Time-Varying Deterministic Kraus Map

• Time-varying deterministic Kraus map is a family of completely positive, trace-preserving maps defined by explicitly time-dependent Kraus operators derived from global unitary dynamics.
• It guarantees deterministic evolution without post-selection by summing over Kraus branches, ensuring both positivity and trace preservation.
• The framework applies to diverse scenarios, from amplitude-damping models to non-Markovian diagnostics, offering insights into open quantum system behavior.

Searching arXiv for the cited Kraus-map papers and related open-system work. arXiv search query: (Maziero, 2015) Kraus representation open quantum systems A time-varying deterministic Kraus map is a family of reduced dynamical maps for an open quantum system in which the system state at time $t$ is obtained from its initial state by an operator-sum representation with explicitly time-dependent Kraus operators, while the evolution remains unconditional and trace-preserving. In the open-system setting treated in “The Kraus representation for the dynamics of open quantum systems” (Maziero, 2015), the system $S$ interacts unitarily with an environment $A$, and the reduced state $\hat{\rho}_t^S$ is written as

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$

In the paper’s notation,

$$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$

with $\hat K_l$ understood as time dependent through the propagator $\hat U(t)$ (Maziero, 2015). In this usage, “deterministic” means trace-preserving: the map describes unconditional reduced dynamics obtained after tracing out the environment, not a post-selected branch associated with a measurement outcome.

1. Definition and basic meaning

Within open quantum theory, the map of interest acts on density operators of the system Hilbert space and takes the form

$$\mathcal E_t:\rho_S(0)\mapsto \rho_S(t)=\sum_k K_k(t)\rho_S(0)K_k^\dagger(t).$$

The defining trace-preservation condition is

$\sum_k K_k^\dagger(t)K_k(t)=I.$

In the notation of (Maziero, 2015), this appears as

$S$0

The same condition is emphasized in “A quantum algorithm for evolving open quantum dynamics on quantum computing devices” (Hu et al., 2019), where the operator-sum evolution is written as

$S$1

The time-varying character enters through the dependence of the Kraus operators on elapsed time. In (Maziero, 2015), this dependence is inherited from the system-environment propagator $S$2. In (Rajagopal et al., 2010), the family $S$3 is the basic object used to distinguish Markovian from non-Markovian behavior through the short-time structure of $S$4. A plausible implication is that “time-varying deterministic Kraus map” is best understood not as a special new class of maps, but as the standard CPTP reduced evolution written explicitly as a one-parameter family $S$5.

The term “deterministic” should not be confused with unitary single-operator evolution. In the present context, the sum over Kraus operators remains essential. What is excluded is stochastic conditioning on monitored outcomes. The index in the operator sum labels environment basis states, Kraus branches, or equivalent decomposition data, but the output state is the nonselective reduced state.

2. Derivation from unitary system-environment dynamics

The derivation in (Maziero, 2015) starts from a standard closed evolution for system plus environment, with an initially factorized state

$S$6

This is the special case of

$S$7

with the environment initially pure. The total state evolves according to

$S$8

where $S$9 is generated by

$A$0

with

$A$1

The reduced state is then

$A$2

To pass from partial trace to operator-sum form, (Maziero, 2015) expands the propagator in the product basis $A$3 and defines the Kraus operators by matrix elements,

$A$4

In standard shorthand this corresponds to

$A$5

understood as an operator on the system Hilbert space. Substituting this definition into the reduced-state expression yields

$A$6

This construction makes the time dependence completely explicit: because every matrix element $A$7 depends on $A$8, each $A$9 is really $\hat{\rho}_t^S$0. The resulting object is therefore a one-parameter family of CPTP maps indexed by time, even though (Maziero, 2015) does not use the notation $\hat{\rho}_t^S$1.

The same system-to-map logic appears in a more model-specific way in the layered spin-star analysis of (Mahdian et al., 2013). There the exact reduced qubit dynamics is first obtained in Bloch form, then rewritten as a superoperator $\hat{\rho}_t^S$2, and only afterward converted into Kraus operators. This suggests two standard routes to a time-varying deterministic Kraus map: direct extraction from the global unitary, as in (Maziero, 2015), or reconstruction from an exactly solved reduced map, as in (Mahdian et al., 2013).

