⚛️ quantum physics

Divisibility of dynamical maps: Schrödinger vs. Heisenberg picture

This paper demonstrates that the divisibility of quantum dynamical maps, a key indicator of non-Markovianity, is not equivalent between the Schrödinger and Heisenberg pictures due to the distinct roles of left and right generators, thereby establishing Heisenberg divisibility as an independent witness of memory effects with a new operational quantifier based on guessing probability.

Original authors: Federico Settimo, Andrea Smirne, Kimmo Luoma, Bassano Vacchini, Jyrki Piilo, Dariusz Chruściński

Published 2026-03-02

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CC BY 4.0

Original authors: Federico Settimo, Andrea Smirne, Kimmo Luoma, Bassano Vacchini, Jyrki Piilo, Dariusz Chruściński

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a movie of a quantum system (like a tiny atom) interacting with its environment (like a noisy room). Physicists usually study this movie in two different ways, or "pictures":

The Schrödinger Picture (The "Actor" View): You watch the state of the atom change over time. It's like watching a character in a movie evolve, get tired, or get excited.
The Heisenberg Picture (The "Camera" View): You keep the atom's state fixed, but you watch the questions (or measurements) you ask the atom change over time. It's like keeping the character still but changing the camera angles, filters, and lighting to see how the story looks different.

For decades, physicists believed these two views were perfectly equivalent. If the story made sense in one view, it made sense in the other. This paper shatters that assumption.

The Core Discovery: Two Different Rules for "Memory"

The paper focuses on a concept called Divisibility. Think of divisibility as a rule about memory.

Divisible (Markovian): The system has "no memory." It's like a drunk person stumbling through a hallway. Where they step next depends only on where they are right now, not on where they stumbled five seconds ago. Information flows out, and it's gone forever.
Indivisible (Non-Markovian): The system has a "memory." It's like a person walking in a maze who remembers a dead end they hit earlier and avoids it. Information flows out, but then flows back in. This "backflow" is the signature of memory.

The Big Twist:
The authors discovered that a quantum process can be Divisible (no memory) in the "Actor" view (Schrödinger) but Indivisible (has memory) in the "Camera" view (Heisenberg), and vice versa.

They are not the same! You can have a system that looks like it has no memory when you watch the atom, but looks like it has a secret memory when you watch the questions you ask it.

The Secret Ingredient: The "Left" vs. "Right" Generators

Why does this happen? The paper introduces a mathematical distinction between a Left Generator and a Right Generator.

Imagine you are pushing a heavy box across a floor.

The Left Generator is like pushing the box from the left side.
The Right Generator is like pushing the box from the right side.

In simple, steady motion (like a semigroup), pushing from the left or right does the exact same thing. But in complex, changing motion (time-dependent dynamics), pushing from the left creates a different result than pushing from the right.

In the Schrödinger picture, the "Left Generator" controls the rules.
In the Heisenberg picture, the "Right Generator" (which is the mirror image of the Schrödinger one) controls the rules.

Because these two "pushes" are different, the rules for memory (divisibility) are different in each picture.

A Real-World Analogy: The Guessing Game

To make this concrete, the authors propose a game involving guessing probabilities.

Scenario A: The Schrödinger Game (Guessing States)

Alice prepares a quantum coin. It's either "Heads" or "Tails."
Bob tries to guess which one it is.
The Rule: If the system is "Divisible" (no memory), Bob's chance of guessing correctly should never get better than it was at the start. If the coin gets mixed with noise, Bob gets worse at guessing. If his luck suddenly improves, it means information flowed back from the environment (Memory!).

Scenario B: The Heisenberg Game (Guessing Measurements)

Alice has a magic box that performs one of two measurements (let's call them "Filter A" or "Filter B").
Bob has to guess which filter Alice used.
The Twist: The authors show that Bob's ability to guess can get better over time even if the Schrödinger game says "no memory."
Meaning: The "Camera" view detects a type of memory that the "Actor" view completely misses.

Why Should You Care?

