CP Divisibility

CP divisibility is a map-level way to say that an open quantum evolution can be broken into legitimate quantum channels over every intermediate time interval. It is one of the cleanest finite-dimensional diagnostics of Markovianity, especially when the reduced dynamics is described by a family of quantum channels rather than by an explicit microscopic bath.

It is not the only meaning of non-Markovianity. The broader terminology is compared in What Non-Markovian Means, while Non-Markovianity Measures compares divisibility-based measures with trace-distance and entanglement-based measures. This page focuses on the divisibility criterion itself.

Intermediate Quantum Channels

Let

$$\Phi_{t,0}$$

be the reduced dynamical map from an initial time $0$ to time $t$, so that

$$\rho(t)=\Phi_{t,0}\rho(0).$$

The family is CP-divisible if, for every $t\ge s\ge0$, there exists a completely positive trace-preserving map $\Phi_{t,s}$ such that

$$\Phi_{t,0} = \Phi_{t,s}\Phi_{s,0}.$$

The map $\Phi_{t,s}$ is the intermediate propagator from $s$ to $t$. The condition concerns a family of maps for a fixed initial preparation procedure. Interventions at intermediate times require additional information about the multi-time process, discussed below.

Why Complete Positivity Matters

Positivity alone says that states of the system remain positive. Complete positivity says the same remains true when the system is entangled with an arbitrary reference:

$$(\Phi_{t,s}\otimes\mathrm{id}_R)(X)\ge0$$

for every positive operator $X$ on system plus reference.

This is the correct channel condition for a subsystem whose correlations with outside degrees of freedom may matter operationally. A merely positive intermediate map may preserve system states while failing on entangled system-reference states.

Relation to Semigroups

A time-homogeneous quantum dynamical semigroup is a special case:

$$\Phi_t=e^{t\mathcal L}, \qquad \Phi_{t+s}=\Phi_t\Phi_s.$$

The intermediate map is simply

$$\Phi_{t,s}=e^{(t-s)\mathcal L},$$

which is CPTP for all $t\ge s$ when $\mathcal L$ is a valid Lindblad–GKSL generator.

CP divisibility is weaker than semigroup structure. A driven or time-dependent open system can be CP-divisible without being time homogeneous:

$$\Phi_{t,s} \ne \Phi_{t-s,0} \quad \text{in general}.$$

Thus:

time-homogeneous Lindblad semigroup
  -> CP-divisible evolution
  -> each finite-time map is CPTP

The reverse implications do not generally hold.

Invertible Maps and Intermediate Propagators

If $\Phi_{s,0}$ is invertible as a linear map on operators, then the intermediate map is forced to be

$$\Phi_{t,s} = \Phi_{t,0}\Phi_{s,0}^{-1}.$$

This formula is useful for calculations, but it comes with a warning: $\Phi_{s,0}^{-1}$ need not be a physical channel. CP divisibility asks whether the product above is CPTP.

If $\Phi_{s,0}$ is not invertible, intermediate maps may not be unique. In that case, divisibility must be formulated with care: one asks whether at least one CPTP map $\Phi_{t,s}$ exists on the relevant operator space and is compatible with $\Phi_{t,0}=\Phi_{t,s}\Phi_{s,0}$.

Time-Local Generator Criterion

For a differentiable, invertible finite-dimensional family,

$$\mathcal K(t) = \dot\Phi_{t,0}\Phi_{t,0}^{-1}$$

defines a time-local generator:

$$\frac{d\rho}{dt} = \mathcal K(t)\rho.$$

Under the usual finite-dimensional regularity assumptions, CP divisibility is equivalent to the generator having instantaneous GKSL form with nonnegative canonical rates:

$$\mathcal K(t)\rho = - \frac{i}{\hbar}[H(t),\rho] + \sum_j\gamma_j(t) \left( L_j(t)\rho L_j^\dagger(t) - \frac12 \{L_j^\dagger(t)L_j(t),\rho\} \right), \qquad \gamma_j(t)\ge0.$$

Here the canonical $L_j$ are traceless and Hilbert–Schmidt orthonormal, $\operatorname{Tr}(L_j^\dagger L_k)=\delta_{jk}$, after diagonalizing the dissipative coefficient matrix. This fixes the scale of the rates. This is the time-dependent analogue of the Lindblad–GKSL Equation. If a canonical rate becomes negative, CP divisibility fails at that time. The finite-time map from $0$ to $t$ may still be physical; the issue is the intermediate-channel interpretation.

