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Quantum Information Processing

Classification of two-qubit states

Paweł Caban, Jakub Rembieliński, Kordian A. Smoliński, Zbigniew Walczak

Published Sep 18, 2015• 6 citations
Publisher

Abstract

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).

DOI
10.1007/s11128-015-1121-y

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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).
Canonical forms of two-qubits under the action of stochastic local operations and classical communications (SLOCC) offer great insight for understanding nonlocality and entanglement shared by them. They also enable geometric picture of two-qubit states within the Bloch ball. It has been shown [Phys. Rev. A 64, 010101(R) (2001)] that an arbitrary two-qubit state gets transformed under SLOCC into one of the two different canonical forms. One of these happens to be the Bell diagonal form of two-qubit states and the other a nondiagonal canonical form is obtained for a family of rank deficient two-qubit states. The method employed by Verstraete et al. [Phys. Rev. A 64, 010101(R) (2001)] required highly nontrivial results on matrix decompositions in nnn-dimensional spaces with an indefinite metric. Here we employ an entirely different approach---inspired by the methods developed by Rao et al. [J. Mod. Opt. 45, 955 (1998)] in classical polarization optics---which leads naturally towards the identification of two inequivalent SLOCC invariant canonical forms for two-qubit states. In addition, our approach results in a simple geometric visualization of two-qubit states in terms of their SLOCC canonical forms.
In 2000, D\"{u}r, Vidal and Cirac indicated that there are infinitely many SLOCC classes for four qubits. Later, Verstraete, Dehaene, and Verschelde proposed nine families of states corresponding to nine different ways of entangling four qubits. And then in 2007 Lamata et al. reported that there are eight true SLOCC entanglement classes of four qubits up to permutations of the qubits. In this paper, we investigate SLOCC classification of the nine families proposed by Verstraete, Dehaene and Verschelde, and distinguish 49 true SLOCC entanglement classes from them.
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.
Determining whether quantum states can be interconverted by a prescribed set of transformations is a fundamental problem in quantum information. Here we develop a general classification framework based on conserved operators (COs), i.e., not necessarily hermitian matrices having the state of interest as an eigenstate. We first show that COs belonging to the algebra generating the allowed transformations can be obtained from the kernel of a quantum covariance matrix. Their number determines the dimension of the accessible orbit, while their algebraic structure and eigenvalues provide further constraints on state convertibility. We establish general bounds on the number of COs and prove that highest-weight states, like separable states in the case of nnn qudits, attain the maximum. We also show how these operators generate parent Hamiltonians having the characterized state as an exact ground state. We then also consider positive zero-eigenvalue COs, which encode forbidden directions of reduced density matrices and incorporate rank-based constraints into the same framework. Applying this approach to multiqubit SLOCC, we obtain complementary classifications from algebraic and rank COs. While for n≤3n\leq 3n≤3 qubits both lead to the usual classes, for n=4n=4n=4 qubits they distinguish 17 and 20 types separately, and 29 when combined. The combined classification yields 287 fixed types and 444 continuous families for n=5n=5n=5 qubits, and 4926 fixed types and 147 continuous families for n=6n=6n=6 qubits, providing, to our knowledge, the most refined exhaustive classification of SLOCC-convertibility classes based on such data.
We construct coefficient matrices of size ${2}^{\ensuremath{\ell}}$ by ${2}^{n\ensuremath{-}\ensuremath{\ell}}$ associated with pure nnn-qubit states and prove the invariance of the ranks of the coefficient matrices under stochastic local operations and classical communication (SLOCC). The ranks give rise to a simple way of partitioning pure nnn-qubit states into inequivalent families and distinguishing degenerate families from one another under SLOCC. Moreover, the classification scheme via the ranks of coefficient matrices can be combined with other schemes to build a more refined classification scheme. To exemplify we classify the nine families of four qubits introduced by Verstraete et al. [Phys. Rev. A 65, 052112 (2002)] further into inequivalent subfamilies via the ranks of coefficient matrices, and as a result, we find 28 genuinely entangled families and all the degenerate classes can be distinguished up to permutations of the four qubits. We also discuss the completeness of the classification of four qubits into nine families.
Pure states of three-qubit quantum systems are represented by eight-dimensional complex vectors, whereas their unitary transformations are described by 8×88 \times 88×8 complex matrices. Motivated by this correspondence, we extend our algebraic model for two-qubit systems to eight-dimensional spaces endowed with commutative and associative multiplication. To each eight-dimensional number we assign a matrix and prove that the resulting correspondence is a homomorphic bijection. We derive conditions under which the associated matrix is unitary and hence represents a three-qubit quantum gate. This leads to a specific algebraic notion of a unitary state: a normalized state whose multiplication matrix is unitary. We construct 16,384 basis spaces; within each fixed space, multiplication generates a commuting family of quantum gates. For any two unitary states in the class considered, we give an explicit transition-gate construction within one of these algebraic families. For real components, the unitary states in a representative basis space form a union of thirty-two four-dimensional manifolds in R8\mathbb{R}^{8}R8. The construction therefore yields new structured classes of three-qubit unitary operators and invariant state sets, while its extension to larger multiqubit systems remains open.