We Might Be Missing Memory: If we only look at quantum systems through the Schrödinger lens (which is the standard way), we might think a system is "memoryless" and simple. But it might actually be full of complex memory effects that only show up when we look at how measurements evolve.
Better Quantum Tech: Understanding these hidden memory effects is crucial for building quantum computers. Noise is the enemy of quantum computers. If we can detect memory effects in the Heisenberg picture, we might find new ways to protect quantum information or even use that "memory" to our advantage.
Two Sides of the Same Coin: The paper teaches us that in the quantum world, "how you look at it" isn't just a matter of perspective; it fundamentally changes the mathematical rules of the game.

The Takeaway

Just because a movie looks like a simple, forgetful story when you watch the actors, doesn't mean the camera angles aren't telling a complex, memory-filled story. To truly understand the quantum world, we must watch both the actors and the camera.

1. Problem Statement

The concept of divisibility of dynamical maps is a cornerstone in the study of open quantum systems and non-Markovianity. Traditionally, divisibility is defined in the Schrödinger picture (evolution of states), where a map $\Phi_t$ is considered divisible if it can be decomposed into a sequence of completely positive (CP) or positive (P) intermediate maps. This property is linked to the non-negativity of the rates in the time-local master equation (ME) and the absence of information backflow.

However, quantum mechanics possesses a fundamental duality between the Schrödinger picture (states) and the Heisenberg picture (observables/effects). While physical predictions are equivalent in both pictures, the authors investigate whether the divisibility properties of the dynamical maps are also equivalent. Specifically, they address the gap in understanding how the time-local generators of the master equation behave under duality and whether a map divisible in one picture must necessarily be divisible in the other.

2. Methodology

The authors employ a rigorous mathematical framework involving:

Time-Local Master Equations (ME): They analyze the general form of the ME for invertible dynamics, distinguishing between left generators ( $L_t$ $L_{t}$ ) and right generators ( $R_t$ $R_{t}$ ).
- Schrödinger ME: $\dot{\Phi}_t = L_t \circ \Phi_t$ (Left) and $\dot{\Phi}_t = \Phi_t \circ R_t$ (Right).
- Heisenberg ME (Dual): $\dot{\Phi}^*_t = \Phi^*_t \circ L^*_t = R^*_t \circ \Phi^*_t$ .
Duality and Time Ordering: They utilize the Hilbert-Schmidt adjoint to map between pictures. A key insight is that time-ordering conventions swap under duality: the chronological time-ordered exponential in the Schrödinger picture becomes anti-chronological in the Heisenberg picture, and vice versa.
Propagator Analysis: They define left ( $\Phi^L_{t,s}$ ) and right ( $\Phi^R_{t,s}$ ) propagators. Crucially, they show that the Heisenberg divisibility is governed by the right propagator of the Schrödinger picture (or the dual of the right generator), not the left propagator which governs Schrödinger divisibility.
Operational Interpretations:
- Schrödinger: Linked to the trace distance ( $D_1$ ) between states and the guessing probability of prepared states.
- Heisenberg: Linked to the operator norm distance ( $D_\infty$ ) between effects (POVM elements) and the guessing probability of which measurement was performed.
Explicit Constructions: The authors construct specific counterexamples using:
- Phase-covariant qubit dynamics.
- Classical stochastic 2-state systems.
- Qubit dynamics involving non-commuting diagonal (dephasing) and orthogonal (rotation) transformations.

3. Key Contributions

A. Inequivalence of Divisibility

The central theoretical result is that Schrödinger divisibility and Heisenberg divisibility are distinct concepts.

A dynamical map $\Phi_t$ can be Schrödinger divisible (governed by $L_t$ ) but Heisenberg indivisible (governed by $R^*_t$ ), and vice versa.
This inequivalence arises because $L_t$ and $R_t$ generally do not commute ( $[L_t, R_t] \neq 0$ ) unless the dynamics is a semigroup or commutative. Consequently, the positivity conditions for the rates of $L_t$ (Schrödinger) and $R_t$ (Heisenberg) are independent.

