Topology of the three-qubit space of entanglement types

Walck, Scott N.; Glasbrenner, J. K.; Lochman, Matthew H.; Hilbert, Shawn A.

doi:10.1103/physreva.72.052324

Cited by 16 publications

(11 citation statements)

References 17 publications

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“…The full classification problem for multipartite entanglement has been a very difficult problem, and complete results exist only for 2-and 3-qubit pure states [3,4,5,6,7]. Progress has been made, however, in understanding aspects of the local unitary orbits and their dimensions [3,8,9,10,11,12].…”

Section: Introductionmentioning

confidence: 99%

Maximum stabilizer dimension for nonproduct states

Walck

Lyons

2007

Phys. Rev. A

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Composite quantum states can be classified by how they behave under local unitary transformations. Each quantum state has a stabilizer subgroup and a corresponding Lie algebra, the structure of which is a local unitary invariant. In this paper, we study the structure of the stabilizer subalgebra for n-qubit pure states, and find its maximum dimension to be n-1 for nonproduct states of three qubits and higher. The n-qubit Greenberger-Horne-Zeilinger state has a stabilizer subalgebra that achieves the maximum possible dimension for pure nonproduct states. The converse, however, is not true: we show examples of pure 4-qubit states that achieve the maximum nonproduct stabilizer dimension, but have stabilizer subalgebra structures different from that of the n-qubit GHZ state.Comment: 6 page

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Section: Introductionmentioning

confidence: 99%

Maximum stabilizer dimension for nonproduct states

Walck

Lyons

2007

Phys. Rev. A

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“…We would like to point out that our result is not a consequence of Walck's [7] result due to the fact that both, the spaces and the equivalent relations involved are different. In fact, one can easily show with arguments based on the dimension of the manifolds involved that the entanglement spaces E n and E R n are different for all n > 2.…”

Section: Introductionmentioning

confidence: 71%

“…The equivalent relation considered here is not new. For the general case of n-qubits with complex amplitudes, the authors [7][8][9] have considered the equivalence relation: Two n-qubit states are equivalent if they are connected by local gates. Since they are considering n-qubits with complex amplitudes there are not restrictions on the local gates, i.e., gates of the form U 1 ⊗ • • • ⊗ U n where each U i is a 2 × 2 unitary matrix.…”

Section: Introductionmentioning

confidence: 99%

“…Each orbit is called an "entanglement type" and the quotient space is called "the space of entanglement types"; the latter denoted as E n . In [7] the author shows that E 2 is a closed interval and the orbits are either a 4-dimensional manifold that contains the unentangled 2-qubit (product) states or, a 3-dimensional manifold that contains the EPR (maximally entangled) states or, 5-dimensional manifolds for all other cases with intermediate entanglement strength. Placing our result in the same perspective, let us denote by E R 2 the quotient space of 2-qubit states with real amplitudes where two 2qubit states are identified if they can be connected with real local gates.…”

Section: Introductionmentioning

confidence: 99%

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Entanglement types for two-qubit states with real amplitudes

Perdómo¹,

Leyton-Ortega²,

Perdomo-Ortiz³

2019

Preprint

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We study the set of two-qubit pure states with real amplitudes and their geometrical representation in the three-dimensional sphere. In this representation, we show that the maximally entangled states -those locally equivalent to the Bell States -form two disjoint circles perpendicular to each other. We also show that taking the natural Riemannian metric on the sphere, the set of states connected by local gates are equidistant to this pair of circles. Moreover, the unentangled, or so called product states, are π/4 units away to the maximally entangled states. This is, the unentangled states are the farthest away to the maximally entangled states. In this way, if we define two states to be equivalent if they are connected by local gates, we have that there are as many equivalent classes as points in the interval [0, π/4] with the point 0 corresponding to the maximally entangled states. The point π/4 corresponds to the unentangled states which geometrically are described by a torus. Finally, for every 0 < d < π/4 the point d corresponds to a disjoint pair of torus. We also show that if a state is d units away from the maximally entangled states, then its entanglement entropy is(1−sin 2d) −1+sin 2d . Finally, we also show how this geometrical interpretation allows to clearly see that any pair of two-qubit states with real amplitudes can be connected with a circuit that only has single-qubit gates and one controlled-Z gate.

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“…For bipartite mixed states the situation is settled for qubits with Wootters' concurrence [10,11]. As the partition number and level structure of the system increases there become increasingly many (in fact exponentially so) different types of entanglement under local unitary operations [12][13][14][15][16][17]. The number of proposed entanglement measures has likewise proliferated [18][19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

Calculation of the convex roof for an open entangled harmonic oscillator system

Landau

Stroud

2010

Phys. Rev. A

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We explicitly calculate the time dependence of entanglement via the convex roof extension for a system of noninteracting harmonic oscillators. These oscillators interact only indirectly with each other by way of a zero-temperature bath. The initial state of the oscillators is taken to be that of an entangled Schrödinger-cat state. This type of initial condition leads to superexponential decay of the entanglement when the initial state has the same symmetry as the interaction Hamiltonian.

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Topology of the three-qubit space of entanglement types

Cited by 16 publications

References 17 publications

Maximum stabilizer dimension for nonproduct states

Maximum stabilizer dimension for nonproduct states

Entanglement types for two-qubit states with real amplitudes

Calculation of the convex roof for an open entangled harmonic oscillator system

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