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Chapman & Hall/CRC eBooks

Kinematics of qubit pairs

Berthold‐Georg Englert, NASSER METWALLY

Published Feb 14, 2002• 0 citations
Publisher

Abstract

Abstract We review some of the important properties of the states of qubit pairs. They are specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3dyadic. There are six classes of families of locally equivalent states in a straightforward scheme for classifying all 2-qubit states; four of the classes consist of two subclasses each. Easy-to-use criteria enable one to check whether a given pair of 3-vectors plus a 3× 3-dyadic specify a 2-qubit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state. We remark on the Hill-Wootters concurrence, discuss the properties of the fundamental Lewenstein-Sanpera decompositions of 2-qubit states, and report a number of examples for which the optimal decomposition is known explicitly.

DOI
10.1201/9781420035377-3

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Abstract We review some of the important properties of the states of qubit pairs. They are specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3dyadic. There are six classes of families of locally equivalent states in a straightforward scheme for classifying all 2-qubit states; four of the classes consist of two subclasses each. Easy-to-use criteria enable one to check whether a given pair of 3-vectors plus a 3× 3-dyadic specify a 2-qubit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state. We remark on the Hill-Wootters concurrence, discuss the properties of the fundamental Lewenstein-Sanpera decompositions of 2-qubit states, and report a number of examples for which the optimal decomposition is known explicitly.
The state of an entangled q-bit pair is specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3 dyadic. There are easy-to-use criteria to check whether a given pair of 3-vectors plus a dyadic specify a 2-q-bit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state. Some progress has been made in the search for analytical expressions for the degree of separability. We report, in particular, the answer in the case of vanishing 3-vectors.
The state of an entangled q-bit pair is specified by 15 numerical parameters that are naturally regarded as the components of two 3-vectors and a 3 × 3-dyadic.There are easy-to-use criteria to check whether a given pair of 3-vectors plus a dyadic specify a 2-q-bit state; and if they do, whether the state is entangled; and if it is, whether it is a separable state.Some progress has been made in the search for analytical expressions for the degree of separability.We report, in particular, the answer in the case of vanishing 3-vectors.
We provide an in-depth study of tripartite entanglement of qudits. We start with a short review of tripartite entanglement invariants, prove a theorem about the complete list of all allowed values of three (out of the total of four) such invariants, and give several bounds on the allowed values of the fourth invariant. After introducing several operations on entangled states (that allow us to build new states from old states) and deriving general properties pertaining to their invariants, we arrive at the decomposition theorem as one of our main results. The theorem relates the algebraic invariants of any entanglement class with the invariants of its corresponding components in each of its direct sum decompositions. This naturally leads to the definition of reducible and irreducible entanglement classes. We explicitly compute algebraic invariants for several families of irreducible classes and show how the decomposition theorem allows computations of invariants for compounded classes to be carried out efficiently. This theorem also allows us to compute the invariants for the infinite number of entanglement classes constructed from irreducible components. We proceed with the complete list of the entanglement classes for three tribits with decompositions of each class into irreducible components, and provide a visual guide to interrelations of these decompositions. We conclude with numerous examples of building classes for higher-spin qudits.
Verstraete, Dehaene and DeMoor showed that each of the two-qubit states can be generated from one of two canonical families of two-qubit states by means of transformations preserving the tensor structure of the state space. Precisely, each of such states can be generated from a three-parameter family of Bell-diagonal states or from three-parameter rank-deficient states. In this paper, we show that this classification of two-qubit states can be refined. In particular, we show that the latter canonical family of states can be reduced to three fixed states and a two-parameter family of two-qubit states. For this family of states, we provide a simple parametrization that guarantees positive semidefiniteness of the states and enables easier calculation of the Wootters concurrence and quantum discord. Moreover, we present a new general parametrization of all two-qubit states generated from the canonical families of states using sets of (pseudo)orthogonal four-vectors (frames). An advantage of the presented approach lies in the fact that the standard conditions for positive semidefiniteness of states are equivalent to (pseudo)orthogonality conditions for four-vectors serving as parameters (and appropriate conditions for parameters of the corresponding canonical family of states).
Quantum entanglement and its paradoxical properties hold the key to an information processing revolution. Much attention has focused recently on the challenging problem of characterizing entanglement. Entanglement for a two qubit system is reasonably well understood; however, the nature and properties of multiple qubit systems are largely unexplored. Motivated by the importance of such systems in quantum computing, we show that typical pure states of N qubits are highly entangled but have decreasing amounts of pairwise entanglement (measured using the Wootter concurrence formula) as N increases. Above six qubits, very few states have any pairwise entanglement and, generally, for a typical pure state of N qubits there is a sharp cut-off where its subsystems of size m become positive partial transpose (i.e. separable or only bound entangled) around N ⪆ 2m + 3, based on numerical analysis up to N = 13.