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Communications in Theoretical Physics

Fringe visibility for two qubits interacting with a macroscopic medium*

Yu-Heng Li, Yanjun Liu, Jing Lu, Lan Zhou

Published Mar 3, 2020• 0 citations
Publisher

Abstract

Abstract We study both the two-particle and single-particle fringe visibility in the generalized version of the Nakazato–Pascazio model where two qubits interact with a finite length one-dimensional array. Both the two-particle and single-particle fringe visibilities are investigated with different initial states of the particles spin. For different initial states of the particles spin, the two-particle fringe visibility either decreases or increases over time, and may even decrease first and increase later. Due to the interaction between the particles and the one-dimensional array, the single-particle fringe visibility increases over time when the initial state of the two particles spin is independent. The single-particle fringe visibility is equal to 0 as the two-particle spin is initially in the completely entangled state or in the singlet state.

DOI
10.1088/1572-9494/ab6905

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Abstract We study both the two-particle and single-particle fringe visibility in the generalized version of the Nakazato–Pascazio model where two qubits interact with a finite length one-dimensional array. Both the two-particle and single-particle fringe visibilities are investigated with different initial states of the particles spin. For different initial states of the particles spin, the two-particle fringe visibility either decreases or increases over time, and may even decrease first and increase later. Due to the interaction between the particles and the one-dimensional array, the single-particle fringe visibility increases over time when the initial state of the two particles spin is independent. The single-particle fringe visibility is equal to 0 as the two-particle spin is initially in the completely entangled state or in the singlet state.
Quantum-mechanical wave-particle duality is quantified in terms of a trade-off relation between the fringe visibility and the which-way distinguishability in an interference experiment. This relation was recently generalized by Banaszek et al. [Nat. Commun. 4, 2594 (2013)] when the particle is equipped with an internal degree of freedom such as spin. Here, we extend the visibility-distinguishability trade-off relation to quantum interference of a particle possessing an internal degree of freedom, when the which-way detector state is entangled with an ancillary system. We introduce an extended which-way distinguishability DE{\mathcal{D}}_{E}DE​ and the associated extended fringe visibility VE{\mathcal{V}}_{E}VE​, satisfying the inequality ${\mathcal{D}}_{E}^{2}+{\mathcal{V}}_{E}^{2}\ensuremath{\le}1$ in this scenario. We illustrate, with the help of three specific examples, that while the which-way information inferred solely from the detector state (without ancilla) vanishes, the extended distinguishability retrievable via measurements on the detector-ancilla entangled state is nonzero. Furthermore, in all the three examples, the extended visibility and the generalized visibility (which was introduced by Banaszek et al.) match identically with each other.
In a two-particle interferometer, one can study the variation of both single- and joint-detection probabilities as functions of the phase shifts of the beams. By combining the usual definition for one-particle fringe visibility vi{\mathit{v}}_{\mathit{i}}vi​ (i=1,2) with a reasonable proposed definition for two-particle fringe visibility v12{\mathit{v}}_{12}v12​, we show that vi2{\mathit{v}}_{\mathit{i}}^{2}vi2​+v122{\mathit{v}}_{12}^{2}v122​\ensuremath{\le}1 or, equivalently, vi{\mathit{v}}_{\mathit{i}}vi​v12{\mathit{v}}_{12}v12​\ensuremath{\le}1/2. Some extensions are discussed.
We study the fringe visibility and the distinguishability of a general Mach–Zehnder interferometer with an asymmetric beam splitter. Both the fringe visibility V and the distinguishability D are affected by the input state of the particle characterized by the Bloch vector S = ( S x , S y , S z ) and the second asymmetric beam splitter characterized by the paramter β . For the total system is initially in a pure state, it is found that the fringe visibility reaches the upper bound and the distinguishability reaches the lower bound when cos β = − S x . The fringe visibility obtain the maximum only if S x = 0 and β = π /2 when the input particle is initially in a mixed state. The complementary relationship V 2 + D 2 ≤ 1 is proved in a general Mach–Zehnder interferometer with an asymmetric beam splitter, and the conditions for the equality are also presented.
Motivated by the Bose et al.-Matletto-Vedral (BMV) proposal for detecting quantum superposition of spacetime geometries, we study a toy model of a quantum entanglement generation between two spins (qubits) mediated by a relativistic free scalar field. After time evolution, spin correlation is generated through the interactions with the field. Because of the associated particle creation into an open system, the quantum state of spins is partially decohered. In this paper, we give a comprehensive study of the model based on the closed time path formalism, focusing on relativistic causality and quantum mechanical complementarity. We calculate various quantities such as spin correlations, entanglement entropies, mutual information and negativity, and study their behaviors in various limiting situations. In particular, we calculate the mutual information of the two spins and compare it with spin correlation functions. In some cases, its inequality relation improves the trade-off relation between the fringe visibility and the distinguishability (which-way information) by giving a stronger upper bound on the visibility. We also discuss why no quantum entanglement can be generated unless both spins are causally affected by one another while spin correlations are generated.
The interaction between an ultrarelativistic particle and a linear array made up of N two-level systems (AgBr molecules) is studied by making use of a modified version of the Coleman-Hepp Hamiltonian. Energy-exchange processes between the particle and the molecules are properly taken into account, and the evolution of the total system is calculated exactly both when the array is initially in the ground state and in a thermal state. In the weak-coupling, macroscopic (N\ensuremath{\rightarrow}\ensuremath{\infty}) limit, the system remains solvable and leads to interesting connections with the Jaynes-Cummings model, which describes the interaction of a particle with a maser. The visibility of the interference pattern produced by the two branch waves of the particle is computed, and the conditions under which the spin array behaves as a ``detector'' are investigated. The behavior of the visibility yields good insights into the issue of quantum measurements: It is found that, in the N\ensuremath{\rightarrow}\ensuremath{\infty} limit, a superselection-rule space appears in the description of the (macroscopic) apparatus. In general, an initial thermal state of the ``detector'' provokes a more substantial loss of quantum coherence than an initial ground state. It is argued that a system increasingly loses coherence as the temperature of the detector increases. The problem of ``imperfect measurements'' is also briefly discussed.