Initial Correlations

Initial correlations are correlations already present between a system $S$ and its environment $E$ at the time where an open-system description begins. They matter because the reduced state $\rho_S(t_0)$ may not contain enough information to predict the later reduced state.

The exact reduced state is always well defined:

$$\rho_S(t) = \operatorname{Tr}_E \left[ U(t,t_0)\rho_{SE}(t_0)U^\dagger(t,t_0) \right].$$

The subtle question is different: can this expression be written as a single quantum channel acting on arbitrary initial system states? Under the usual factorized assumption, yes. With initial system-environment correlations, not in general.

This page is the canonical home for the factorized-initial-state caveat behind Reduced Dynamics, the inhomogeneous terms in Nakajima–Zwanzig Projection and Time-Convolutionless Master Equations, and the preparation-domain warnings in the Approximation Checklist.

Factorized Initial States

A factorized initial state has the form

$$\rho_{SE}(t_0) = \rho_S(t_0)\otimes\eta_E,$$

where $\eta_E$ is a fixed environment reference state independent of $\rho_S(t_0)$.

Then the reduced evolution is a map on the system state alone:

$$\Phi_{t,t_0}(\rho) = \operatorname{Tr}_E \left[ U(t,t_0)(\rho\otimes\eta_E)U^\dagger(t,t_0) \right].$$

This map is completely positive and trace preserving. If

$$\eta_E = \sum_r q_r \lvert r\rangle\langle r\rvert,$$

then a Kraus representation is

$$K_{\mu r} = \sqrt{q_r}\, \langle\mu|U(t,t_0)|r\rangle_E, \qquad \Phi_{t,t_0}(\rho) = \sum_{\mu,r} K_{\mu r}\rho K_{\mu r}^\dagger .$$

This is the standard route from a unitary system-environment model to a Kraus representation. The crucial assumption is not merely that the environment exists. It is that the same environment state $\eta_E$ is paired with every admissible input $\rho$.

What Initial Correlations Change

An initially correlated state is any state for which

$$\rho_{SE}(t_0) \ne \rho_S(t_0)\otimes\eta_E$$

for the intended environment reference state. Correlations may be classical, quantum, weak, strong, short lived, or hidden in an equilibrium state. The common feature is that $\rho_S(t_0)$ alone is not a complete description of the preparation.

Two total states may have the same reduced state:

$$\operatorname{Tr}_E\rho_{SE}^{(1)}(t_0) = \operatorname{Tr}_E\rho_{SE}^{(2)}(t_0) = \rho_S(t_0),$$

but after the same joint unitary,

$$\operatorname{Tr}_E \left[ U\rho_{SE}^{(1)}(t_0)U^\dagger \right] \ne \operatorname{Tr}_E \left[ U\rho_{SE}^{(2)}(t_0)U^\dagger \right].$$

In that situation no map of $\rho_S(t_0)$ alone can represent both preparations. The reduced state did not carry the missing information.

This is not a failure of quantum mechanics. It is a failure of a proposed reduced description to include the preparation variables that influence the future.

Assignment Maps

One way to describe correlated initial conditions is to introduce an assignment map

$$\mathcal A: \rho_S \longmapsto \rho_{SE},$$

which assigns a compatible total state to each system state in some set. Consistency means

$$\operatorname{Tr}_E\mathcal A(\rho_S) = \rho_S.$$

Once an assignment is specified, the reduced evolution is

$$\Phi_{\mathcal A}(\rho_S) = \operatorname{Tr}_E \left[ U\,\mathcal A(\rho_S)\,U^\dagger \right].$$

For the factorized case,

$$\mathcal A_0(\rho_S) = \rho_S\otimes\eta_E,$$

and $\Phi_{\mathcal A_0}$ is a channel on the full system state space.

For genuinely correlated assignments, one cannot generally demand all of the following properties on the full state space:

• linearity of $\mathcal A$;
• consistency under partial trace;
• positivity of $\mathcal A(\rho_S)$ for every density operator $\rho_S$;
• fixed nontrivial system-environment correlations.

This obstruction is often associated with the Pechukas–Alicki discussion of reduced dynamics. The lesson for practice is modest but important: a correlated assignment may be meaningful on a restricted compatibility domain, but extending it to all system density operators can introduce nonphysical states or maps that are not completely positive.

Compatibility Domains

A compatibility domain is the set of system states for which the preparation procedure supplies a legitimate total state. It is the natural domain of a reduced model with initial correlations.

For example, a preparation apparatus might only produce total states of the form

$$\rho_{SE} = \sum_j p_j \rho_S^{(j)}\otimes\rho_E^{(j)}.$$

The allowed reduced states are then

$$\rho_S = \sum_j p_j\rho_S^{(j)}$$

with the same probabilities $p_j$ that determine the environment state. The map from $\rho_S$ to $\rho_{SE}$ may be well defined only on the convex set generated by the prepared $\rho_S^{(j)}$.

