical map is known.

For qubits, antipodal Bloch-sphere states are often good candidates. For dephasing, transverse antipodal states reveal coherence revivals. For amplitude damping, energy-basis states reveal population return.

Common Mistakes

• Saying that no observed backflow means the dynamics is definitely Markovian.
• Testing only one arbitrary pair of states.
• Confusing trace-distance backflow with thermodynamic heat backflow.
• Ignoring uncertainty when differentiating noisy trace-distance data.
• Calling an increase in a classical plotting variable information backflow without relating it to distinguishability.
• Forgetting that CP divisibility is stronger than trace-distance monotonicity.
• Treating the BLP measure as the only meaningful non-Markovianity measure.

Exercises

Guessing probability

Two states have trace distance $D=0.4$. What is the optimal success probability for distinguishing them with equal priors?

Solution

For equal priors,

$$P_{\mathrm{guess}} = \frac12(1+D).$$

Thus

$$P_{\mathrm{guess}} = \frac12(1+0.4) = 0.7.$$

Dephasing backflow

A dephasing model has $|\eta(1)|=0.2$ and $|\eta(2)|=0.35$. What happens to the trace distance between the $\lvert+\rangle$ and $\lvert-\rangle$ initial states?

Solution

For the transverse antipodal pair,

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

Therefore

$$D(1)=0.2, \qquad D(2)=0.35.$$

The trace distance increases, so this interval shows information backflow and rules out P divisibility, hence CP divisibility, on that interval.

Markovian amplitude damping

For Markovian amplitude damping with $q(t)=e^{-\Gamma t}$, show that the trace distance between initially excited and ground states is monotone.

Solution

For the pair $\lvert e\rangle\langle e\rvert$ and $\lvert g\rangle\langle g\rvert$,

$$D(t)=q(t)=e^{-\Gamma t}.$$

Since $\Gamma\ge0$,

$$\frac{dD}{dt} = - \Gamma e^{-\Gamma t} \le0.$$

Thus there is no trace-distance backflow for this pair. In this standard semigroup model, no pair shows backflow.

Backflow and CP divisibility

Why does trace-distance backflow rule out CP divisibility?

Solution

If the evolution were CP-divisible, then for every $t\ge s$ there would be a CPTP intermediate map $\Phi_{t,s}$ such that

$$\rho_j(t)=\Phi_{t,s}\rho_j(s), \qquad j=1,2.$$

CPTP maps contract trace distance:

$$D(\rho_1(t),\rho_2(t)) \le D(\rho_1(s),\rho_2(s)).$$

An observed increase contradicts this contraction property. Therefore the evolution cannot be CP-divisible over that interval.

Cross-Links

• Non-Markovian Dynamics
• CP Divisibility
• Non-Markovianity Measures
• Completely Positive Maps
• Choi Matrix
• Dephasing Channel
• Amplitude-Damping Channel
• Time-Convolutionless Master Equations
• Approximation Checklist

References

• H.-P. Breuer, E.-M. Laine, and J. Piilo, “Measure for the degree of non-Markovian behavior of quantum processes in open systems,” Physical Review Letters 103, 210401 (2009).
• H.-P. Breuer, E.-M. Laine, J. Piilo, and B. Vacchini, “Colloquium: Non-Markovian dynamics in open quantum systems,” Reviews of Modern Physics 88, 021002 (2016).
• Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quantum non-Markovianity: characterization, quantification and detection,” Reports on Progress in Physics 77, 094001 (2014).
• D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Physical Review A 83, 052128 (2011).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018).