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Chinese Physics B

The stabilizer for n -qubit symmetric states

Xian Shi

Published Oct 1, 2018• 3 citations
Publisher

Abstract

The stabilizer group for an $n$-qubit state $\ket{\phi}$ is the set of all invertible local operators (ILO) $g=g_1\otimes g_2\otimes \cdots\otimes g_n,$ $ g_i\in \mathcal{GL}(2,\mathbb{C})$ such that $\ket{\phi}=g\ket{\phi}.$ Recently, G. Gour $et$ $al.$ \cite{GKW} presented that almost all $n$-qubit state $\ket{\psi}$ own a trivial stabilizer group when $n\ge 5.$ In this article, we consider the case when the stabilizer group of an $n$-qubit symmetric pure state $\ket{\psi}$ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state $\ket{\phi}$ is nontrivial when $n\le 4$. Then we present a class of $n$-qubit symmetric states $\ket{\phi}$ with the trivial stabilizer group. At last, we prove that an $n$-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5, which confirms the main result of \cite{GKW} partly.

DOI
10.1088/1674-1056/27/10/100311

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The stabilizer group for an nnn-qubit state ∣ϕ⟩\ket{\phi}∣ϕ⟩ is the set of all invertible local operators (ILO) g=g1⊗g2⊗⋯⊗gn,g=g_1\otimes g_2\otimes \cdots\otimes g_n,g=g1​⊗g2​⊗⋯⊗gn​, gi∈GL(2,C) g_i\in \mathcal{GL}(2,\mathbb{C})gi​∈GL(2,C) such that ∣ϕ⟩=g∣ϕ⟩.\ket{\phi}=g\ket{\phi}.∣ϕ⟩=g∣ϕ⟩. Recently, G. Gour etetet al.al.al. \cite{GKW} presented that almost all nnn-qubit state ∣ψ⟩\ket{\psi}∣ψ⟩ own a trivial stabilizer group when n≥5.n\ge 5.n≥5. In this article, we consider the case when the stabilizer group of an nnn-qubit symmetric pure state ∣ψ⟩\ket{\psi}∣ψ⟩ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state ∣ϕ⟩\ket{\phi}∣ϕ⟩ is nontrivial when n≤4n\le 4n≤4. Then we present a class of nnn-qubit symmetric states ∣ϕ⟩\ket{\phi}∣ϕ⟩ with the trivial stabilizer group. At last, we prove that an nnn-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5, which confirms the main result of \cite{GKW} partly.
The stabilizer group for an nnn-qubit state ∣ϕ⟩\ketϕ∣ϕ⟩ is the set of all invertible local operators (ILO) g=g1⊗g2⊗⋯⊗gn,g=g_1\otimes g_2\otimes \cdots\otimes g_n,g=g1​⊗g2​⊗⋯⊗gn​, $ g_i\in \mathcal{GL}(2,\mathbb{C})$ such that ∣ϕ⟩=g∣ϕ⟩.\ketϕ=g\ketϕ.∣ϕ⟩=g∣ϕ⟩. Recently, G. Gour etetet al.al.al. \cite{GKW} presented that almost all nnn-qubit state ∣ψ⟩\ketψ∣ψ⟩ own a trivial stabilizer group when n≥5.n\ge 5.n≥5. In this article, we consider the case when the stabilizer group of an nnn-qubit symmetric pure state ∣ψ⟩\ketψ∣ψ⟩ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state ∣ϕ⟩\ketϕ∣ϕ⟩ is nontrivial when n≤4n\le 4n≤4. Then we present a class of nnn-qubit symmetric states ∣ϕ⟩\ketϕ∣ϕ⟩ with the trivial stabilizer group. At last, we prove that an nnn-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5, which confirms the main result of \cite{GKW} partly.
The stabilizer group of an n-qubit state |ψ is the set of all matrices of the form g=g1⊗⋯⊗gn, with g1,…,gn being any 2 × 2 invertible complex matrices that satisfy g|ψ=|ψ. We show that for 5 or more qubits, except for a set of states of zero measure, the stabilizer group of multipartite entangled states is trivial, that is, containing only the identity element. We use this result to show that for 5 or more qubits, the action of deterministic local operations and classical communication (LOCC) can almost always be simulated simply by local unitary (LU) operations. This proves that almost all n-qubit states with n≥5 can neither be reached nor be converted into any other (n-partite entangled), LU-inequivalent state via deterministic LOCC. We also find a simple and elegant expression for the maximal probability to convert one multi-qubit entangled state to another for this generic set of states.
We refine recent local unitary entanglement classification for symmetric pure states of nnn qubits (that is, states invariant under permutations of qubits) using local unitary stabilizer subgroups and Majorana configurations. Stabilizer subgroups carry more entanglement distinguishing power than do the stabilizer subalgebras used in our previous work. We extend to mixed states recent results about local operations on pure symmetric states by showing that if two symmetric density operators are equivalent by a local unitary operation, then they are equivalent via a local unitary operation that is the {\em same} in each qubit. A geometric consequence, used in our entanglement classification, is that two symmetric pure states are local unitary equivalent if and only if their Majorana configurations can be interchanged by a rotation of the Bloch sphere.
We classify local unitary equivalence classes of symmetric states via a classification of their local unitary stabilizer subgroups. For states whose local unitary stabilizer groups have a positive number of continuous degrees of freedom, the classification is exhaustive. We show that local unitary stabilizer groups with no continuous degrees of freedom are isomorphic to finite subgroups of the rotation group SO(3)SO(3)SO(3), and give examples of states with discrete stabilizers.
The study of state transformations under local operations and classical communication (LOCC) plays a crucial role in entanglement theory. While this has been long ago characterized for pure bipartite states, the situation is drastically different for systems of more parties: generic pure qudit states cannot be obtained from nor transformed to any state (i.e., they are isolated), which contains a different amount of entanglement. We consider here the question of LOCC convertibility for permutation-symmetric pure states of an arbitrary number of parties and local dimension, a class of clear interest both for physical and mathematical reasons and for which the aforementioned result does not apply given that it is a zero-measure subset in the state space. While it turns out that generic nnn-qubit symmetric states are also isolated, we consider particular families for which we can determine to be, on the contrary, endowed with a rich local stabilizer, a necessary requirement for LOCC convertibility to be possible. This allows us to identify classes in which LOCC transformations among permutation-symmetric states are possible. Notwithstanding, we provide several results that indicate severe obstructions to LOCC convertibility in general even within these highly symmetrical classes. In the course of the study of LOCC transformations, we also characterize the local symmetries of symmetric states.
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.