научная статья по теме CONSTRAINTS ON INVARIANT POLYNOMIALS FOR A PAIR OF ENTANGLED QUBITS Физика

Текст научной статьи на тему «CONSTRAINTS ON INVARIANT POLYNOMIALS FOR A PAIR OF ENTANGLED QUBITS»

ЯДЕРНАЯ ФИЗИКА, 2011, том 74, № 6, с. 919-925

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

CONSTRAINTS ON SU(2) ® SU(2) INVARIANT POLYNOMIALS FOR A PAIR OF ENTANGLED QUBITS

©2011 V. Gerdt1)*, A. Khvedelidze1)>2)**, Yu. Palii1) 3)***

Received July 19,2010

We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ® SU(2) group on the space of density matrices P+. Since elements of P+ are Hermitian, nonnegative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, g R15. We define P+ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ® SU(2) invariants is proposed and the well-known Peres—Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ® SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz—Mahler entanglement matrices.

1. INTRODUCTION

Attempting to understand the nature of quantum information we strongly rely on a knowledge from the classical background [1, 2]. A fundamental unit of quantum information, "qubit", was introduced by analogy with the classical binary alternatives as the information associated with a 2-level quantum-mechanical system4). Follow further this correspondence, the quantum relative of the classical n-bit string is composite object constructed as quantum superposition of n-qubit states. In the same way, for information processing on quantum level, instead of the classical logical gates the manipulations based on the unitary transformations of n-qubit density matrix p

P

p' = UpU\ U G U(2n),

(1)

are used. However, exactly at this place the quantum-classical analogies cease to work. While classical gates act transitively on all n-bit strings, i.e., an arbitrary n-bit string can be transformed to a fixed

'■'Laboratory of Information Technologies, Joint Institute for Nuclear Research, Dubna, Russia.

2)Department of Theoretical Physics, A.Razmadze Mathematical Institute, Tbilisi, Georgia.

3)Institute of Applied Physics, Chisinau, Moldova. E-mail: gerdt@jinr.ru

E-mail: akhved@jinr.ru E-mail: palii@jinr.ru

4)Qubits in question could be of quite different kinds, e.g.,

correspond to spin-l/2 degrees of freedom of an atom or a single photon's helicity.

n-bit string applying logical operations on individual bits, quantum local transformations act nontran-sitively on a multiqubit states [3]5). The qubits states that are not related by local transformations are "different" as far as their nonlocal properties are concerned. This means that regardless of the actual physical nature of gates and qubits nonlocal properties of composed system are encoded in equivalence relations between states provided by local transformations. In mathematical terms local unitary transformations acting on the state space partition it into equivalent classes — orbits. Each orbit is a representative of state with definite nonlocal characteristics and the set of all orbits of n qubits forms the "entanglement space", En [3,4]. Therefore, quantification and classification of all possible nonlocalities in multiqubit system reduces to the analysis of En. For the case of 2-qubits in pure state the entanglement space represents the closed interval [0,1], parameterized by the concurrence [5]. However, for mixed states, especially if the number of qubits is more than two, the constructive description of geometry and topology of En becomes very challenging. The canonical way to describe the entanglement space can be adopted from the classical theory of invariants [6]; to separate orbits, giving coordinates for points in En, the polynomial functions of the density matrix elements which are invariant under the local operations

5)The local operations, acting independently on each of n qubits, are defined as unitary transformations of the form SU(2) ® SU(2) SU(2). The complementary ones,

SU(2n )/SU (2) ® SU(2) SU(2) represent the non-

local transformations.

—>

920

GERDT et al.

can be used6). A series of important results in this direction has been obtained since Linden and Popescu addressed this issue to characterize the entanglement by polynomial invariants [8]. Particularly, for a mixed state of 2-qubits a complete set of fundamental local invariants that are homogeneous polynomials, from which all others can be constructed as the sums of products, has been determined [9]. Furthermore, the algebraic structure of the corresponding polynomial ring, c[R15]^(2)®^(2), that is necessarily Cohen-Macaulay type, has been recently identified [10].

All above studies were based on the assumption that the action of the local group is a linear representation on a vector space. However, according to the definition, the density matrix of n-

level system p e P+(R"2-1), where P+ (R"2-1) is a space of n x n Hermitian, semi-definite matrices with trace normalized to unity, Tr(p) = 1. The requirement of positive semi-definiteness means that space P+(R15) in question is a nonlinear subset of R15. Therefore semi-definiteness implies that the local group SU(2) ® SU(2) acts not on elements of vector space but on a certain semi-algebraic variety, given by the set of polynomial inequalities in elements of the density matrix. Consequently, values that local invariants can have are not arbitrary ones but are constrained. This in turn constraints characteristics of the entanglement space En and requires therefore the detailed analysis.

