The Mathematical Language of Markovianity

“Markovian” can refer to a homogeneous channel semigroup, divisibility by intermediate channels, or absence of memory in a multitime experiment. These properties answer different mathematical questions. The canonical What Non-Markovian Means article surveys their physical uses; this page examines the algebraic conditions that remain meaningful when reduced maps are not invertible.

Required background. Quantum Dynamical Semigroups fixes the homogeneous case. Helpful background. CP Divisibility gives the finite-dimensional time-local generator criterion.

Divisibility is an existence statement

Let $\Phi_t$ be channels from initial states to states at time $t$, with $\Phi_0=\mathrm{id}$. CP divisibility means that for every $t\ge s\ge0$ there exists a channel $V_{t,s}$ on the full system trace class such that

$$\Phi_t=V_{t,s}\Phi_s.$$

Every conservative quantum dynamical semigroup is CP divisible, with $V_{t,s}=\Phi_{t-s}$. The converse fails because a divisible family can depend on absolute time.

In finite dimensions, if $\Phi_s$ is invertible as a linear superoperator, the only possible intermediate map is $V_{t,s}=\Phi_t\Phi_s^{-1}$. Its complete positivity is a property to check; the inverse itself usually is not positive. If $\Phi_s$ is not invertible, the formula has no meaning on the full space, and existence of a channel extension becomes the central problem.

The definition stated here asks for channel factorizations of the one-time maps. If a theory also specifies a consistent two-time evolution family obeying a cocycle law on all inputs, that is additional structure to identify explicitly, especially when channel extensions are nonunique.

Lost linear directions cannot reappear

Factorization by any linear map, even one not positive, requires

$$\ker\Phi_s\subseteq\ker\Phi_t\qquad(t\ge s).$$

Indeed, if $\Phi_s(X)=0$, then $\Phi_t(X)=V_{t,s}(0)=0$. Equivalently, two initial inputs that have become identical at $s$ cannot become distinguishable later under a map acting only on their common intermediate state.

This kernel condition defines a unique linear map on the range:

$$\widetilde V_{t,s}(\Phi_sX)=\Phi_tX.$$

It is well defined exactly when the kernel inclusion holds. That does not prove it is a channel on the full state space. One must still establish complete positivity on the relevant ordered subspace and a trace-preserving extension. The reachable range need not contain every state; arbitrary continuation outside it can violate positivity.

In infinite dimensions, additional continuity issues enter even for injective maps. An inverse on the range can be unbounded or the range can be nonclosed. For example, damping each excited level $|n\rangle$ to $|0\rangle$ with survival probability $e^{-nt}$ multiplies $|n\rangle\langle0|$ by $e^{-nt/2}$. The inverse, where defined, multiplies these unit trace-norm matrix units by $e^{nt/2}$ and is unbounded. An inverse formula alone cannot establish a bounded intermediate channel.

Worked examples with a reset channel

Fix a state $\tau$ and let $R_\tau(X)=\operatorname{Tr}(X)\tau$. For $0\le f(t)\le1$ define

$$\Phi_t=f(t)\mathrm{id}+(1-f(t))R_\tau.$$

When $f(s)>0$ and $f(t)\le f(s)$, a channel factorization is

$$V_{t,s}=\frac{f(t)}{f(s)}\mathrm{id} +\left(1-\frac{f(t)}{f(s)}\right)R_\tau.$$

Taking time in fixed units, $f(t)=e^{-t^2}$ gives a CP-divisible family. It is not a homogeneous semigroup because $(t+s)^2\ne t^2+s^2$ in general. Its instantaneous reset rate is $2t$, so an interval’s effect depends on its starting time.

In contrast, choose $f(t)=\cos^2t$. Every individual $\Phi_t$ is a channel and the family is smooth. At $s=\pi/2$, however, $\Phi_s=R_\tau$ erases every traceless direction, while $\Phi_\pi=\mathrm{id}$ restores all of them. The kernel condition fails, so no linear factorization from $s$ to $\pi$ exists. In particular, the family is not CP divisible.

No matrix inverse or potentially singular time-local generator was needed to reach this conclusion. The noninvertible time is precisely where the kernel test is most useful.

One-time divisibility does not determine multitime memory

A family $\Phi_t$ tells how an initial state evolves when the subsequent protocol is fixed. It does not specify the response to every possible measurement, reset, or control operation inserted at an intermediate time.

Different system–environment processes can agree on these one-time maps but differ after interventions. Correlations stored between different parts of an environment can generate correlated outputs at successive times even when the one-time channels admit CP factorizations. Milz and collaborators give explicit constructions establishing this distinction.

Likewise, contractivity of trace distance under every $V_{t,s}$ is a necessary consequence of CP divisibility, but absence of a revival in one chosen pair of states is not a proof of divisibility. Tests involving all inputs, ancillas, invertibility, or intervention histories must be stated with their own quantifiers.

Exercises

Absorbing information loss. Suppose $\Phi_s=R_\tau$ and the family is divisible at all later times. What form must every $\Phi_t$ have?

Solution

Factorization gives $\Phi_t(X)=\operatorname{Tr}(X)V_{t,s}(\tau)$. Thus every later map is another replacement channel. Its output state may change with time, but no dependence on the initial traceless component can return.

Nonunique extensions. If $\Phi_s=R_\tau=\Phi_t$, exhibit two different channels $V_{t,s}$ when the Hilbert space has dimension at least two.

Solution

Both $V_{t,s}=\mathrm{id}$ and $V_{t,s}=R_\tau$ satisfy the factorization. They agree on the one-dimensional reachable range and generally differ elsewhere. A propagator inferred only from reachable states therefore need not be uniquely determined.

References

• Milz, Simon, M. S. Kim, Felix A. Pollock, and Kavan Modi. “Completely positive divisibility does not mean Markovianity.” Physical Review Letters 123, 040401 (2019). Preprint. Explicit separation between divisibility and multitime memory.
• Rivas, Ángel, Susana F. Huelga, and Martin B. Plenio. “Entanglement and non-Markovianity of quantum evolutions.” Physical Review Letters 105, 050403 (2010). Preprint. Complete positivity of intermediate evolution and ancilla-assisted diagnostics.
• Rivas, Ángel, Susana F. Huelga, and Martin B. Plenio. “Quantum non-Markovianity: characterization, quantification and detection.” Reports on Progress in Physics 77, 094001 (2014). Preprint. Comparison of mathematical criteria and their hypotheses.