3. Positivity, trace preservation, and the meaning of “deterministic”

The existence of an operator-sum form does not by itself explain why the map is physically admissible. In (Maziero, 2015), positivity and trace preservation are both established from the construction. For positivity, for any $\hat{\rho}_t^S$3,

$\hat{\rho}_t^S$4

where $\hat{\rho}_t^S$5. For trace preservation,

$\hat{\rho}_t^S$6

and the derivation yields

$\hat{\rho}_t^S$7

This is the precise sense in which the map is deterministic. The reduced state is the unconditional output obtained by tracing out the environment. No measurement is performed on the environment, and no branch

$\hat{\rho}_t^S$8

is selected. The same distinction is stressed in (Hu et al., 2019), where the algorithm simulates the density-matrix evolution by summing over Kraus branches rather than sampling a stochastic unraveling.

A related clarification appears in (Rajagopal et al., 2010). There the family

$\hat{\rho}_t^S$9

is treated as the deterministic reduced dynamics of an open system, while Markovianity or non-Markovianity is diagnosed by the time structure of the Kraus operators rather than by any stochastic interpretation. This helps dispel a common misconception: a multi-Kraus representation does not imply randomness at the dynamical-map level.

4. Explicit amplitude-damping realization

The canonical explicit example in (Maziero, 2015) is a two-level atom interacting with the vacuum electromagnetic field. The phenomenological joint unitary action is

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$0

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$1

with $$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$2 and

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$3

Using the general matrix-element definition of the Kraus operators, the paper derives

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$4

with $$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$5. In standard time-dependent notation,

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$6

These operators satisfy

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$7

so they constitute an explicit time-varying deterministic Kraus map. The reduced state is

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$8

For an initial qubit state

$$\rho_S(t)=\mathcal E_t(\rho_S(0))=\sum_k K_k(t)\,\rho_S(0)\,K_k^\dagger(t).$$9

the Bloch vector transforms as

$$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$0

This shows explicitly that the channel is not unital: the longitudinal component is shifted by $$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$1, while the transverse components are contracted by $$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$2. A plausible implication is that the amplitude-damping family is the paradigmatic example of a time-varying deterministic Kraus map with relaxation toward a preferred state.

A closely related amplitude-damping example also appears in (Hu et al., 2019), where the open-system evolution is written in operator-sum form and simulated through Sz.-Nagy dilation. That work treats the family of channels at sampled times rather than as an analyzed continuous-time semigroup, reinforcing the interpretation of time dependence as parameter dependence of the Kraus operators.

5. Coherence decay and state-space action

In (Maziero, 2015), the amplitude-damping map is used to study quantum coherence in the computational basis $$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$3. The $$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$4-coherence of the initial state is

$$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$5

After the map,

$$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$6

Since $$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$7,

$$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$8

and the off-diagonal terms obey

$$\hat{\rho}_{t}^{S}=\sum_{l}\hat{K}_{l}\hat{\rho}^{S}\hat{K}_{l}^{\dagger},$$9

The paper emphasizes that coherence decreases monotonically and tends to zero asymptotically as $\hat K_l$0. This provides a direct operational reading of the time-varying Kraus family: the channel parameter $\hat K_l$1 encodes both population relaxation and coherence damping.

A different, but structurally parallel, state-space action appears in the exactly solvable layered spin-star model of (Mahdian et al., 2013). There the reduced qubit dynamics is

$\hat K_l$2

and the resulting Kraus operators are

$\hat K_l$3

$\hat K_l$4

This is again a time-dependent deterministic Kraus map, but now of Pauli-diagonal, unital type (Mahdian et al., 2013). The contrast with amplitude damping is instructive: time-varying deterministic Kraus maps may be unital or nonunital depending on the underlying open-system model.

