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arXiv:1503.05421 · Physical Review A

Determination of stabilizer states

Xia Wu, Ying‐Hui Yang, Yukun Wang, Qiaoyan Wen, Su‐Juan Qin, Fei Gao

Published Jul 2, 2015• 5 citations
arXivHTMLPDF

Abstract

The determination of many special types of quantum states has been studied thoroughly, such as the generalized $|\text{GHZ}\ensuremath{\rangle}$ states, stochastic local operations and classical communication equivalent $|W\ensuremath{\rangle}$ states, and Dicke states. In this paper, we study another special entanglement state, that is, stabilizer states, which play an important role in quantum error correcting codes, multipartite purification, and so on. We show that all $n$-qubit stabilizer states are uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices for systems which are the supports of $n$ independent generators of the corresponding stabilizer formalisms.

DOI
10.1103/physreva.92.012305

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The determination of many special types of quantum states has been studied thoroughly, such as the generalized ∣GHZ\ensuremath⟩|\text{GHZ}\ensuremath{\rangle}∣GHZ\ensuremath⟩ states, stochastic local operations and classical communication equivalent ∣W\ensuremath⟩|W\ensuremath{\rangle}∣W\ensuremath⟩ states, and Dicke states. In this paper, we study another special entanglement state, that is, stabilizer states, which play an important role in quantum error correcting codes, multipartite purification, and so on. We show that all nnn-qubit stabilizer states are uniquely determined (among arbitrary states, pure or mixed) by their reduced density matrices for systems which are the supports of nnn independent generators of the corresponding stabilizer formalisms.
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.
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 stabilizer formalism allows the efficient description of a sizeable class of pure as well as mixed quantum states of n -qubit systems. That same formalism has important applications in the field of quantum error correcting codes, where mixed stabilizer states correspond to projectors on subspaces associated with stabilizer codes. In this paper, we derive efficient reduction procedures to obtain various useful normal forms for stabilizer states. We explicitly prove that these procedures will always converge to the correct result and that these procedures are efficient in that they only require a polynomial number of operations on the generators of the stabilizers. On one hand, we obtain two single-party normal forms. The first, the row-reduced echelon form, is obtained using only permutations and multiplications of generators. This form is useful to calculate partial traces of stabilizer states. The second is the fully reduced form, where the reduction procedure invokes single-qubit operations and CNOT operations as well. This normal form allows for the efficient calculation of the overlap between two stabilizer states, as well as of the Uhlmann fidelity between them, and their Bures distance. On the other hand, we also find a reduction procedure of bipartite stabilizer states, where the operations involved are restricted to be local ones. The two-party normal form thus obtained lays bare a very simple bipartite entanglement structure of stabilizer states. To wit, we prove that every bipartite mixed stabilizer state is locally equivalent to a direct product of a number of maximally entangled states and, potentially, a separable state. As a consequence, using this normal form we can efficiently calculate every reasonable bipartite entanglement measure of mixed stabilizer states.
We present a method for multipartite entanglement purification of any stabilizer state shared by several parties. In our protocol each party measures the stabilizer operators of a quantum error-correcting code on his or her qubits. The parties exchange their measurement results, detect or correct errors, and decode the desired purified state. We give sufficient conditions on the stabilizer codes that may be used in this procedure and find that Steane's seven-qubit code is the smallest error-correcting code sufficient to purify any stabilizer state. An error-detecting code that encodes two qubits in six can also be used to purify any stabilizer state. We further specify which classes of stabilizer codes can purify which classes of stabilizer states.
It has been proven by Rana et al. that the nnn-qubit generic Dicke states $|\mathrm{GD}(n,\ensuremath{\ell})\ensuremath{\rangle}$ are uniquely determined, among arbitrary states, by their $(\ensuremath{\ell}+1)$-partite reduced density matrices, and $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$ number of them which have one party common to all are sufficient. We show that among arbitrary states, Dicke-type states are also uniquely determined by their $(\ensuremath{\ell}+1)$-partite reduced density matrices. Here, by Dicke-type states, we mean all multiqubit pure states that are stochastic local operations and classical communication equivalent to the nnn-qubit standard Dicke state $|D(n,\ensuremath{\ell})\ensuremath{\rangle}$. Moreover, we find two kinds of subsets of the $(\ensuremath{\ell}+1)$-partite reduced density matrices set, both of which can determine the Dicke-type states. More importantly, the number of elements in these two subsets is much smaller than $(\begin{array}{c}n\ensuremath{-}1\\ \ensuremath{\ell}\end{array})$.
It has been known that, among pure states, NNN-qubit WWW states cannot be uniquely determined by their arbitrary $(N\ensuremath{-}1)$ bipartite reduced density matrices. Parashar and Rana proved that among arbitrary states, $(N\ensuremath{-}1)$ bipartite reduced density matrices that the pairs of qubits constitute a star graph or a line graph can uniquely determine stochastic local operations and classical communication (SLOCC) equivalent WWW states, and we generalize this conclusion into tree graph. In this paper, we show that all SLOCC equivalent WWW states can be uniquely determined (among pure, mixed states) by their $(N\ensuremath{-}1)$ bipartite reduced density matrices, if the $(N\ensuremath{-}1)$ pairs of qubits constitute a tree graph on NNN vertices, where each pair of qubits represents an edge.