Theory of a Pair of Quantum Bits

Abstract

In this paper we present a thorough study of the theory of a pair of qubits, whose Hilbert space can be identified with ℂ2 ⊗ ℂ2. Given an hermitian operator ρ of trace 1 in ℂ2 ⊗ ℂ2 we focus on the following Problems: Problem 1: Find conditions that guarantee that ρ is a state, that is, positive semidefinite. Problem 2: Find conditions that guarantee that a given state ρ is separable, or that ρ is a convex combination of products of one-particle states. The language we develop for our investigation makes use of the observation that ℂ2 ⊗ ℂ2 carries representations of the special unitary group SU(2) in two dimensions and of the direct product of this group by itself. We introduce a new type of observable called Bell observable (section 5) and a new measure of entanglement called concurrence, which is closely related to the concurrence introduced by Wootters (Physical Review Letters (1998) 80, 2245–2248) (section 8). The work has been inspired by the works of Wootters (Physical Review Letters (1997) 78, 5022–5025; Physical Review Letters (1998) 80, 2245–2248) and members of the Horodecki family (cf Horodecki and Horodecki, Physical Review A (1996) 54, 1838–1843; Horodecki et al., Physics Letters A (1996a) 223, 1–8; Physics Letters A (1996b) 222, 21–25) and reproduces some of their results.