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Quantum tomography for non-iid sources

Leonardo Zambrano

Abstract

Quantum state and process tomography are typically analyzed under the assumption that devices emit independent and identically distributed (i.i.d.) states or channels. In realistic experiments, however, noise, drift, feedback, or adversarial behavior violate this assumption. We show that projected least-squares tomography remains statistically optimal even under fully adaptive state and channel preparation. Specifically, we prove that the sample complexity for reconstructing the time-averaged state or channel matches the optimal i.i.d. scaling for non-adaptive, single-copy measurements. For rank-$r$ states, the sample complexity is $\mathcal{O}(d r^2/ε^2)$ to achieve accuracy $ε$ in trace distance, while for process tomography it is $\mathcal{O}(d^6/ε^2)$ to achieve accuracy $ε$ in diamond distance. Thus, dropping the i.i.d. assumption does not increase the fundamental sample complexity of quantum tomography, but only changes the interpretation of the reconstructed object.

Quantum tomography for non-iid sources

Abstract

Quantum state and process tomography are typically analyzed under the assumption that devices emit independent and identically distributed (i.i.d.) states or channels. In realistic experiments, however, noise, drift, feedback, or adversarial behavior violate this assumption. We show that projected least-squares tomography remains statistically optimal even under fully adaptive state and channel preparation. Specifically, we prove that the sample complexity for reconstructing the time-averaged state or channel matches the optimal i.i.d. scaling for non-adaptive, single-copy measurements. For rank- states, the sample complexity is to achieve accuracy in trace distance, while for process tomography it is to achieve accuracy in diamond distance. Thus, dropping the i.i.d. assumption does not increase the fundamental sample complexity of quantum tomography, but only changes the interpretation of the reconstructed object.
Paper Structure (18 sections, 6 theorems, 63 equations)

This paper contains 18 sections, 6 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Robust projected least-squares quantum state tomography
  3. Non-i.i.d. physical model
  4. Estimation protocol
  5. Non-asymptotic guarantees
  6. Robust projected least-squares quantum process tomography
  7. Estimation protocol
  8. Non-asymptotic guarantees
  9. Conclusions
  10. Statistical tools
  11. Proof of Theorem \ref{['thm:main_result']}
  12. Martingale structure and unbiasedness
  13. Concentration bounds for global 2-design measurements
  14. Concentration bounds for local 2-design measurements
  15. Trace-norm conversion and final bounds
  16. ...and 3 more sections

Key Result

Theorem 1

Let the source emit an arbitrary adaptive sequence of states with time-average $\bar{\rho}_N$. Fix a target rank $r$, accuracy $\epsilon > 0$, and failure probability $\delta \in (0,1)$. With probability at least $1-\delta$, the estimation error satisfies provided that the number of samples $N$ satisfies: For global complex projective $2$-design measurements, For tensor products of local $2$-des

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Theorem 3: Matrix Freedman inequality tropp2011freedman
  • Lemma 1
  • Theorem 4
  • proof
  • Theorem 5
  • proof