Topological phase structure of entangled qudits

Khoury, A. Z.; Oxman, L. E.

doi:10.1103/physreva.89.032106

Cited by 16 publications

(17 citation statements)

References 47 publications

(71 reference statements)

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“…For SU(d) transformations the dynamical phase vanishes identically. This has been demonstrated in three other previous theoretical works [19,20,24]. Fractional topological phases are observed through polarization-controlled two-photon interference [26,27].…”

supporting

confidence: 70%

“…Mukunda and Simon added an important contribution to the understanding of geometric phases in terms of a kinematic approach [28]. When applied to a pair of d-dimensional entangled systems (qudits A and B), following a cyclic evolution under local unitary operations, this approach allows the derivation of a general formula, N), where C is the I-concurrence, C m its maximum value (C m = 2(d − 1)/d) and Φ A(B) is a phase contribution dependent on the whole evolution history of qudit A(B) [19,20]. For example, for qubits, this phase contribution encompasses the usual Bloch sphere solid angle.…”

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confidence: 99%

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Experimental observation of fractional topological phases with photonic qudits

et al. 2016

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Geometrical and topological phases play a fundamental role in quantum theory. Geometric phases have been proposed as a tool for implementing unitary gates for quantum computation. A fractional topological phase has been recently discovered for bipartite systems. The dimension of the Hilbert space determines the topological phase of entangled qudits under local unitary operations. Here we investigate fractional topological phases acquired by photonic entangled qudits. Photon pairs prepared as spatial qudits are operated inside a Sagnac interferometer and the two-photon interference pattern reveals the topological phase as fringes shifts when local operations are performed. Dimensions d = 2, 3 and 4 were tested, showing the expected theoretical values.Geometrical phases were introduced long ago in a seminal work on polarization transformations in classical optics [1,2]. Later, geometrical and topological features were demonstrated to play a fundamental role in the realm of quantum theory [3,4]. Tomita and Chiao verified experimentally the existence of Berry's phase and its topological properties at the classical level in helically wound optical fiber [5]. Geometric phases were measured by using coincidence detection of photon pairs produced in parametric down-conversion in conjunction with a Michelson interferometer [6,7], by changing adiabatically the polarization state of the photon pairs [8], with single photons in a mixed state of polarization by using a Mach-Zenhder interferometer [9] or in a polarization pure state by using a polarimetric technique [10]. More recently, they have been proposed as a robust tool for implementing unitary gates for quantum computation [11,12]. From this perspective, the geometric phase on entangled bipartite systems and the role of entanglement in its topological nature have been discussed both theoretically [13][14][15] and experimentally [16][17][18] for twoqubit systems. Later, they were generalized to pairs of qudits of any dimension [19,20] and to multiple qubits [21], showing that the dimension of the Hilbert space plays a crucial role in determining fractional topological phases in both cases. A recent review on this rich subject can be found in Ref. [22]. Although experimental schemes for demonstrating these fractional values have been proposed [23, 24], they have not been implemented so far. In a broader context, fractional phases may be revealed in quantum Hall systems, related to different homotopy classes in the configuration space of anyons. This nontrivial topology has been conjectured to be a possible resource for fault tolerant quantum computation [25].In this work we present experimental results of fractional topological phase measurements on entangled qu-dits. A qudit is a quantum state that belongs to a d-dimensional Hilbert space (d > 2). A quantum system in a general qudit state can be written in terms of d states that form a basis in the d-dimensional Hilbert space. Photonic qudits with dimensions d = 3 and 4 and qubits (d = 2) were encoded on the transverse ...

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confidence: 99%

Experimental observation of fractional topological phases with photonic qudits

et al. 2016

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“…The topological two‐qubit phases have been generalized by Oxman and Khoury to pairs of d ‐level systems (qudits) and pairs of systems with different dimension . For equal dimension, they found the allowed phase factors to be the d th roots of unity

e^{i f} = e^{i q (2 π / d)}, q = 0, 1, \dots, d - 1,

and being related to topological properties of SU( d ).…”

Section: Entanglementmentioning

confidence: 99%

Geometric phases in quantum information

Sjöqvist

2015

Int J of Quantum Chemistry

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The rise of quantum information science has opened up a new venue for applications of the geometric phase (GP), as well as triggered new insights into its physical, mathematical, and conceptual nature. Here, we review this development by focusing on three main themes: the use of GPs to perform robust quantum computation, the development of GP concepts for mixed quantum states, and the discovery of a new type of topological phases for entangled quantum systems. We delineate the theoretical development as well as describe recent experiments related to GPs in the context of quantum information.

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“…The former corresponds to the metaplectic group of unitary operations, M S , generated by quadratic Hamiltonians in the position and momentum canonical conjugate operators [22,59], which are characterized by a symplectic matrix S [59]. The second tool is the total phase acquired by the Gaussian stateρ G through the evolution, given by [60]:…”

Section: Introductionmentioning

confidence: 99%

Unified framework to determine Gaussian states in continuous-variable systems

et al. 2017

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Gaussian states are the backbone of quantum information protocols with continuous variable systems, whose power relies fundamentally on the entanglement between the different modes. In the case of global pure states, knowledge of the reduced states in a given bipartition of a multipartite quantum system bears information on the entanglement in such bipartition. For Gaussian states, the reduced states are also Gaussian, so there determination requires essentially the experimental determination of their covariance matrix. Here, we develop strategies to determine the covariance matrix of an arbitrary n−mode bosonic Gaussian state through measurement of the total phase acquired when appropriate metaplectic evolutions, associated with quadratic Hamiltonians, are applied. Simply one-mode metaplectic evolutions, such rotations, squeezing and shear transformations, in addition to a single two-mode rotation, allows to determine all the covariance matrix elements of a n−mode bosonic system. All the single-mode metaplectic evolutions are applied conditionally to a state in which an ancilla qubit is entangled with the n-mode system. The ancillary system provides, after measurement, the value of the total phase of each evolution. The proposed method is experimentally friendly to be implemented in the most currently used continuous variable systems.

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Topological phase structure of entangled qudits

Cited by 16 publications

References 47 publications

Experimental observation of fractional topological phases with photonic qudits

Experimental observation of fractional topological phases with photonic qudits

Geometric phases in quantum information

Unified framework to determine Gaussian states in continuous-variable systems

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