Measuring the entanglement-breaking index of a qubit collision channel on a quantum processor: protocol and worked example

Technical Summary: Measuring the Entanglement-Breaking Index of a Qubit Collision Channel

Problem Statement
The entanglement-breaking (EB) index, $n_{EB}(\Phi)$, is defined as the smallest number of self-compositions ($n$) required for a quantum channel $\Phi$ to destroy all entanglement with any reference system. While the theoretical properties of eventually EB channels are well-studied, experimental verification has been limited. Prior work (Cuevas et al.) established an index of 2 for a specific photonic channel but did not scan for larger indices or compare direct multi-round measurements against the composition of single-round maps. Furthermore, existing experimental records lack a protocol to locate indices greater than 2 by scanning the $n$-round channel directly.

Methodology
The paper proposes a protocol to measure $n_{EB}$ directly on a programmable quantum processor for qubit channels realized as collision models. The system qubit interacts with a sequence of ancillas (bath) prepared in states diagonal in a known basis. The methodology relies on four key technical components:

1. Exact Bath Programming via Weighting: Instead of preparing mixed states physically, the protocol prepares pure basis states for the ancillas and reconstructs the desired mixed bath state $\tau(p)$ in the analysis phase. By weighting the measurement outcomes of different ancilla configurations ($b$) according to their exact probabilities $w_p(b)$, the bath polarization $p$ is set without sampling error (Lemma 1).
2. Direct Multi-Round Tomography: The protocol performs single-qubit process tomography on the $n$-round channel for various $n$. It does not require a reference qubit; instead, it reconstructs the affine pair $(A_n, c_n)$ of the channel and computes the minimum eigenvalue, $\lambda_{\min}$, of the partially transposed Choi matrix.
3. Statistical Decision Rules: Readings are pre-registered at a threshold of five standard deviations ($5\sigma$). A channel is classified as:
   • NPT (Non-Positive Partial Transpose): $\lambda_{\min} \le -5\sigma$ (entanglement preserved).
   • PPT (Positive Partial Transpose): $\lambda_{\min} \ge 5\sigma$ (entanglement broken).
   • Unresolved: Between the thresholds.
     Due to the ideal property of EB channels, the sequence of readings must be monotone (NPT up to $n_{EB}-1$, then PPT). A non-monotone sequence indicates memory effects or circuit inconsistencies.
4. Consistency Check (Composed vs. Direct): The protocol compares the direct measurement of the $n$-round channel against the composition of the measured single-round map (adjusted for reset errors). This tests the Markovian assumption inherent in collision models.

Key Contributions

• Protocol Design: A practical method to measure $n_{EB}$ for any qubit collision channel with diagonal ancillas, utilizing mid-circuit resets to reuse qubits.
• Noise Analysis: Detailed characterization of how depolarizing noise and ancilla reset errors affect the decision margins. The author demonstrates that noise typically "eats" the NPT side (making it harder to detect entanglement) while "thickening" the PPT side, necessitating careful budgeting of shots.
• Pilot Rule: A dynamic shot-allocation strategy where a pilot run measures single-round maps and reset errors to predict the required shots for subsequent rounds, ensuring the experiment fits within a processor time budget while securing statistical significance.
• Correction of Previous Work: The paper corrects specific claims from a prior version (arXiv:2609.09350v2), including the gate count for heavy-hex compilation (8 CZ gates per round, not 10) and the clarification that an index of 2 was previously measured, though larger indices had not been located.

Worked Example and Results
The protocol is demonstrated on a three-qubit feedback loop compiled for a heavy-hex superconducting processor (IBM Kingston).

• Targets: Three certified ideal indices were targeted: $n_{EB}=3$ for an X-bath ($p=1$), and $n_{EB}=2$ for a Z-bath ($p=1$) and an unpolarized bath ($p=0$).
• Simulation: Simulations using the FakeKingston noise model (both nominal and stressed scenarios) were performed.
  • In the nominal scenario, the pilot identified a weak PPT reading for the X-bath at $n=3$, triggering a "raise" of shots to 16,384. The final result successfully recovered the index of 3.
  • In the stressed scenario (higher depolarization and reset errors), the protocol still recovered all three indices ($3, 2, 2$) without shot raising, as noise thickened the PPT margins.
• Consistency: The "composed vs. direct" test showed no significant deviation ($|D_n| < 2.1\sigma$), confirming the memoryless nature of the simulated rounds.
• Cost: The estimated processor time for the full run is a few hundred seconds (approx. 397 seconds including pilot raises).

Significance and Claims
The paper claims to provide the first protocol capable of locating an entanglement-breaking index larger than 2 through a direct scan of rounds, rather than inferring it from a single two-round measurement. It emphasizes that the method is robust against specific noise types (depolarization, reset errors) provided the "composed vs. direct" test passes.

The author modestly notes that while the protocol is ready for execution on current hardware, the "certified ideal indices" serve as comparison values rather than pass/fail criteria for the hardware itself, as coherent errors could shift the index in either direction. The paper concludes that while the machinery can theoretically certify indices under time-dependent baths or in "thermal valleys" (where entanglement breaks faster at intermediate polarizations), the shot requirements for such complex scenarios (potentially $>10^5$ shots per setting) currently place them out of reach for near-term measurement, serving instead as a map for future hardware capabilities.