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arXiv:quant-ph/0209125 · International Journal of Foundations of Computer Science

SEPARABILITY OF PURE N-QUBIT STATES: TWO CHARACTERIZATIONS

Philippe Jorrand, Mehdi Mhalla

Published Oct 1, 2003• 10 citations
arXivHTMLPDF

Abstract

Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.

DOI
10.1142/s0129054103002035

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Given a pure state |ψN>∈ℋN of a quantum system composed of n qubits, where ℋN is the Hilbert space of dimension N=2n, this paper answers two questions: what conditions should the amplitudes in |ψN> satisfy for this state to be separable (i) into a tensor product of n qubit states |ψ2>0⊗ |ψ2>1 ⊗⋯⊗ |ψ2>n-1, and (ii), into a tensor product of two subsystems states |ψP> ⊗ |ψQ> with P=2p and Q=2q such that p+q=n? For both questions, necessary and sufficient conditions are proved, thus characterizing at the same time families of separable and entangled states of n qubit systems. A number of more refined questions about separability in n qubit systems can be studied on the basis of these results.
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In this paper, we study the distinguishability of multipartite quantum states by separable operations. We first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations. An analytical version of this condition is derived for the case of(D-1) pure states, whereDis the total dimension of the state space under consideration. A number of interesting consequences of this result are then carefully investigated. Remarkably, we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication. Before our work, only a class of 3 otimes 3 nonlocal separable operations was known [Bennett , Phys. Rev. A 59, 1070 (1999)]. We also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states, i.e., has an orthogonal Schmidt number not less than three, thus generalize the recent work about indistinguishable bipartite subspaces [Watrous, Phys. Rev. Lett. 95, 080505 (2005)]. Notably, we obtain an explicit construction of indistinguishable subspaces of dimension 7 (or 6) by considering a composite quantum system consisting of two qutrits (resp., three qubits), which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8.
Quantum computing operates in three stages: (i) preparation of the initial states of the n qubits of a register, (ii) step by step transformation of the state of this register by unitary operators which compose the quantum program, and (iii), measurement of all or some of the qubits of the register. The existence of quantum algorithms which are exponentially less complex than their classical counterparts for some classes of problems, stems from entangled states established by multi-qubit unitary operators within the quantum program. A register of n qubits is a quantum system composed of n quantum subsystems. The state |/spl psi/> of a quantum system composed of two quantum subsystems A and B is said to be entangled when |/spl psi/> is not reducible to a pair composed of a state |/spl psi/> of A and a state |/spl psi//sub B/> of B: such situations have no counterpart in the classical world. In quantum theory, such a pair of states is denoted by a tensor product: |/spl psi/> is entangled if it cannot be factorized into the tensor product |/spl psi//sub A/>/spl ominus/|/spl psi//sub B/>. This paper establishes conditions according to which it is possible to tell whether or not the state of a register of n qubits is entangled. The state of a single qubit is a vector /spl alpha/|0>+/spl beta/|1> of unit norm in a 2-dimensional vector space, where |0> and |1> are the two basis states and where a and are complex amplitudes. Then, if both A and B are qubits, the most general form of the state of a register composed of the 2 qubits A and B is also a vector of unit norm, but now in a 4-dimensional space: |/spl psi/>=/spl alpha/|00>+/spl beta/|01+/spl gamma/|10>+/spl sigma/|11>. It is straightforward to prove that |/spl psi/> can be factorized into |/spl psi//sub A/>/spl ominus/|/spl psi//sub B/> if and only if /spl alpha//spl delta/=/spl beta//spl gamma/. In such a case, |/spl psi/> is said to be separable, i.e. not entangled. This paper generalizes this form of condition to registers of n qubits. If |/spl psi/> is the state of a register of n qubits, two different questions about the separability of |/spl psi/> are answered: (i) is |/spl psi/> separable into a product |/spl psi//sub 1/>/spl ominus/|/spl psi//sub 2/>/spl ominus/L/spl ominus/|/spl psi//sub n/> of n single qubit states? and (ii), is |/spl psi/> separable into a product |/spl psi//sub A/>/spl ominus/|/spl psi//sub B/> of the states of two subregisters A and B, respectively containing p and q adjacent qubits with p+q=n? For both questions, necessary and sufficient conditions are given.
After introducing the partially separable concept, we proved the equivalence between the partial separability of a given mmm-partite subsystem with mmm qubits and the purity of states of this mmm-partite subsystem for a pure state in multipartite systems with arbitrary finite n(>m)n(>m)n(>m) qubits. Furthermore, we give out the operational realizations (corollaries) of our theorem, which are the sufficient and necessary criterions of partial separability of states and can be used to classification of states. Our results are helpful to understand and describe quantum entanglement in multipartite systems.
An algebraic approach to quantum non-separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such a general formulation has some advantages. Using algebraic formalism, we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non-separable pure states. As a universal measure of non-separability of pure states, we propose to take the so-called total correlation. This quantity depends on the state as well as on the algebraic partition. Its numerical value is given by the norm of the corresponding correlation matrix.
Entanglement between n particles is a generalization of the entanglement between two particles, and a state is considered entangled if it cannot be written as a mixture of tensor products of the n particles' states. We present the key notion of semi-separability , used to investigate n -particle entanglement by looking at two-party entanglement between its various subsystems. We provide necessary conditions for n -particle separability (that is, sufficient conditions for n -particle entanglement). We also provide necessary and sufficient conditions in the case of pure states. By surprising examples, we show that such conditions are not sufficient for separability in the case of mixed states, suggesting entanglement of a strange type.