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Acta Polytechnica

Multi-Particle Universal Processes

Jiří Novotný, M. Štefaňák, Vojtech Kostak

Published Jan 5, 2005• 0 citations
Publisher

Abstract

We generalize bipartite universal processes to the subclass of multi-particle universal processes from one to N particles. We show how the general statement for a multi-particle universal process can be constructed. The one-parameter family of processes generating totally anti-symmetric states has been generalized to a multi-particle regime and its entanglement properties have been studied. A view is given on the complete positivity and the possible physical realization of universal processes.

DOI
10.14311/766

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We generalize bipartite universal processes to the subclass of multi-particle universal processes from one to N particles. We show how the general statement for a multi-particle universal process can be constructed. The one-parameter family of processes generating totally anti-symmetric states has been generalized to a multi-particle regime and its entanglement properties have been studied. A view is given on the complete positivity and the possible physical realization of universal processes.
Universal two-particle entanglement processes are analyzed in arbitrary dimensional Hilbert spaces. On the basis of this analysis the class of possible optimal universal entanglement processes is determined whose resulting output states do not contain any separable states. It is shown that these processes form a one-parameter family. For all Hilbert space dimensions larger than two the resulting optimally entangled output states are mixtures of anti-symmetric states which are freely entangled and which also preserve information about input states. Within this one-parameter family there is only one process by which all information about any input state is destroyed completely.
Within the class of all possible universal (covariant) two-particle quantum processes in arbitrary dimensional Hilbert spaces those universal quantum processes are determined whose output states optimize the recently proposed entanglement measure of Vidal and Werner. It is demonstrated that these optimal entanglement processes belong to a one-parameter family of universal entanglement processes whose output states do not contain any separable components. It is shown that these optimal universal entanglement processes generate antisymmetric output states and, with the single exception of qubit systems, they preserve information about the initial input state.
Entanglement is a powerful resource for processing quantum information. In this context pure, maximally entangled states have received considerable attention. In the case of bipartite qubit-systems the four orthonormal Bell-states are of this type. One of these Bell states, the singlet Bell-state, has the additional property of being antisymmetric with respect to particle exchange. In this contribution we discuss possible generalizations of this antisymmetric Bell-state to cases with more than two particles and with single-particle Hilbert spaces involving more than two dimensions. We review basic properties of these totally antisymmetric states. Among possible applications of this class of states we analyze a new quantum key sharing protocol and methods for comparing quantum states.
A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles 1,21,21,2 with a Hilbert space Cd1⊗Cd2\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}Cd1​⊗Cd2​, a UE exists when min⁡(d1,d2)≥3\min{(d_1,d_2)}\geq 3min(d1​,d2​)≥3 and (d1,d2)≠(3,3)(d_1,d_2)\neq (3,3)(d1​,d2​)=(3,3). It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic systems whose wave functions sit in the symmetric/antisymmetric subspace of Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d}Cd⊗Cd. The development of a theory of UEs for these types of systems needs considerably different approaches from that used for UEs of distinguishable systems. This is because the general entanglement of identical particle systems cannot be discussed in the usual way due to the effect of (anti)-symmetrization which introduces "pseudo entanglement" that is inaccessible in practice. We show that, unlike the distinguishable particle case, UEs exist for bosonic/fermionic systems with Hilbert spaces which are symmetric (resp. antisymmetric) subspaces of Cd⊗Cd\mathbb{C}^{d}\otimes\mathbb{C}^{d}Cd⊗Cd if and only if d≥3d\geq 3d≥3 (resp. d≥8d\geq 8d≥8). To prove this we employ algebraic geometry to reason about the different algebraic structures of the bosonic/fermionic systems. Additionally, due to the relatively simple coherent state form of unentangled bosonic states, we are able to give the explicit constructions of two bosonic UEs. Our investigation provides insight into the entanglement properties of systems of indisitinguishable particles, and in particular underscores the difference between the entanglement structures of bosonic, fermionic and distinguishable particle systems.
Analyzing the properties of entanglement in many-particle spin-1/2 systems is generally difficult because the system's Hilbert space grows exponentially with the number of constituent particles, N. Fortunately, it is still possible to investigate a many-particle entanglement when the state of the system possesses sufficient symmetry. In this paper, we present a practical method for efficiently computing various bipartite entanglement measures for states in the symmetric subspace and perform these calculations for $N\ensuremath{\sim}{10}^{3}.$ By considering all possible bipartite splits, we construct a picture of the multiscale entanglement in large symmetric systems. In particular, we characterize dynamically generated spin-squeezed states by comparing them to known reference states (e.g., Greenberger-Horne-Zeilinger and Dicke states), and families of states with near-maximal bipartite entropy. We quantify the trade-off between the degree of entanglement and its robustness to particle loss, emphasizing that substantial entanglement need not be fragile.
We introduce and study bipartite quantum states that are invariant under the local action of the cyclic sign group. Due to symmetry, these states are sparse and can be parameterized by a triple of vectors. Their important semi-definite properties, such as positivity and positivity under partial transpose (PPT), can be simply characterized in terms of these vectors and their discrete Fourier transforms. We study in detail the entanglement properties of this family of symmetric states, showing that it contains PPT entangled states. For states that are diagonal in the Dicke basis, deciding separability is equivalent to a circulant version of the complete positivity problem. In local dimension d ≤ 5, we completely characterize these sets and construct entanglement witnesses; some partial results are also obtained for d = 6, 7. We construct a new family of states for which the properties of PPT and separability can be characterized for all dimensions, generalizing results from from the literature. Our results show that these states have a rich entanglement structure, even in the bosonic subspace.