Sample-optimal tomography of quantum states (1508.01797v2)

Abstract: It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error $\epsilon$ in trace distance required $O(dr^2/\epsilon^2)$ copies for a $d$-dimensional density matrix of rank $r$. Here, we give a theoretical measurement scheme (POVM) that requires $O (dr/ \delta ) \ln (d/\delta) $ copies of $\rho$ to error $\delta$ in infidelity, and a matching lower bound up to logarithmic factors. This implies $O( (dr / \epsilon^2) \ln (d/\epsilon) )$ copies suffice to achieve error $\epsilon$ in trace distance. We also prove that for independent (product) measurements, $\Omega(dr^2/\delta^2) / \ln(1/\delta)$ copies are necessary in order to achieve error $\delta$ in infidelity. For fixed $d$, our measurement can be implemented on a quantum computer in time polynomial in $n$.

• The paper introduces a novel POVM-based measurement scheme that significantly reduces the quantum state copies required for accurate estimation.
• It establishes nearly optimal upper and lower bounds, showing that O((dr/ε)ln(d/ε)) copies suffice to achieve an error ε in trace distance.
• The study outlines a practical method for implementing the measurement on quantum computers, paving the way for scalable quantum state characterization.

Sample-optimal Tomography of Quantum States

The paper addresses a central problem in quantum information theory: determining the number of copies of an unknown mixed quantum state that are necessary and sufficient to accurately characterize the state. The authors present novel results in quantum state tomography by providing both upper and lower bounds on the number of required quantum state copies for achieving certain error levels.

The primary contribution of this work is the introduction of a theoretical measurement scheme, based on a Positive Operator-Valued Measure (POVM), which significantly optimizes the number of copies needed for quantum state estimation. Specifically, they demonstrate that O((dr/ε)ln(d/ε)) copies suffice to achieve an error ε in trace distance for a d-dimensional density matrix of rank r, a result that is near-optimal up to logarithmic factors. This finding refines the previously known bound of O(dr/ε) copies, providing a more sample-efficient approach.

Moreover, the authors present a matching lower bound for independent (product) measurements, showing that (dr/ε)/ln(1/ε) copies are necessary for achieving the same error in infidelity. The paper provides mathematical proofs and detailed analysis for these bounds, offering a comprehensive understanding of the trade-offs in quantum state estimation.

In addition to the theoretical results, the authors outline a method for implementing their proposed measurement on a quantum computer, highlighting the potential for practical applications. They note that their POVM can be realized in polynomial time for fixed dimensionality, which is pertinent for scalable quantum technologies.

This research carries significant implications both theoretically and practically. Theoretically, it advances the understanding of quantum state estimation by clarifying the sample complexity and measurement strategies involved. Practically, it informs the development of experimental protocols and algorithms for efficiently characterizing quantum states, which is vital for the progression of quantum computing, communication, and cryptography.

Future developments could explore the potential separation in performance between adaptive measurements and collective measurements, as well as further improvements in the efficiency of implementing the measurement schemes on quantum computers. Overall, this paper illuminates critical aspects of quantum tomography, paving the way for improved techniques in quantum state characterization.