Qubit Ensembles from Directed Distinguishability with Uncalibrated Loss

Technical Summary: Qubit Ensembles from Directed Distinguishability with Uncalibrated Loss

Problem Statement
Conventional quantum state tomography specifies an ensemble of quantum states relative to calibrated measurements. This paper investigates whether an ensemble can be determined solely by relational information—specifically, how strongly every preparation can be distinguished from every other—without relying on calibrated measurement settings. The authors focus on the directed max-relative entropy ($D_{ij} = D_{\max}(\rho_i \| \rho_j)$) as the primary metric of distinguishability. While the symmetric sum of these entropies corresponds to Hilbert's projective metric (which defines hyperbolic geometry but fails to fix state purities due to Lorentz boost ambiguities), the central question is whether the antisymmetric sector of the directed matrix provides the missing normalization to uniquely reconstruct the ensemble.

Methodology
The authors develop a theoretical framework and an experimental validation pipeline based on the following components:

1. Theoretical Reconstruction:

   • For a single qubit, states are represented as $\rho_i = (I + \mathbf{r}_i \cdot \boldsymbol{\sigma})/2$. The directed max-relative entropy is decomposed into a symmetric part ($H_{ij}$) and an antisymmetric part ($A_{ij}$).
   • The paper establishes that for four generic faithful qubit states, the complete directed matrix determines the ensemble up to a common unitary or antiunitary transformation.
   • A scalar parameter $\tau$ is introduced to classify compatible Euclidean Gram matrices. The authors prove that the endpoint of this scalar family ($\tau = q$) uniquely minimizes the embedding dimension (fitting the states into the 3D Bloch ball) and maximizes the entropy of every state.
   • Sparse Extension: Once four "anchor" states are reconstructed, the remaining $n-4$ states in an ensemble can be determined using only five directed distances per new state (four to the anchors and one reverse distance).

2. Acquisition and Nuisance Cancellation:

   • The protocol addresses "uncalibrated loss" by utilizing the dual cone identity of the max-relative entropy.
   • One-port analysis: If the same setting loss applies to all preparations, the ratio of click probabilities cancels unknown efficiency.
   • Paired measurements: To remove restrictions on shared loss, the authors propose a "track swapping" method. By exchanging outcomes between channels and calculating a double ratio of counts, the method cancels arbitrary common-mode throughput and unequal static channel gains, provided additive backgrounds are subtracted.

3. Experimental Validation:

   • The authors applied the inverse protocol directly to public frequency tables from a superconducting-qubit experiment (Aloy et al.).
   • The reconstruction path involved converting experimental frequencies directly to a directed divergence matrix, then to reconstructed qubit geometry, without intermediate density matrix estimation or standard tomography.
   • Tests were conducted at 5 and 10 $\mu$s delays using 100 preparations against 100 binary measurements.

Key Results

• Uniqueness: The complete directed max-relative entropy matrix uniquely fixes four generic faithful qubit states (up to unitary/antiunitary transformation). For fewer than four states, a scalar ambiguity remains.
• Stability: The inverse map is locally analytic for invertible four-state anchors. However, the system exhibits non-uniform stability; near affine degeneracy (planar limits), noise amplification can be significant (up to ~22x relative to directed-entry noise), though cycle projection reduces this to ~6.9x.
• Finite-Setting Performance:
  • Using pooled data from 100 measurement directions, the median four-state Gram error was approximately $2.4 \times 10^{-2}$.
  • Under strict split-sample selection (to control maximization bias), the median error rose to $3.5 \times 10^{-2}$.
  • Reducing measurement directions from $K=100$ to $K=20$ increased errors significantly (e.g., from 0.023 to 0.070 at 5 $\mu$s), demonstrating the dependence on the number of measurement directions.
• Nuisance Cancellation: Independent full-Stokes data confirmed that the proposed paired-swap acquisition method successfully cancels common-mode throughput and suppresses false polarization by a factor of 36–50 relative to a single setting.

Claims and Significance
The paper claims to establish a method for calibration robustness rather than a universal sample advantage. Key assertions include:

• Relational Fixation: Directed information, specifically the antisymmetric sector of the max-relative entropy, supplies the normalization datum absent in symmetric projective geometry, fixing generic qubit ensembles at the four-state threshold.
• Robustness: The protocol allows for the reconstruction of qubit geometries without calibrated measurements, effectively canceling specific nuisance parameters (setting loss, common-mode throughput, and unequal channel gains) through specific acquisition rules.
• Limitations: The authors explicitly state that the experimental analysis demonstrates a "finite-setting realization" of the reconstruction pipeline, not a loophole-free measurement of the continuum maximum. They do not claim universal sample efficiency or that the method works for all ensembles without the specific four-state anchor condition.
• Scope: The work provides a theoretical classification of compatible realizations via a scalar parameter and an end-to-end test of a reconstruction pipeline that bypasses traditional tomography, offering a new perspective on how relational distinguishability defines quantum states.