Below to gain some insight into this important issue we discuss the question: How does semi-definiteness requirement affect the local invariants of bipartite quantum system?7)

In the present article this problem is analyzed for the simplest bipartite system, pair of qubits. Below we define P+ (R15) explicitly as solution of the system of linear and second-order polynomial inequalities in the Casimir invariants of the enveloping algebra of SU(4) group. Having in mind this result the special integrity basis for local SU(2) ® SU(2) polynomial invariants is constructed, with minimal number of elements constrained by requirement of positive semi-definiteness.

Our plan is as follows. We start with preliminaries, introducing the basic notions and present one vivid example. Sections 3 and 4 contain main results; the special, optimal basis for local polynomial scalars is

6)Note that usage of polynomial functions is a reasonable restriction since according to the Schwarz theorem [7] any C^ class function, that is invariant under finite-dimensional linear orthogonal representation of a compact group, is C function of invariant polynomials.

7)Also this question has been studied for many years (see, e.g., [11 — 13] and references therein) the elaboration of effi-

cient practical computational methods is still under question.

described and inequalities in global SU(4) scalars that guarantee the semi-definiteness of density matrices are presented. In Section 4 the well-known separability criterion for two qubit density matrices, the Peres-Horodecki condition, is reformulated in the form of polynomial inequalities in local invariants.

2. PRELIMINARIES

In this section the basic conventions and terminology are presented for the reader's convenience.

2.1. Mixed States for One and Two Qubits

Irrelevant to realization of a qubit its mixed state is described by analog of the density matrix of a nonrelativistic spin-1/2

p = \(l + cx-a). (2)

Here, a are the standard Pauli matrices and a is defined as mathematical expectation:

a = (a) = Tr (pa).

If p2 = p, a qubit is in "pure state", otherwise in the "mixed" one. For pure states a parameterizes points on the so-called Bloch 2-sphere, a2 = 1, while for mixed states the positive semi-definiteness of the density matrix is provided inside the Bloch ball a2 < < 1.

The generic form for an arbitrary mixed 2-qubit state is given by decomposition [14]:

p = ^ [I4 + a • a <g> I2 + + I2 0 a • b + Cij ai 0 aj ]

(3)

The state is characterized by 15 expectation values: a, b and cij, i,j = 1,2,3. The parameters a, b are related with density matrices pA and pB of individual qubit's (A, B), extracted from p by taking the partial traces:

pA = Trg p, pB = Ta p.

(4)

The coefficients cij are entries of the so-called "correlation matrix", C := \ \cij||8).

Similarly to the one-qubit case, using the expansion for traceless part of p over the basis X = = {Ai,A2, ...,A15} of the su(4) algebra, a density matrix for 2-qubits can be characterized by 15-dimensional Bloch vector £ = {{1,{2,...,{15} G G R15 [15]:

p = ](h + \/6£ • aV (5)

8)Under the local transformations, acting on each qubit independently, parameters a and b transform as 3-vectors, while cij as dyadic.

a b

ß

-0.5

-1.0 1.0

-1.0 -1.0

Fig. 1. The positivity domain for po (a) and p1B (b).

2.2. The Entangled Mixed States

The useful quantity for measure of correlation in composed bipartite system is the so-called Schlienz—Mahler matrix [16]:

M = p — pA ® Pb .

(6)

The invariants constructed with the aid of (6) are aimed to describe the correlations between subsystems in the the so-called "entangled states". The mixed entangled states are states complementary to those representable in the following separable form [3]:

Psep = (1 + ak ' ak"> ° ^ i1 + A-' ak) (7)

with

y^^k = 1, Wk > 0.

pTB = I ® Tp,

where T is transposition operation. Under the transposition the Pauli matrices change as T(o1, o2,o3) ^ ^ (01, —02,03).

2.4. An Example: aßY States

Here we present an illustrative example of 3-parameter family of density matrices showing how the non-negativity of density matrices constraints the moduli space.

Consider 3-parameter family of density matrices of the following form:

Po

pTb

The well-known test for detecting entanglement states for 2-qubits is based

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