On the distinction between distinguishability of states and witness of non-Markovianity of dynamical maps

This paper challenges the conventional association between information backflow and non-P-divisibility in qubit dynamics by demonstrating that the generalized trace distance criterion is neither tight nor faithful for individual state pairs, while also clarifying the specific conditions under which it offers advantages over the standard trace distance measure.

The Gist

The Big Picture: Tracking "Lost" Information

Imagine you have a secret message (a quantum state) written on a piece of paper. You hand this paper to a friend (the environment) who plays with it for a while and then hands it back.

In the world of quantum physics, scientists want to know: Did the friend keep some of the secret, or did they give it back?

• If the friend keeps the secret, the information flows out of your system. This is like "Markovian" behavior (forgetful).
• If the friend suddenly hands the secret back to you, the information flows in. This is called "Non-Markovian" behavior (memory).

For a long time, scientists used a specific ruler to measure this. They looked at how "different" two pieces of paper were. If the difference between them grew larger over time, they knew information had flowed back.

The New, "Smarter" Ruler

The paper discusses a newer, more sophisticated ruler called the Generalized Trace Distance (GTD).

• The Old Ruler (BLP): Only measured how different the shapes of the papers were. It ignored where the papers were located on the table.
• The New Ruler (GTD): Measures both the shape and the location. It was created because sometimes, even if the shapes stay the same, the location changes in a way that proves information is flowing back.

The paper acknowledges that the New Ruler is excellent at its main job: telling us if the entire system is acting memory-less or memory-full. If the New Ruler detects a change, the system is definitely "Non-Markovian."

The Problem: The Ruler Lies About Specific Pairs

The authors of this paper argue that while the New Ruler is great for judging the whole system, it is not trustworthy when you look at just one specific pair of papers (two specific quantum states).

They found two ways the New Ruler gets confused when applied to individual pairs:

1. The "False Negative" (The Ruler Misses the Action)

The Analogy: Imagine two runners on a track. They are running away from each other, clearly getting further apart. You expect the referee to shout, "They are separating!" But the referee looks at his fancy stopwatch and says, "No, the distance hasn't changed."

What the paper says: There are cases where two quantum states clearly evolve and become more distinguishable (the distance between them grows). However, the New Ruler fails to register this change. It says "no information flow" even though the states are clearly moving apart. The ruler is "blind" to this specific type of movement.

2. The "False Positive" (The Ruler Sees Ghosts)

The Analogy: Imagine two identical twins standing still. They are indistinguishable. But then, someone paints a tiny, invisible dot on one of them and changes the probability of who is who. Suddenly, the referee's fancy machine screams, "These two are totally different!" But in reality, they are still standing there looking the same.

What the paper says: There are cases where two quantum states are effectively identical (indistinguishable). However, because of a mathematical quirk involving "biased probabilities" (like weighting one state more than the other), the New Ruler calculates a distance and claims they are different. It creates an illusion of information flowing back when, in reality, nothing has happened.

The Main Conclusion

The authors are not saying the New Ruler is useless. They are making a very specific distinction:

1. Map-Level (The Big Picture): If you look at the entire system, the New Ruler is the best tool. It perfectly tells you if the system has memory (Non-P-divisibility).
2. State-Level (The Micro View): If you look at specific pairs of states to see if information is flowing back for them, the New Ruler is unreliable. It can miss real changes and invent fake ones.

The Takeaway:
Think of the New Ruler like a weather satellite. It is perfect for telling you if a storm is happening over the whole country (the map). But if you use it to tell you exactly what the wind speed is in your specific backyard (a specific pair of states), it might give you the wrong answer. You can't assume that because the "system" has memory, every single pair of particles inside it is experiencing a flow of information.

Summary of Claims

• Non-P-divisibility is the strongest definition of quantum memory.
• The Generalized Trace Distance (GTD) is the perfect tool to detect if a system as a whole has this memory.
• However, GTD is not a faithful witness for individual pairs of states. It can fail to see real information flow (false negative) or claim there is flow when there isn't (false positive).
• For Unital Dynamics (where the center of the quantum "sphere" doesn't move), the old ruler and the new ruler give the same results. The problem only happens in Non-Unital Dynamics (where the center moves).