Outside that set, asking whether the reduced map is positive or completely positive may be asking the wrong question. The preparation never supplies those inputs. A model should state the domain on which it claims to act.

Complete Positivity: What Fails and What Does Not

Complete positivity is the correct requirement for operations that may act on one part of a larger entangled state without hidden dependence on that larger state. It is not a blanket statement that every formula obtained by tracing an initially correlated environment must be a channel on all system states.

With initial correlations:

• the exact total evolution remains unitary or completely positive on $SE$;
• the exact reduced state remains positive and normalized;
• a universal CPTP map on arbitrary $\rho_S$ may not exist;
• a reduced map may exist only on a restricted preparation domain;
• a non-CP extension outside that domain does not by itself show that the physical preparation is impossible.

Conversely, lack of complete positivity in an approximate reduced equation is not automatically harmless. If the equation is advertised as a channel on arbitrary inputs, or is used on a system entangled with a reference, complete positivity must be checked. The point is to match the mathematical claim to the preparation scenario.

Inhomogeneous Terms in Projection Methods

Projection-operator derivations make the role of initial correlations explicit. With a projection $\mathcal P$ and $\mathcal Q=\mathcal I-\mathcal P$, the exact Nakajima–Zwanzig equation contains

$$\mathcal I(t) = \mathcal P\mathcal L(t) \mathcal G_Q(t,t_0) \mathcal Q\rho(t_0).$$

This term vanishes when

$$\mathcal Q\rho(t_0)=0,$$

meaning the initial total state lies in the range of the chosen projection. For the standard factorizing projection,

$$\mathcal P X = \operatorname{Tr}_E(X)\otimes\rho_E,$$

this includes states of the form $\rho_S(t_0)\otimes\rho_E$. Initial correlations, bath deviations, or a poor projection choice can make $\mathcal Q\rho(t_0)$ nonzero.

The corresponding time-convolutionless equation can also contain an inhomogeneous term. Dropping it is an approximation, not a mathematical identity.

Slippage of Initial Conditions

In weak-coupling Markovian theory, short-time transients often occur on the bath memory time $\tau_E$. During this initial layer, the system and environment build correlations that a later Markovian equation does not explicitly track.

One practical response is slippage of initial conditions. Instead of applying a Markovian master equation directly to the bare state $\rho_S(t_0)$, one applies it after a short transient:

$$\rho_S(t_0) \longmapsto \rho_S(t_0+\tau_E) \equiv \mathcal S\rho_S(t_0),$$

and then evolves the slipped state with the reduced Markovian generator. The map $\mathcal S$ is not a universal cure. It is a way of acknowledging that the effective equation is not intended to resolve the microscopic initial layer.

Slippage is most relevant when:

• the bath memory time is short but not negligible;
• the derived Markovian generator gives unphysical behavior at very early times;
• the actual preparation is close to, but not exactly, the factorized reference state;
• one cares about long-time relaxation more than microscopic transients.

It is not a license to ignore strong coupling, long memory, or a preparation that is far outside the model domain.

Physical Sources

Initial correlations appear naturally in common situations.

Interacting equilibrium

If the system and environment equilibrate while coupled, the joint thermal state is

$$\rho_{SE}^{\beta} = \frac{ e^{-\beta(H_S+H_E+H_I)} }{ \operatorname{Tr} e^{-\beta(H_S+H_E+H_I)} }.$$

Unless $H_I=0$ or a special limit applies, this state is not the product of a system Gibbs state and an environment Gibbs state.

Prior measurements and conditioning

A measurement on $S$ can condition the environment if $S$ and $E$ were previously correlated. Even if the post-measurement reduced state of $S$ looks simple, the environment state may now depend on the outcome or on the preparation setting.

Strongly coupled modes

A narrow cavity mode, defect, phonon coordinate, or reaction coordinate can remain correlated with the system over the time scale of interest. Treating that mode as part of the environment may create apparent initial-correlation and memory effects. Including it in the system can produce a cleaner reduced description for the remaining bath; see Reaction-Coordinate Mapping for the constructive version of this boundary change.

Diagnostic Questions

Before using a channel or master equation, ask:

• How was $\rho_S(t_0)$ prepared?
• Was the interaction $H_I$ off during preparation, or was the system already coupled to the environment?
• Is the environment reference state independent of the chosen system input?
• Could two different preparations produce the same $\rho_S(t_0)$ but different environment states?
• Is the map meant to act on all density operators or only on a compatibility domain?
• Are initially irrelevant variables $\mathcal Q\rho(t_0)$ being dropped?
• Is an initial slip or enlarged system boundary needed?
• Will the reduced map be applied to a system entangled with an external reference?

The answers determine whether the channel language is appropriate, whether a preparation map must be included, or whether the system boundary should be redrawn.

Common Mistakes

Treating the factorized state as automatic

The product form $\rho_S\otimes\rho_E$ is a modeling assumption. It can be excellent for independently prepared systems and large reservoirs, but it is not guaranteed by tracing out an environment.

