Nonlocal characteristics and argand diagram of two-qubit gates

Abstract

Nonlocal characteristics of a two-qubit gate are determined by its nonlocal part. The squared eigenvalues of the nonlocal part of a two-qubit gate exist on the unit circle in the complex plane. We show that two sets of chords, the chords connecting the squared eigenvalues with each other and those connecting a squared eigenvalue with the complex conjugate of others in the unit circle, can be used to describe the nonlocal characteristics of two-qubit gates. Lengths of both sets of chords are proportional to the amount of entanglement contained in certain pure states. The entangling power of a two-qubit gate can be expressed in terms of the squared lengths of the first set of chords. Similarly, we show that the gate typicality of a two-qubit gate can be expressed in terms of the squared lengths of the second set of chords and the linear entropy of a two-qubit gate can be expressed using the squared lengths of both sets of chords. Perfect entanglers are known to transform some product states into maximally entangled states. The convex hull of the squared eigenvalues of the nonlocal part of perfect entanglers contain the zero. We analyse the simplices containing the zero in the convex hull of the squared eigenvalues of the nonlocal part of perfect entanglers to construct a pair of orthonormal product states that can be transformed into maximally entangled states by the nonlocal part of perfect entanglers and divide the region of perfect entanglers in the Weyl chamber into three tetrahedral regions and eight bounding planes based on the uniqueness of the simplices containing the zero.

arXiv comment8 pages. Comments are welcome

References in corpus (30)

• Entanglement of Formation of an Arbitrary State of Two Qubits
• Elementary gates for quantum computation
• Two-Bit Gates are Universal for Quantum Computation
• Demonstration of a small programmable quantum computer with atomic qubits
• Local Distinguishability of Multipartite Orthogonal Quantum States
• A simple all-microwave entangling gate for fixed-frequency superconducting qubits
• Optimal Creation of Entanglement Using a Two--Qubit Gate
• On the Entangling Power of Quantum Evolutions
• A geometric theory of non-local two-qubit operations
• Nonlocal properties of two-qubit gates and mixed states and optimization of quantum computations
• Entanglement of Quantum Evolutions
• Statistical distinguishability between unitary operations
• A Universal Two--Bit Gate for Quantum Computation
• Quantum dynamics as a physical resource
• Entanglement Is Not Necessary for Perfect Discrimination between Unitary Operations
• Characterization of two-qubit perfect entanglers
• Quantum entanglement of unitary operators on bi-partite systems
• Unitary quantum gates, perfect entanglers and unistochastic maps
• Quantum Instruction Set Design for Performance
• Entangling power and local invariants of two-qubit gates
• Can local dynamics enhance entangling power?
• Metric Structure of the Space of Two-Qubit Gates, Perfect Entanglers and Quantum Control
• Entangling capacity and distinguishability of two-qubit unitary operators
• Construction and local equivalence of dual-unitary operators: from dynamical maps to quantum combinatorial designs
• Entanglement measures of bipartite quantum gates and their thermalization under arbitrary interaction strength
• Discrimination of two-qubit unitaries via local operations and classical communication
• Measures of operator entanglement of two-qubit gates
• Capability of local operations and classical communication for distinguishing bipartite unitary operations
• Optimal Simulation of a Perfect Entangler
• Characteristics, Implementation and Applications of Special Perfect Entangler Circuits