Technical Summary

Technical Summary: On the Distinction Between Distinguishability of States and Witness of Non-Markovianity of Dynamical Maps

Problem Statement
The paper addresses a conceptual gap in the characterization of quantum non-Markovianity. While non-P-divisibility is established as the strongest divisibility-based notion of non-Markovianity, and the Generalized Trace Distance (GTD) criterion is known to be an optimal map-level witness for non-P-divisibility, the authors question the validity of associating information backflow directly with state distinguishability at the individual state-pair level. Specifically, the paper investigates whether the GTD criterion serves as a "tight" (no false negatives) and "faithful" (no false positives) indicator of information flow for specific pairs of evolving states, or if this association holds only when optimizing over all possible state pairs.

Methodology
The authors employ a theoretical framework based on qubit dynamics represented as affine transformations of the Bloch vector ($\vec{r}' = T\vec{r} + \vec{c}$). They utilize two primary measures:

1. BLP Measure: Based on the standard trace distance ($D = \frac{1}{2}\|\rho_1 - \rho_2\|_1$) for equiprobable states.
2. GBLP Measure: Based on the generalized trace distance ($D = \|\Delta_p(\rho_1, \rho_2)\|_1$) involving biased probabilities ($p, q$) and the Helstrom matrix, which accounts for translation vectors ($\vec{c}$) in non-unital dynamics.

The methodology involves:

• Deriving the relationship between P-divisibility and the monotonicity of the generalized trace distance.
• Proving theorems regarding the equivalence of BLP and GBLP conditions for unital versus non-unital dynamics.
• Constructing specific counter-examples to test the "tightness" and "faithfulness" of the GTD criterion when applied to fixed pairs of states rather than optimized ensembles.
• Analyzing the behavior of the distance metric in regimes where the Bloch vector magnitude $|\vec{w}(t)|$ crosses the threshold $|p-q|$.

Key Contributions and Results

1. Equivalence in Unital Dynamics: The authors prove (Theorem 1) that for qubit unital dynamics (where the translation vector $\vec{c} = 0$), the BLP and GBLP conditions are equivalent. Consequently, for unital maps, the generalized measure offers no advantage over the standard trace distance in witnessing non-Markovianity.

2. Necessity of GBLP for Non-Unital Dynamics: The paper establishes (Theorem 2) that the BLP and GBLP criteria differ only in the case of non-unital, non-P-divisible dynamics. In these cases, the standard BLP measure may fail to detect non-Markovianity (falsely indicating Markovianity) because it is insensitive to the translation component of the dynamics, whereas GBLP correctly identifies the non-P-divisibility.

3. Failure of Tightness (False Negatives): The authors demonstrate that the GTD criterion is not a tight witness for individual state pairs. They provide examples of non-Markovian dynamics (both unital and non-unital) where the distinguishability of a specific pair of states manifestly increases (indicating information backflow), yet the GTD criterion fails to register this increase. This occurs when the evolution of the weighted Bloch vector $\vec{w}(t)$ causes the distance metric to plateau or behave non-monotonically in a way that masks the underlying distinguishability changes, particularly when $|\vec{w}(t)| < |p-q|$.

4. Failure of Faithfulness (False Positives): More critically, the paper shows the GTD criterion is not a faithful witness. The authors construct a scenario involving a composition of a depolarizing channel and a shift channel. In this case, two states that become identical (indistinguishable) at an intermediate time $\tau$ are subsequently separated by a shift. The GTD measure yields a non-zero distance at time $t > \tau$ due to the bias in preparation probabilities ($p \neq q$), falsely suggesting information backflow or distinguishability where none physically exists between the evolved states. This indicates that the GTD increase reflects preparation information rather than dynamical information backflow.

Significance and Claims
The paper argues that while the GBLP condition is an optimal witness for the non-P-divisibility of a dynamical map (when optimized over all state pairs), it is neither a tight nor a faithful indicator of information backflow or distinguishability when applied to individual pairs of states.

The authors conclude that there is a subtle but crucial distinction between:

• Map-level witnessing: Associating non-P-divisibility with the existence of some pair of states showing increased distinguishability (where GBLP is optimal).
• State-level witnessing: Associating the information flow of a specific pair of states with the GTD criterion (where GBLP fails).

The significance of this work lies in cautioning against the direct interpretation of GTD increases for specific state pairs as definitive evidence of information backflow. The paper suggests that the association of information flow with divisibility must be applied carefully at the microscopic level of state pairs, as the generalized measure can yield misleading results (both false negatives and false positives) in specific dynamical contexts, particularly those involving non-unital noise and biased state preparations. The authors note that their analysis is specific to qubits but suggests that similar geometric constraints may apply to higher-dimensional systems (e.g., qutrits with embedded qubit subspaces).