6. Dynamical scope, non-Markovianity, and limitations

The construction in (Maziero, 2015) is deliberately limited in scope. It assumes initial system-environment independence and does not claim that arbitrary correlated initial states admit the same deterministic CPTP Kraus form. The paper also does not develop semigroup composition, CP divisibility, Markovianity criteria, or a time-local master equation for the reduced map. Its time dependence enters through the global propagator and, in the worked example, through the phenomenological decay parameter $\hat K_l$5.

This limitation is important because a family of CPTP maps indexed by time need not satisfy any stronger dynamical property. The paper (Rajagopal et al., 2010) makes this distinction explicit. There, both Markovian and non-Markovian evolutions are written as time-dependent Kraus maps, but the short-time scaling of the Kraus operators distinguishes the two cases. In the Markovian case,

$\hat K_l$6

and

$\hat K_l$7

which yields the LGKS structure. In non-Markovian examples, the nontrivial Kraus operators scale instead as $\hat K_l$8, and the resulting reduced evolution fails to exhibit the standard semigroup infinitesimal form (Rajagopal et al., 2010).

The layered spin-star analysis of (Mahdian et al., 2013) further sharpens the point. That paper constructs an exact time-dependent reduced map and its Kraus representation, and then derives exact TCL and NZ equations from it. Yet it explicitly does not claim CP-divisibility for all intermediate propagators. This suggests that “time-varying deterministic Kraus map” should be interpreted minimally as a family of CPTP maps at each time, not as a guarantee of semigroup structure or divisibility.

A still stronger non-Markovian construction appears in (Wonderen et al., 2018), where the reduced dynamics is obtained microscopically for a system linearly coupled to a thermal bosonic reservoir. There the exact map is represented by an infinite Kraus expansion with explicitly time-dependent Kraus matrices determined by a hierarchy and solved in Laplace space via continued fractions. The perturbative truncations are designed to conserve positivity and probability at every order (Wonderen et al., 2018). This suggests that time-varying deterministic Kraus maps remain well defined even far beyond the Markovian regime, although their explicit structure can become highly nonlocal in time.

7. Related constructions and broader uses

Several other lines of work clarify the broader role of time-dependent deterministic Kraus maps. “Kraus Mapping for atom-cavity and reservoir system” (Mejía et al., 2015) gives an explicit continuous-time channel

$\hat K_l$9

for a three-level effective atom-cavity-reservoir model, together with a discrete short-time Kraus map

$\hat U(t)$0

whose repeated application reproduces the Lindblad evolution in the small-step limit. This provides an explicit bridge between finite-time time-dependent Kraus operators and repeated infinitesimal deterministic updates.

On the implementation side, (Hu et al., 2019) shows how a known Kraus family $\hat U(t)$1 can be simulated on quantum hardware by embedding each non-unitary Kraus operator into a unitary through Sz.-Nagy dilation. The method is applied to an amplitude-damping example over sampled times from $\hat U(t)$2 to $\hat U(t)$3 ps with time step of $\hat U(t)$4 ps. A plausible implication is that once a time-varying deterministic Kraus map is known analytically or numerically, it can be treated as a directly implementable computational object rather than merely a formal representation.

The gate-noise study (Morazotti et al., 2021) supplies a different perspective. There, the noisy evolution associated with the entangling action of the $\hat U(t)$5 gate under dephasing is described by a time-dependent process matrix $\hat U(t)$6, diagonalized to produce Kraus operators

$\hat U(t)$7

This is again a family of deterministic CPTP maps $\hat U(t)$8, now in an explicitly non-Markovian, control-dependent setting (Morazotti et al., 2021).

Taken together, these constructions indicate that the phrase “time-varying deterministic Kraus map” refers most precisely to a family of unconditional reduced evolutions

$\hat U(t)$9

with

$$\mathcal E_t:\rho_S(0)\mapsto \rho_S(t)=\sum_k K_k(t)\rho_S(0)K_k^\dagger(t).$$0

typically derived either from a global system-environment unitary, from an exactly solved reduced map, or from a process matrix. The family may be Markovian or non-Markovian, unital or nonunital, finite-rank or infinite, but its defining characteristics are explicit time dependence, complete positivity, and unconditional trace-preserving reduced dynamics.