Declaring non-CP reduced maps unphysical without checking the domain

A non-CP extension outside a compatibility domain may only show that the extension was too ambitious. The physical preparation can still be legitimate.

Using a correlated equilibrium state with a factorized derivation

Starting from a joint thermal state of $H_S+H_E+H_I$ and then applying a derivation that assumes $\rho_S\otimes\rho_E$ mixes two different preparation assumptions.

Dropping the inhomogeneous term by habit

In projection methods, $\mathcal I(t)$ vanishes only when $\mathcal Q\rho(t_0)=0$. Initial correlations are exactly one reason this condition can fail.

Hiding the preparation in notation

Writing $\Phi(\rho_S)$ can conceal dependence on how $\rho_S$ was prepared. If the environment state changes with the preparation, include that dependence explicitly.

Exercises

Same reduced state, different futures

Let $S$ and $E$ be qubits. Define

$$\rho_{SE}^{(1)} = \frac14 I_S\otimes I_E$$

and

$$\rho_{SE}^{(2)} = \frac12 \lvert 00\rangle\langle 00\rvert + \frac12 \lvert 11\rangle\langle 11\rvert .$$

Both have $\rho_S=I/2$. Let $U$ flip $S$ if $E=1$:

$$U\lvert s,e\rangle = \lvert s\oplus e,e\rangle.$$

Compute the final reduced state of $S$ for each initial total state.

Solution

The maximally mixed product state is invariant under any unitary, so

$$\operatorname{Tr}_E \left[ U\rho_{SE}^{(1)}U^\dagger \right] = \frac12 I_S.$$

For the correlated state,

$$\lvert 00\rangle \longmapsto \lvert 00\rangle, \qquad \lvert 11\rangle \longmapsto \lvert 01\rangle.$$

Both final components have $S=0$. Hence

$$\operatorname{Tr}_E \left[ U\rho_{SE}^{(2)}U^\dagger \right] = \lvert 0\rangle\langle 0\rvert .$$

The same initial reduced state led to different final reduced states. Therefore $\rho_S(t_0)$ alone did not determine the future.

Product assignment gives a channel

Let $\mathcal A_0(\rho)=\rho\otimes\eta_E$. Show that

$$\Phi(\rho) = \operatorname{Tr}_E \left[ U\mathcal A_0(\rho)U^\dagger \right]$$

is completely positive and trace preserving.

Solution

Diagonalize $\eta_E=\sum_r q_r\lvert r\rangle\langle r\rvert$ and trace in an environment basis $\{\lvert\mu\rangle\}$. Define

$$K_{\mu r} = \sqrt{q_r} \langle\mu|U|r\rangle_E.$$

Then

$$\Phi(\rho) = \sum_{\mu,r} K_{\mu r}\rho K_{\mu r}^\dagger,$$

which is completely positive. Trace preservation follows from

$$\sum_{\mu,r}K_{\mu r}^\dagger K_{\mu r} = I_S,$$

using $\sum_\mu\lvert\mu\rangle\langle\mu\rvert=I_E$, $U^\dagger U=I_{SE}$, and $\sum_r q_r=1$.

Inhomogeneous term

In the Nakajima–Zwanzig equation, why does the term

$$\mathcal P\mathcal L(t) \mathcal G_Q(t,t_0) \mathcal Q\rho(t_0)$$

vanish for a projection-compatible initial state?

Solution

A projection-compatible initial state lies in the range of $\mathcal P$, so

$$\mathcal P\rho(t_0)=\rho(t_0).$$

Therefore

$$\mathcal Q\rho(t_0) = (\mathcal I-\mathcal P)\rho(t_0) = 0.$$

The inhomogeneous term contains this factor, so it vanishes.

Compatibility-domain reasoning

Why can a reduced map be meaningful on a compatibility domain even if a linear extension to all density operators is not completely positive?

Solution

The compatibility domain contains only the states that the preparation procedure can actually produce together with their associated environment states. A linear extension to all density operators may assign invalid total states to inputs outside this domain. Failure of complete positivity for that extension does not necessarily describe a physical experiment. It shows that the extension should not be interpreted as a universal channel.

References

• E. B. Davies, Quantum Theory of Open Systems, Academic Press (1976).
• P. Pechukas, “Reduced Dynamics Need Not Be Completely Positive,” Physical Review Letters 73, 1060-1062 (1994).
• R. Alicki, “Comment on ‘Reduced Dynamics Need Not Be Completely Positive’,” Physical Review Letters 75, 3020 (1995).
• P. Pechukas, “Reply to ‘Comment on Reduced Dynamics Need Not Be Completely Positive’,” Physical Review Letters 75, 3021 (1995).
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press (2002).
• A. Shaji and E. C. G. Sudarshan, “Who’s afraid of not completely positive maps?”, Physics Letters A 341, 48-54 (2005).
• Á. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction, Springer (2012).
• K. Modi, “Operational approach to open dynamics and quantifying initial correlations,” Scientific Reports 2, 581 (2012).