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.
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.
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.
A state ψ = α/00] + β/01] + γ/10] +δ/11] of a system of two qubits is separable if the equality among pair-wise products αδ = βγ holds. This paper generalize this form of condition for distinguishing among separable and entangled states of systems of n qubits. Given a pure state /ψN] of a quantum system composed of n qubits, where N = 2n, this paper defines minimal sets of equalities among pair-wise products of amplitudes of /ψN] for characterizing two forms of separability of /ψN]: (i) into a tensor product of n qubit states /ψ2]0 x/ψ2]1 x...x/ψ2]n-1, and (ii), into a tensor product of 2 subsystems states /ψp]x/ψQ] with P=2p and Q=2q such that p+q=n.
To begin with, we need a quantum register made of many qubits to store information. Recall that a classical computer also requires memory to store information. The simplest way to realize a qubit physically is to use a two-level quantum system. For example, an electron, a spin 1/2 nucleus or two mutually orthogonal polarization states (horizontal and vertical, for example) of a single photon can be a qubit. We may also employ a two-dimensional subspace, such as the ground state and the first excited state, of a multi-dimensional Hilbert space, such as atomic energy levels. In the latter case, special care must be taken to avoid leakage of the state to the other part of the Hilbert space. In any case, the two states are identified as the basis vectors, |0〉 and |1〉, of the Hilbert space so that a general single qubit state takes the form |ψ〉 = α|0〉 + β|1〉, where |α|2 + |β|2 = 1. A multiqubit state is expanded in terms of the tensor products of these basis vectors. Each qubit must be separately addressable. Moreover it should be scalable up to a large number of qubits. The two-dimensional vector space of a qubit may be extended to be three-dimensional (qutrit) or, more generally, d-dimensional (qudit). A system may be made of several different kinds of qubits. Qubits in an ion trap quantum computer, for instance, may be defined as: (1) hyperfine/Zeeman sublevels in the electronic ground state of ions (2) a ground state and an excited state of a weakly allowed optical transition and (3) normal mode of ion oscillation. A similar scenario is also proposed for Josephson junction qubits, in which two flux qubits are coupled through a quantized LC circuit. Simultaneous usage of several types of qubits may be the most promising way to achieve a viable quantum computer.
Entanglement can be understood as an extraordinary degree of correlation between states of quantum systems – a correlation that cannot be given an explanation in terms of something like a common cause. Entanglement can occur between two or more quantum systems, and the most interesting case is when these correlations occur between systems that are space-like separated, meaning that changes made to one system are immediately correlated with changes in a distant system even though there is no time for a signal to travel between them. In this case one says that quantum entanglement leads to non-local correlations, or non-locality. More precisely, entanglement can be defined in the following way. Consider two particles, A and B, whose (pure) states can be represented by the state vectors ψA and ψB. Instead of representing the state of each particle individually, one can represent the composite two-particle system by another wavefunction, Ψ AB. If the two particles are unentangled , then the composite state is just the tensor product of the states of the components: Ψ AB = ψ A ⊗ ψ B; the state is then said to be factorable (or separable). If the particles are entangled, however, then the state of the composite system cannot be written as such a product of a definite state for A and a definite state for B. This is how an entangled state is defined for pure states: a state is entangled if and only if it cannot be factored: Ψ AB ≠ ψ A ⊗ ψ B.
As we have seen, even a single qubit is a surprisingly interesting object. However, the real power of quantum information processing begins with systems of two or more qubits. Before studying these in detail we need to expand our notation a little. Consider a system of two qubits, labeled a and b , each of which has two basis states, |0⧽ and |1⧽. The whole system then has four basis states, which can be written as |0 a 0 b ⧽, |0 a 1 b ⧽, |1 a 0 b ⧽ and |1 a 1 b ⧽, and can be found in any general superposition of these states, so that it occupies a four-dimensional Hilbert space. In the same way, a system of three qubits inhabits an eight-dimensional Hilbert space, and so on. This exponential increase in the size of the Hilbert space with a linear increase in the number of qubits underlies the power of quantum computers. Direct products The size of the Hilbert spaces involved can also be a huge problem, however, making it difficult to describe states of systems with many qubits. A partial solution is to note that some states can be described in a simpler way, using the concept of direct products . These states, in which the individual qubits can in principle be discussed separately, make up a tiny minority of the states accessible to a multi-qubit system, but include many important states, most notably the basis states. States of this kind are said to be separable , and states which are not separable are said to be entangled . Entangled states are much more interesting than separable ones, but it is wise to begin with the simpler case.
Abstract In quantum computational logics meaning of sentences are repre-sented by quantum information quantities. Any n-qubit sequence is identified with a unit vector of the tensor product of the complex Hilbert space ⊗nC2. Let us split the set of the vectors of its computational basis in the two sets: Bn1: = {i: ‖ i À= | x1,.., xn 〉 and xn = 1}; Bn0: = {i: ‖ i À= | x1,.., xn 〉 and xn = 0}. Any unit vector | ψ 〉 can be expressed in the form: | ψ 〉 = ∑i∈B0 ai ‖ iÀ +∑j∈B1 bj ‖ j À where by the Born rule,∑ i∈B0 | ai |2 + j∈B1 | bj |2 = 1 A key role in the definition of semantics is played by the probability-value of | ψ〉, defined as follows: p( | ψ〉): = ∑j∈B1 | bj |2 Moreover a function g: [0, 1]2 7 → [0, 1] is said quantum simulable iff ∃n ≥ 1, a unitary operator Ug: ⊗n+2C2 7 → ⊗n+2C2 and a quregister | χ 〉 ∈ ⊗nC2 such