Geometry of entangled states, Bloch spheres and Hopf fibrations

Mosseri, Rémy; Dandoloff, Rossen

doi:10.1088/0305-4470/34/47/324

Cited by 169 publications

(301 citation statements)

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“…For pure states, the concurrence, which is an established measure of entanglement, is C = 2|α 1 α 4 −α 2 α 3 | [20]. (The separability criterion α 1 α 4 − α 2 α 3 = 0 has also been derived from a Hilbert space geometrical viewpoint in [33].) Our measure Γ, on the other hand, is Γ = 2||α 1 α * 4 | − |α 2 α * 3 || ≤ C. Equality is only achieved if either the complex vectors α 1 α 4 and α 2 α 3 are parallel, or if at least one of the coefficients α 1 , .…”

Section: An Experimental Implementationmentioning

confidence: 99%

Measurable entanglement criterion for two qubits

2003

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We propose a directly measurable criterion for the entanglement of two qubits. We compare the criterion with other criteria, and we find that for pure states, and some mixed states, it coincides with the state's concurrence. The measure can be obtained with a Bell state analyser and the ability to make general local unitary transformations. However, the procedure fails to measure the entanglement of a general mixed two-qubit state.

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Section: An Experimental Implementationmentioning

confidence: 99%

Measurable entanglement criterion for two qubits

2003

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“…The role of entanglement in the phase evolution of two-qubit systems was investigated in refs. [13,14], and the topological nature of the corresponding geometric phases was investigated both theoretical [15][16][17] and experimentally in the context of spin-orbit transformations on a paraxial laser beam [18] and in nuclear magnetic resonance [19]. In a recent work, we investigated the crucial role played by the dimension of the Hilbert space on the topological phases acquired by entangled qudits [20,21].…”

Section: Introductionmentioning

confidence: 99%

Topological phase structure of entangled qudits

Khoury¹,

Oxman²

2014

Phys. Rev. A

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We discuss the appearance of fractional topological phases on cyclic evolutions of entangled qudits. The original result reported in Phys. Rev. Lett. 106, 240503 (2011) is detailed and extended to qudits of different dimensions. The topological nature of the phase evolution and its restriction to fractional values are related to both the structure of the projective space of states and entanglement. For maximally entangled states of qudits with the same Hilbert space dimension, the fractional geometric phases are the only ones attainable under local SU(d) operations, an effect that can be experimentally observed through conditional interference.

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“…It can be shown that in that case, the π phase also occur. However, the relation between non MES and the SO(3) group is more complex [6], as well as the geometrical interpretation of their time evolution. …”

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Topological Phase for Entangled Two-Qubit States

Milman¹,

Mosseri²

2003

Phys. Rev. Lett.

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103

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We present an unambiguous characterization of the rotation group SO(3) biconnectedness topology using two-qubit maximally entangled states. We show how to generate cyclic evolutions of these states, which are in one-to-one correspondence to closed paths in SO(3). The difference between the well known two classes of such paths translates into the gain of a global phase of π for one class and no phase change for the other. We propose a simple quantum optics interference experiment to demonstrate this topological phase shift. [2]. In addition, it is known that any quantum computing protocol involving many qubits can be implemented by concatenation of one or two-qubit gates [2]. This is why much attention has been paid to two-qubit systems, their properties and characterizations.In the present work, we intend to show that two-qubit MES can also be used to display, even experimentally, the well known double-connectedness of the SO(3) rotation group. Since the latter is often evoked to explain the minus sign multiplying a spin 1/2 state after a 2π rotation, we shall first argue that this is to some respect an ambiguous statement. Indeed, take one qubit state |Ψ(t = 0) = α |0 + β |1 subject to the Hamiltonian H =h ω 2σ z (an effective magnetic field along the z axis). At time t, the state reads |Ψ(t) = e −iωt 2 α |0 + e iωt 2 β |1 . Let us turn to the Bloch sphere representation of |Ψ(t) . The three coordinates are given by the expectation values of the Pauli matrices. As is well known, the expectation value of the spin precesses around the effective magnetic field. At t = 2π/ω it has experienced a full 2π rotation, and a π phase for the wave function. The π phase has been clearly demonstrated on several systems, starting with beautiful experiment on spin 1/2 neutrons [3]. On general grounds, the change of global phase γ under a cycle decomposes into its dynamical part, γ d (as derived from the time dependent Schrödinger equation) and its geometrical part, the Berry phase γ g , whose origin relies on the Hilbert space peculiar geometry. The latter phase, γ g , equals half of the area bounded by the representing part of the Bloch sphere [4]. The dynamical phase γ d is simply related to the time average of the Hamiltonian.In the above case of one qubit precessing in a magnetic field, we have γ d = −π cos θ and γ g = −π(1 − cos θ), so the expected global phase γ = γ d + γ g = −π for an initial state with |α| = cos θ/2 and |β| = sin θ/2. Let us remark here that this π phase is present even for θ = 0, where it is of pure dynamical nature (with no precession and therefore no rotation at all). We may therefore question the geometrical interpretation of the π phase. It would be more correct to state that this phase has a geometrical and a dynamical component, being purely geometrical only when θ = π/2, when the initial state is orthogonal to the magnetic field. However, even in that case, we find problems in relating the phase to the double connectedness of SO(3). Indeed, the latter property relates paths on the SO(3) manifol...

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Geometry of entangled states, Bloch spheres and Hopf fibrations

Cited by 169 publications

References 5 publications

Measurable entanglement criterion for two qubits

Measurable entanglement criterion for two qubits

Topological phase structure of entangled qudits

Topological Phase for Entangled Two-Qubit States

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