[Paper Review] Non-completely positive maps: properties and applications

Christopher J. Wood|arXiv (Cornell University)|Nov 17, 2009

Quantum Information and Cryptography38 references3 citations

TL;DR

This thesis investigates non-completely positive (non-CP) maps in open quantum systems with initial system-environment correlations, demonstrating that standard quantum process tomography (SQPT) can yield non-CP maps due to state preparation protocols. It proposes a modified preparation method using single measurements followed by unitary rotations to ensure CP evolution, and shows that statistical noise in tomography can falsely indicate non-CP behavior, which can be mitigated by increasing state copies. The work provides a framework for identifying and distinguishing true non-CP dynamics from noise-induced artifacts.

ABSTRACT

We investigate the evolution of open quantum systems in the presence of initial correlations with an environment. Here the standard formalism of describing evolution by completely positive trace preserving (CPTP) quantum operations can fail and non-completely positive (non-CP) maps may be observed. A new classification of correlations between a system and environment using quantum discord is explored. However, we find quantum discord is not a symmetric quantity between exchange of systems and this leads to ambiguity in classifications - states which are both quantum and classically correlated depending on the order of the two systems. State preparation in quantum process tomography is investigated with regard to non-CP maps. In SQPT the preparation procedure can influence the complete-positivity of the reconstructed quantum operation if our system is initially correlated with an environment. We examine a recently proposed preparation procedures using projective measurements, and propose our own protocol that uses a single measurement followed by unitary rotations. The former can give rise to non-CP evolution while the later will always give rise to a CP map. State preparation in AAPT was found always to give rise to CP evolution. We examine the effect of statistical noise in process tomography and find it can result in the identification of a non-CP when the evolution should be CP. The variance of the distribution for reconstructed processes is found to be inversely proportional to the number of copies of a state used to perform tomography. Finally, we detail an experiment using currently available linear optics QC devices to demonstrate non-CP maps arising in SQPT.

Motivation & Objective

• To investigate the emergence of non-completely positive (non-CP) maps in open quantum systems when initial correlations exist between system and environment.
• To analyze the impact of state preparation procedures on the complete-positivity of reconstructed quantum operations in standard quantum process tomography (SQPT).
• To propose a new state preparation protocol using projective measurements followed by unitary rotations that guarantee CP evolution in SQPT.
• To examine how statistical noise in process tomography can lead to false identification of non-CP maps and propose methods to distinguish such artifacts from true non-CP dynamics.
• To demonstrate experimentally observable non-CP maps using current linear optics quantum computing platforms.

Proposed method

• Uses the quantum discord formalism to classify correlations between system and environment, though notes its asymmetry across system exchange.
• Applies standard quantum process tomography (SQPT) and ancilla-assisted process tomography (AAPT) to reconstruct quantum operations under initial correlations.
• Proposes a novel state preparation protocol involving a single projective measurement followed by unitary rotations to preserve complete positivity in SQPT.
• Analyzes the variance of reconstructed process distributions under statistical noise, showing it scales inversely with the number of state copies used in tomography.
• Investigates bilinear process tomography as a potential method for probing non-CP dynamics.
• Designs a feasible linear optics experiment to demonstrate non-CP maps in SQPT using current technology.

Experimental results

Research questions

• RQ1 Can initial system-environment correlations lead to non-completely positive maps in standard quantum process tomography?
• RQ2 How does the choice of state preparation protocol affect the complete-positivity of reconstructed quantum operations in SQPT?
• RQ3 Can statistical noise in process tomography falsely identify a CP map as non-CP, and if so, how can this be mitigated?
• RQ4 Does ancilla-assisted process tomography (AAPT) always yield completely positive maps under initial correlations?
• RQ5 Can a proposed single-measurement + unitary rotation protocol ensure that reconstructed maps remain completely positive in SQPT?

Key findings

• State preparation via projective measurement can result in non-completely positive maps in SQPT, while the proposed single-measurement + unitary rotation protocol always yields completely positive maps.
• Ancilla-assisted process tomography (AAPT) consistently produces completely positive maps regardless of initial correlations.
• The variance of the distribution of reconstructed quantum processes under statistical noise is inversely proportional to the number of state copies used in tomography.
• By increasing the number of copies, the distributions of CP and true non-CP processes can be distinguished with high accuracy.
• A linear optics experiment using current technology can demonstrate non-CP maps in SQPT, validating the theoretical findings.
• Quantum discord is found to be asymmetric under system exchange, leading to ambiguous classification of correlations as classical or quantum depending on the order of systems.