B. Operational Interpretation of Heisenberg Divisibility

The paper provides a novel operational interpretation for violations of Heisenberg P-divisibility:

Scenario: Alice prepares a "black box" performing one of two measurements (effects $E$ or $F$ ). Bob must guess which measurement was performed using a single state preparation.
Metric: The success probability is determined by the operator distance $D_\infty(E, F) = \|E - F\|_\infty$ .
Result: If Heisenberg P-divisibility holds, $D_\infty$ decreases monotonically (information flows to the environment). If violated, $D_\infty$ exhibits revivals, indicating a backflow of information regarding the distinguishability of effects.

C. Quantifiers of Non-Markovianity

The authors introduce a measure for the violation of Heisenberg P-divisibility, $N_H(\Phi^*)$ , analogous to the trace-distance-based measure $N_S(\Phi)$ in the Schrödinger picture:
$N_H(\Phi^*) = \sup_{E,F} \int_{\dot{\eta}_H(t)>0} \dot{\eta}_H(t) dt$
where $\eta_H(t) = D_\infty(\Phi^*_t[E], \Phi^*_t[F])$ .

D. Connection to POVM Properties

They demonstrate that Heisenberg indivisibility manifests as non-monotonic behavior in:

Incompatibility: The degree to which two POVMs cannot be jointly measured.
Sharpness: How close a POVM is to a projective measurement.
These quantities are contractive under CPU (Completely Positive Unital) maps; thus, their revival signals Heisenberg non-Markovianity.

4. Key Results

Counterexamples:
- Phase-Covariant Dynamics: The authors show a specific time-dependent phase-covariant qubit dynamics where the rates for the left generator ( $\gamma_\pm$ ) are non-negative (Schrödinger CP-divisible), but the rates for the right generator ( $\xi_\pm$ ) become negative (Heisenberg P-indivisible).
- Classical Stochastic Dynamics: Even in classical 2-state systems, Schrödinger and Heisenberg divisibility are not equivalent. However, in the classical 2-state case, Heisenberg divisibility was found to imply Schrödinger divisibility (a specific property of 2-state systems not holding generally in quantum mechanics).
- Orthogonal Transformations: They constructed a qubit dynamics where the Bloch sphere undergoes a rotation (unitary evolution) superimposed on dephasing.
  - If the rotation is applied after dephasing (in the operator ordering), the Schrödinger picture sees a simple rotation (divisible), but the Heisenberg picture sees a non-monotonic evolution of effects (indivisible).
  - Conversely, if the rotation is applied before dephasing, the Heisenberg picture is divisible, but the Schrödinger picture is not.
Non-Markovianity without State Expansion: The paper highlights that non-Markovianity (memory effects) can exist even if the image of the Bloch sphere does not expand (i.e., no state distinguishability revival), provided the surface of the sphere moves non-trivially, affecting the distinguishability of effects.

5. Significance

Redefining Non-Markovianity: The work challenges the sufficiency of Schrödinger-picture divisibility as the sole witness of memory effects. It proves that a system can appear Markovian (divisible) regarding state evolution while exhibiting strong memory effects regarding observable evolution.
Operational Completeness: By linking Heisenberg divisibility to the guessing probability of effects and the properties of POVMs (compatibility/sharpness), the paper provides a complete operational framework for characterizing non-Markovianity in terms of measurement resources.
Implications for Quantum Control and Metrology: Since many quantum tasks (like parameter estimation or steering) rely on the properties of observables (POVMs) rather than just state preparation, ignoring Heisenberg divisibility could lead to an incomplete understanding of noise and memory effects in these protocols.
Theoretical Clarity: It clarifies the role of left vs. right generators in time-local master equations, showing that the choice of picture fundamentally alters the mathematical conditions for divisibility due to the non-commutativity of time-dependent generators.

In conclusion, the paper establishes that Heisenberg divisibility is an independent witness of memory effects, necessitating a dual-perspective approach to fully characterize non-Markovian quantum dynamics.