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.
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.
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.
Product testing, i.e., deciding whether a pure multipartite quantum state is fully unentangled across a specified tensor decomposition, serves as a bridge between quantum property testing, unentangled quantum proof systems, and tensor optimization. Despite being a fundamental property testing task and having many applications, the product test's exact (worst-case) acceptance probability curve has yet to be fully determined. In this work, we determine this curve exactly. Let
ω be the maximum squared overlap of the input with a product state, and let
PTn(ω) be the largest possible acceptance probability of the product test over all
n-partite pure states with product overlap
ω, allowing arbitrary finite local dimensions. We prove that, for every $n\ge 2 $ and every $ω\in(0,1] $,
PTn(ω)=21(1+mω2+(1−mω)2), where $m=\lfloor1/ω\rfloor $. The formula recovers the previously known tight section of the curve for $ω\ge 1/2 $, resolves all low-overlap regimes $ω<1/2 $, and implies $\mathrm{PT}_n(ω)\to 1/2 $ as
ω→0 answering an open problem in [Soleimanifar and Wright, SODA 2022]. As a complexity-theoretic application, our results improve the one-shot soundness parameter in the Harrow-Montanaro reduction from
QMA(k) to
QMA(2). Our techniques, built upon those of Soleimanifar and Wright, allow us to resolve these open questions while remaining surprisingly elementary.
We study separability properties in a 5-dimensional set of states of quantum systems composed of three subsystems of equal but arbitrary finite Hilbert space dimension. These are the states, which can be written as linear combinations of permutation operators, or, equivalently, commute with unitaries of the form UxUxU. We compute explicitly the following subsets: (1) triseparable states, which are convex combinations of triple tensor products, (2) biseparable states, which are separable for a twofold partition of the system, and (3) states with positive partial transpose with respect to such a partition.