Quantum tomography for non-iid sources

1. Problem Statement

Quantum State Tomography (QST) and Quantum Process Tomography (QPT) are fundamental tools for characterizing quantum devices. Standard theoretical analyses rely heavily on the independent and identically distributed (i.i.d.) assumption, which posits that a source emits $N$ copies of a fixed quantum state $\rho$ (or applies a fixed channel $\mathcal{E}$).

However, in realistic experimental settings, this assumption is frequently violated due to:

• Drift: Thermal fluctuations, laser instabilities, and calibration errors cause the state to change over time.
• Adaptive Feedback: In closed-loop control or adaptive experiments, the state prepared at time $t$ depends explicitly on previous measurement outcomes.
• Adversarial Behavior: In security contexts, a source may strategically adapt to revealed information.

Under non-i.i.d. conditions, the sequence of states $\{\rho_1, \dots, \rho_N\}$ cannot be modeled as identical copies. Consequently, standard statistical tools (e.g., Matrix Bernstein inequalities) that rely on independent increments fail, and the statistical guarantees of existing tomography protocols break down. The core problem is to determine if Projected Least-Squares (PLS) tomography remains statistically optimal when the i.i.d. assumption is dropped, and if so, what the sample complexity is for reconstructing the time-averaged state or channel.

2. Methodology

The author proposes a rigorous framework extending PLS tomography to the non-i.i.d. regime by modeling the experiment as an interaction between an experimenter and an adaptive environment (or adversary).

A. Physical Model

• Adaptive Source: The source prepares a state $\rho_t$ at round $t$ based on the entire history $F_{t-1}$ (previous settings, outcomes, and hidden variables).
• Conditional Independence: Crucially, once $\rho_t$ is fixed, the measurement outcome $Y_t$ is governed solely by the Born rule: $P(Y_t=k | F_{t-1}) = \text{tr}(\Pi_k \rho_t)$. The source cannot influence the specific outcome beyond quantum probabilistic constraints.
• Objective: Reconstruct the time-averaged state $\bar{\rho}_N = \frac{1}{N}\sum_{t=1}^N \rho_t$ (or the time-averaged channel $\bar{\mathcal{E}}_N$).

B. Estimation Protocol (PLS)

The protocol utilizes single-copy, non-adaptive measurements (the measurement settings are fixed or randomized independently of the state preparation).

1. Data Collection: Measure $\rho_t$ using an Informationally Complete (IC) POVM.
2. Linear Estimation: Map each outcome $Y_t$ to a single-shot matrix estimator $\hat{\rho}_t$.
   • For global 2-designs: $\hat{\rho}_t = (d+1)P_k - I$.
   • For local 2-designs (e.g., Pauli measurements): $\hat{\rho}_t = \bigotimes_j (3P_{k_j} - I)$.
   • Compute the average linear estimator: $\hat{L}_N = \frac{1}{N}\sum \hat{\rho}_t$.
3. Projection: Project $\hat{L}_N$ onto the set of valid density matrices $\mathcal{D}(\mathbb{C}^d)$ to obtain the final estimator $\hat{\rho}_{\text{PLS}}$.

C. Statistical Analysis: Matrix Martingales

The key theoretical innovation is replacing independence-based bounds with Matrix Martingale techniques.

• Unbiasedness: The single-shot estimator is conditionally unbiased: $\mathbb{E}[\hat{\rho}_t | F_{t-1}] = \rho_t$.
• Martingale Difference: The error term $X_t = \hat{\rho}_t - \rho_t$ forms a martingale difference sequence ($\mathbb{E}[X_t | F_{t-1}] = 0$).
• Concentration: The total error is a sum of conditionally zero-mean fluctuations. The author applies the Matrix Freedman Inequality (a concentration inequality for martingales) to bound the deviation of the estimator, controlling cumulative errors based on their predictable variance rather than strict independence.

3. Key Contributions

1. Extension of PLS to Non-i.i.d. Regime: The paper proves that PLS tomography remains statistically optimal even when the source is fully adaptive or adversarial.
2. Optimal Sample Complexity: It demonstrates that dropping the i.i.d. assumption does not increase the fundamental sample complexity scaling required to achieve a target accuracy $\epsilon$.
3. Generalization to Process Tomography: The framework is extended to Quantum Process Tomography (QPT) via the Choi-Jamiołkowski isomorphism, treating time-varying channels as time-varying bipartite states.
4. Rigorous Non-Asymptotic Bounds: The paper provides explicit, non-asymptotic sample complexity bounds with high probability ($1-\delta$) for both state and process tomography under adaptive conditions.

4. Main Results

A. Quantum State Tomography (QST)

For estimating the time-averaged state $\bar{\rho}_N$ with rank $r$ and accuracy $\epsilon$ (in trace distance):

• Global 2-Design Measurements:
  $$N \geq \frac{64 d r^2 (1 + \epsilon/24r)}{\epsilon^2} \log\left(\frac{2d}{\delta}\right)$$
• Local 2-Design (Pauli) Measurements:
  $$N \geq \frac{32 r^2 (3^n + 2^n \epsilon/12r)}{\epsilon^2} \log\left(\frac{2^{n+1}}{\delta}\right)$$
• Result: The scaling is $O(dr^2/\epsilon^2)$, matching the optimal i.i.d. rates. The error bound includes a term dependent on the residual spectral mass $\Sigma_r$, ensuring robustness for low-rank states.

B. Quantum Process Tomography (QPT)

For estimating the time-averaged channel $\bar{\mathcal{E}}_N$ with accuracy $\epsilon$ (in diamond distance):

• Global 2-Design:
  $$N \geq \frac{32 d^6 (1 + \epsilon/24)}{\epsilon^2} \log\left(\frac{2d^2}{\delta}\right)$$
• Local 2-Design:
  $$N \geq \frac{24^{n+3} (3^{2n} + \epsilon/6)}{\epsilon^2} \log\left(\frac{2^{2n+1}}{\delta}\right)$$
• Result: The scaling is $O(d^6/\epsilon^2)$, which is optimal for process tomography and matches the i.i.d. case.

5. Significance and Implications

• Robustness of Standard Protocols: The results provide a rigorous justification for using standard PLS estimators on data collected from noisy, drifting, or actively controlled quantum hardware without needing to assume stability.
• Interpretation Shift: While the sample complexity remains unchanged, the interpretation of the reconstructed object shifts from a single static state $\rho$ to the time-averaged trajectory $\bar{\rho}_N$. This is a physically meaningful quantity for characterizing device performance over time.
• Comparison to Alternatives: The paper contrasts this approach with methods based on Quantum de Finetti theorems. While de Finetti theorems can handle non-i.i.d. sources, they typically incur a much higher sample complexity penalty (scaling as $\tilde{O}(d^6/\epsilon^6)$) and target a different effective state. The proposed PLS approach retains the optimal $O(dr^2/\epsilon^2)$ scaling.
• Practical Impact: This work bridges the gap between theoretical guarantees and experimental reality, validating that temporal correlations and adaptive drift do not fundamentally hinder the accuracy of quantum tomography, provided the correct statistical tools (martingale concentration) are applied.

In conclusion, the paper establishes that the statistical optimality of projected least-squares tomography is robust against the violation of the i.i.d. assumption, ensuring that high-fidelity characterization of quantum devices is possible even in the presence of significant drift and adaptive behavior.