[Paper Review] Cohering Power of Unitary Operations and De-cohering of Quantum operations

Zhengjun Xi, Ming‐Liang Hu+2 more|arXiv (Cornell University)|Oct 22, 2015

Quantum Information and Cryptography8 references4 citations

TL;DR

This paper introduces the cohering power of unitary operations as the maximum coherence generated when acting on incoherent states, reducing optimization to a discrete set based on the reference basis. It also defines de-cohering power of quantum channels via coherence decay on maximally coherent states, analyzing 1- and 2-qubit systems using relative entropy, and explores connections between coherence and quantum discord via incoherent operations.

ABSTRACT

In this paper, we consider the cohering power of unitary operation, which is defined by the maximum coherence caused by an unitary operation acting on incoherent states. We obtain that the optimizations can be reduced to a simple discrete set that depends only on the fixed reference basis. Then, we investigate that the de-cohering power of quantum channel, which is defined by the maximum decay of coherence caused by quantum channel acting on the maximally coherent states. Based on the relative entropy, we discuss the cohering power of 1-qubit and 2-qubit unitary operations and the de-cohering power of quantum channel on 1-qubit system. Finally, we study the relation between the coherence and the discord via incoherent operation.

Motivation & Objective

• To define and quantify the cohering power of unitary operations as the maximal coherence they generate from incoherent states.
• To investigate the de-cohering power of quantum channels by measuring the maximal coherence loss on maximally coherent states.
• To analyze the cohering power of 1-qubit and 2-qubit unitary operations and de-cohering power of 1-qubit channels using the relative entropy of coherence.
• To explore the relationship between quantum coherence and quantum discord through the action of incoherent operations.

Proposed method

• Define cohering power as the maximum relative entropy of coherence generated by a unitary operation acting on incoherent states.
• Reduce the optimization of cohering power to a finite discrete set determined solely by the fixed reference basis.
• Define de-cohering power as the maximum decrease in coherence, measured via relative entropy, when a quantum channel acts on maximally coherent states.
• Apply the relative entropy of coherence to compute cohering and de-cohering powers for 1-qubit and 2-qubit systems.
• Use incoherent operations to examine the interplay between coherence and quantum discord.
• Analyze the structure of optimal states and operations via basis-dependent optimization.

Experimental results

Research questions

• RQ1 What is the maximal coherence that a unitary operation can generate from incoherent states, and how can this be optimized?
• RQ2 How can the de-cohering power of a quantum channel be quantified in terms of coherence decay on maximally coherent states?
• RQ3 What are the cohering powers of 1-qubit and 2-qubit unitary operations under the relative entropy measure?
• RQ4 How does the de-cohering power of a 1-qubit quantum channel relate to its structure and action on maximally coherent states?
• RQ5 What is the relationship between quantum coherence and quantum discord when incoherent operations are applied?

Key findings

• The optimization of cohering power is reduced to a finite discrete set dependent only on the reference basis, simplifying computation.
• The de-cohering power of a quantum channel is defined as the maximum relative entropy of coherence lost when acting on maximally coherent states.
• For 1-qubit and 2-qubit unitary operations, the cohering power is analytically characterized using the relative entropy of coherence.
• The de-cohering power of 1-qubit channels is quantified via the same relative entropy measure, enabling comparison across channels.
• Incoherent operations are shown to influence both coherence and quantum discord, suggesting a structural link between the two measures.
• The results demonstrate that coherence generation and degradation are fundamentally tied to the unitary or channel's action on specific state classes.