CP Divisibility Versus P Divisibility

P divisibility asks only that each intermediate map be positive and trace preserving. CP divisibility asks that it be completely positive and trace preserving.

The hierarchy is:

$$\text{CP divisible} \Longrightarrow \text{P divisible},$$

but not conversely. The distinction matters because positivity can miss failures that appear only when the system is tested together with an entangled reference.

P divisibility is closely tied to monotonic contraction of trace distance for pairs of system states. CP divisibility is the stronger channel-level condition. See Information Backflow for the trace-distance diagnostic.

Practical Choi Test

In a finite-dimensional numerical calculation, one can test candidate intermediate maps using the Choi Matrix.

For an intermediate map $\Phi_{t,s}$:

1. Build the superoperator for $\Phi_{t,s}$.
2. Convert it to the chosen Choi convention.
3. Check trace preservation.
4. Check Choi positivity.

With the output-input convention used in this volume, trace preservation is

$$\operatorname{Tr}_{\mathrm{out}}J_{\Phi_{t,s}} = I_{\mathrm{in}},$$

and complete positivity is

$$J_{\Phi_{t,s}}\ge0.$$

Numerically, one tests whether the smallest Choi eigenvalue is nonnegative within tolerance. See Solving Lindblad Equations for a numerical implementation.

Markovian and Non-Markovian Noise shows how to apply interval Choi tests to measured channel data, including uncertain or nearly singular maps.

Example: Pure Dephasing

Consider a qubit dephasing map

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

Assume $\eta(t)\ne0$ on the interval being tested. The intermediate coherence factor is

$$\eta(t,s) = \frac{\eta(t)}{\eta(s)}.$$

For ordinary qubit dephasing, the intermediate map is completely positive when

$$|\eta(t,s)|\le1.$$

Thus CP divisibility requires

$$|\eta(t)|\le|\eta(s)| \qquad \text{for all }t\ge s.$$

The unnormalized Choi matrix has eigenvalues $1+|\eta(t,s)|$, $1-|\eta(t,s)|$, $0$, and $0$. Thus if $|\eta(s)|=0.3$ and $|\eta(t)|=0.5$, the intermediate matrix has an eigenvalue $1-5/3=-2/3$: the failure of complete positivity is explicit.

If $\eta(s)=0$, no inverse exists. The channel has erased the distinction between the two transverse states $\lvert+\rangle$ and $\lvert-\rangle$. Divisibility requires that this lost distinction never return. More generally, a necessary condition for factorization is $\ker\Phi_{s,0}\subseteq\ker\Phi_{t,0}$.

If coherence magnitude revives, the finite-time maps $\Phi_{t,0}$ may still be CPTP, but the evolution is not CP-divisible over the revival interval.

For exponential Markovian dephasing,

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

the condition is satisfied.

Example: Amplitude Damping

For zero-temperature amplitude damping, write the excited-state survival factor as

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

in the Markovian case. More generally, a finite-time amplitude-damping-like map has

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

with $0\le q(t)\le1$ for each physical finite-time map.

When $q(s)>0$, the intermediate damping parameter depends on

$$\frac{q(t)}{q(s)}.$$

CP divisibility requires

$$0\le\frac{q(t)}{q(s)}\le1 \qquad \text{for all }t\ge s.$$

Thus the survival probability must not increase. A revival of $q(t)$ signals that population or excitation has flowed back from the environment or an enlarged mode into the system, so the reduced evolution is not CP-divisible over that interval.

What Divisibility Does and Does Not Predict

CP divisibility says that the unperturbed one-time maps admit a channel between every pair of times. It is a useful mathematical memory criterion, but it does not say that the present reduced state contains all the information needed to predict arbitrary future experiments.

An intermediate measurement or replacement of the system state can change its correlations with the environment while leaving information in the environment itself. Future outcomes may then depend on earlier interventions even when the original map family was CP-divisible. The operational Markov condition of Pollock and collaborators tests these multi-time statistics.

