Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States

Abstract

We prove for any pure three-quantum-bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the threequantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties. The Schmidt decomposition [1,2] allows one to write any pure state of a bipartite system as a linear combination of biorthogonal product states or, equivalently, of a nonsuperfluous set of product states built from local bases. For two quantum bits (qubits) it readsHere jii͘ ϵ ji͘ A ≠ ji͘ B , both local bases ͕ji͖͘ A,B depend on the state jC͘, the relative phase has been absorbed into any of the local bases, and the state j00͘ has been defined by carrying the larger (or equal) coefficient. A larger value of u means more entanglement. The only entanglement parameter, u, plus the hidden relative phase, plus the two parameters which define each of the two local bases are the six parameters of any two-qubit pure state, once normalization and global phase have been disposed of. Very many results in quantum information theory have been obtained with the help of the Schmidt decomposition: its simplicity reflects the simplicity of bipartite systems as compared to N-partite systems. Much of its usefulness comes from it not being superfluous: to carry one entanglement parameter one needs only two orthogonal product states built from local bases states, no more, no less.The aim of this work is to generalize the Schmidt decomposition of (1) to three qubits. It is well known [2] that its straightforward generalization, that is, in terms of triorthogonal product states, is not possible (see also [3]). Nevertheless, having a minimal canonical form in which to cast any pure state, by performing local unitary transformations, will provide a new tool for quantifying entanglement for three qubits, a notoriously difficult problem. It will lead to a complete classification of exceptional states which, as we will see, is much more complex than in the two-qubit case. The generalization to N quantum dits (d-state systems) is not completely straightforward and will be given elsewhere.Linden and Popescu [4] and Schlienz [5] showed that for any pure three-qubit state the number of entanglement parameters is five and, using repeatedly the two-qubit

Few-Body Entanglement Manipulation

“…In contrast to the two-qubit case where the MES is given by {|Φ + 2 } the MES for three qubits, M ES 3 , contains infinitely many states. It is characterized by 3 parameters, whereas an arbitrary three-qubit state is characterized by 5 parameters (up to LUs) [25,26]. Therefore, M ES 3 is of measure zero.…”

Universal Subspaces for Local Unitary Groups of Fermionic Systems

“…(a) If M = 2K ≥ 6 is even, then the complex subspace spanned by all BSOVs e ijk except the K of them with indexes (1, 3, 6), (1,4,6) and…”

“…Acin et al showed that five LBPS are enough to span a universal subspace for three qubits [4]. For one such choice given by {|000 , |100 , |101 , |110 , |111 }, they can further restrict the regions for the superposition coefficients such that it gives a canonical form for three qubits [6].…”

“…They studied the property of subspaces spanned by LBPS such that every three-qubit pure state is LU equivalent to some states in such a subspace. This kind of subspace can hence be called 'universal subspace' [4,6].…”

“…Still, the first step toward the understanding of multipartite entanglement is to study the orbits of pure quantum states under LU. Acin et al started from looking at the simplest nontrivial case of a three qubit system [4,5]. They introduced the concept of local basis product states (LBPS), which are three-qubit product states from a fixed set of single particle orthonormal basis.…”