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arXiv:0808.3862 · Physical Review A

Local entanglability and multipartite entanglement

Caroline Kruszynska, Barbara Kraus

Published May 6, 2009• 71 citations
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Abstract

We introduce a physically motivated classification of pure quantum states describing $n$ qubits. We characterize all multipartite states which can be maximally entangled to local auxiliary systems using controlled operations. A state has this property if and only if one can construct out of it an orthonormal basis by applying independent local unitary operations. This implies that those states can be used to encode locally the maximum amount of $n$ independent bits. Examples of these states are the so-called stabilizer states, which are used for quantum error correction and one-way quantum computing. We give a simple characterization of these states and construct a complete set of commuting unitary observables which characterize the state uniquely. Furthermore we show how these states can be prepared and discuss their applications.

DOI
10.1103/physreva.79.052304

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We introduce a physically motivated classification of pure quantum states describing nnn qubits. We characterize all multipartite states which can be maximally entangled to local auxiliary systems using controlled operations. A state has this property if and only if one can construct out of it an orthonormal basis by applying independent local unitary operations. This implies that those states can be used to encode locally the maximum amount of nnn independent bits. Examples of these states are the so-called stabilizer states, which are used for quantum error correction and one-way quantum computing. We give a simple characterization of these states and construct a complete set of commuting unitary observables which characterize the state uniquely. Furthermore we show how these states can be prepared and discuss their applications.
Stabilizer states constitute a set of pure states which plays a dominant role in quantum error correction, measurement-based quantum computation, and quantum communication. Central in these applications are the local symmetries of these states. We characterize all local, invertible (unitary and nonunitary) symmetries of arbitrary stabilizer states and provide an algorithm which determines them. We demonstrate the usefulness of these results by showing that the additional local symmetries find applications in entanglement theory and quantum error correction. More precisely, we study a central problem in entanglement theory, which is concerned with the existence of transformations via local operations among pure states. We demonstrate that the identified symmetries enable additional transformations from a stabilizer state to some other multipartite pure state. Furthermore, we demonstrate how the identified symmetries can be used to construct stabilizer codes with diagonal transversal gates.
The necessary and sufficient conditions for the equivalence of arbitrary nnn-qubit pure quantum states under local unitary (LU) operations derived in [B. Kraus, Phys. Rev. Lett. 104, 020504 (2010)] are used to determine the different LU-equivalence classes of up to five-qubit states. Due to this classification new parameters characterizing multipartite entanglement are found and their physical interpretation is given. Moreover, the method is used to derive examples of two nnn-qubit states (with n>2n>2n>2 arbitrary) which have the properties that all the entropies of any subsystem coincide; however, the states are neither LU equivalent nor can be mapped into each other by general local operations and classical communication.
Stabilizer states constitute a set of pure states which plays a dominant role in quantum error correction, measurement-based quantum computation, and quantum communication. Central in these applications are the local symmetries of these states. We characterize all local, invertible (unitary and nonunitary) symmetries of arbitrary stabilizer states and provide an algorithm which determines them. We demonstrate the usefulness of these results by showing that the additional local symmetries find applications in entanglement theory and quantum error correction. More precisely, we study a central problem in entanglement theory, which is concerned with the existence of transformations via local operations among pure states.We demonstrate that the identified symmetries enable additional transformations from a stabilizer state to some other multipartite pure state. Furthermore, we demonstrate how the identified symmetries can be used to construct stabilizer codes with diagonal transversal gates.
In this paper, we provide a characterization of multipartite entanglement in terms of equivalence classes of states under local unitary transformation (LU) by demonstrating a simple method to construct all homogenous polynomials that are invariant under local unitary group(LUIPs) for any fixed degree. We give an upper bound on the degree of the LUIP such that the ring of LUIPs can be generated by LUIPs with degree lower than the bound. Our characterization can directly generate an algorithm whose output is a generating set of LUIPs. By employing the concept of LUIPs, we prove that multipartite entanglement is additive in the sense that two multipartite states are LU equivalent if and only if nnn-copies of these two states are LU equivalent for some nnn. The method for studying LU equivalence is also used to classify the different types of multipartite mixed entanglement according to equivalence classes of states under stochastic local operations and classical communication (SLOCC), where the pure states case was previously studied by Gour and Wallach using another approach.
Maximally entangled states (MES) represent a valuable resource in quantum information processing. In NNN-qubit systems the MES are NNN-GHZ states [i.e., the collection of $|{\mathrm{GHZ}}_{N}\ensuremath{\rangle}=\frac{1}{\sqrt{2}}(|00\dots{}0\ensuremath{\rangle}+|11\dots{}1\ensuremath{\rangle})]$ and its local unitary (LU) equivalences. While it is well known that such states are uniquely stabilized by NNN commuting observables, in this article we consider the minimum number of noncommuting observables needed to characterize an NNN-qubit MES as the unique common eigenstate. Here, we prove, rather surprisingly, that in this general case any NNN-GHZ state can be uniquely stabilized by only two observables. Thus, for the task of MES certification, only two correlated measurements are required with each party observing the spin of his or her system along one of two directions.
We characterize the entanglement contained in a pure three-qubit state via operational entanglement measures. To this end, we derive a new decomposition for arbitrary three-qubit states which is characterized by five parameters (up to local unitary operations). We show that these parameters are uniquely determined by bipartite entanglement measures. These quantities measure the entanglement required to generate the state following a particular preparation procedure and have a clear physical meaning. Moreover, we show that the classification of states obtained in this way is strongly related to the one obtained when considering general local operations and classical communication.