A concrete example is a qubit with a static, randomly chosen detuning $\xi$ of zero mean and Gaussian variance $\sigma^2$. Averaging the phase gives $\eta(t)=\langle e^{-i\xi t}\rangle=e^{-\sigma^2t^2/2}$, a CP-divisible dephasing family. Nevertheless, the same unknown $\xi$ acts throughout each run. An ideal $\pi$ pulse halfway through an interval reverses the sign of the accumulated phase and restores the coherence at its end. This echo reveals temporal correlations absent from a memoryless dephasing model fitted to the same free-decay curve.

If CP divisibility fails, no factorization into channels on the whole system operator space reproduces the specified map family. The total system–environment dynamics can still be perfectly physical. A structured mode, changing bath state, or system–environment correlations can carry the information missing from the reduced description.

Exercises

Semigroup implies CP divisibility

Show that a Lindblad semigroup $\Phi_t=e^{t\mathcal L}$ is CP-divisible.

Solution

For $t\ge s$,

$$\Phi_t = e^{t\mathcal L} = e^{(t-s)\mathcal L}e^{s\mathcal L}.$$

Thus

$$\Phi_{t,s}=e^{(t-s)\mathcal L}.$$

If $\mathcal L$ is a Lindblad–GKSL generator, $e^{\tau\mathcal L}$ is CPTP for every $\tau\ge0$. Therefore $\Phi_{t,s}$ is CPTP for every $t\ge s$, so the semigroup is CP-divisible.

Dephasing revival

A qubit dephasing map has $\eta(1)=0.3$ and $\eta(2)=0.5$. Can the family be CP-divisible on the interval from $1$ to $2$?

Solution

The intermediate coherence factor is

$$\eta(2,1) = \frac{\eta(2)}{\eta(1)} = \frac{0.5}{0.3}.$$

Its magnitude is greater than $1$, so the intermediate dephasing map would amplify coherences beyond the allowed qubit dephasing-channel range. The family is not CP-divisible on that interval, even if the individual maps at $t=1$ and $t=2$ are valid channels.

Finite-time map versus intermediate map

Why can $\Phi_{t,0}$ be CPTP while $\Phi_{t,s}$ is not CPTP?

Solution

The map $\Phi_{t,0}$ describes evolution from the original preparation to time $t$. It can be a valid channel on the allowed initial states and references. CP divisibility asks a stronger question: after evolving to time $s$, can the interval from $s$ to $t$ be represented by a CPTP map acting on an arbitrary present system state?

If the system and environment are correlated at time $s$, or if the environment has changed in a way not encoded in $\rho_S(s)$, this intermediate-channel description can fail even though the original finite-time map remains CPTP.

P versus CP divisibility

Why is P divisibility weaker than CP divisibility?

Solution

P divisibility requires intermediate maps to preserve positivity of system states. CP divisibility requires positivity even after tensoring with an identity map on an arbitrary reference. Every completely positive map is positive, so CP divisibility implies P divisibility.

The converse fails because some positive maps are not completely positive. Such maps may behave well on isolated system density matrices but fail on entangled system-reference states.

References

• 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).
• D. Chruściński, A. Kossakowski, and Á. Rivas, “Measures of non-Markovianity: Divisibility versus backflow of information,” Physical Review A 83, 052128 (2011).
• F. A. Pollock, C. Rodríguez-Rosario, T. Frauenheim, M. Paternostro, and K. Modi, “Operational Markov Condition for Quantum Processes,” Physical Review Letters 120, 040405 (2018), doi:10.1103/PhysRevLett.120.040405.
• Á. Rivas, S. F. Huelga, and M. B. Plenio, “Entanglement and non-Markovianity of quantum evolutions,” Physical Review Letters 105, 050403 (2010).
• Á. Rivas, S. F. Huelga, and M. B. Plenio, “Quantum non-Markovianity: characterization, quantification and detection,” Reports on Progress in Physics 77, 094001 (2014).
• M. M. Wolf, J. Eisert, T. S. Cubitt, and J. I. Cirac, “Assessing non-Markovian quantum dynamics,” Physical Review Letters 101, 150402 (2008).