Optimal entangled coherent states in lossy quantum-enhanced metrology

An entangled two-mode coherent state is studied within the framework of $2\times 2$ dimensional Hilbert space. An entanglement concentration scheme based on joint Bell-state measurements is worked out. When the entangled coherent state is embedded in vacuum environment, its entanglement is degraded but not totally lost. It is found that the larger the initial coherent amplitude, the faster entanglement decreases. We investigate a scheme to teleport a coherent superposition state while considering a mixed quantum channel. We find that the decohered entangled coherent state may be useless for quantum teleportation as it gives the optimal fidelity of teleportation less than the classical limit 2/3.

By introducing a four-mode unitary operator U = exp[−iλ(X 1 P 2 + X 2 P 3 + X 3 P 4 + X 4 P 1)], we show how a four-mode coherent entangled state can be generated by entangling a two bipartite coherent entangled state. The corresponding squeezed vacuum state U|0000⟩ in four-mode Fock space is derived by virtue of the technique of integration within ordered production of operators, which exhibits the standard squeezing for the four-mode quadratures. A new ideal quantum mechanical representation |α, β, γ⟩ is constructed from U|0000⟩ in the limit of infinite squeezing, which possesses the properties of both coherent and entangled states. The entanglement involved in |α, β, γ⟩ is explained. A scheme for generating |α, β, γ⟩ is presented.

Based on the coherent entangled state |α,x⟩ we introduce the squeezed entangled state (SES). Then wepropose a teleportation protocol for the SES by usingEinstein–Podolsky–Rosen entangled state |η⟩as a quantum channel. The calculation is greatly simplified by virtue of theSchmidt decompositions of both |α,x⟩ and|η⟩. Any bipartite states that can be expandedin terms of |α,x⟩ may be teleported in thisway due to the completeness of |α,x⟩.

We analyze entanglement properties of entangled coherent state (ECS), |α, 0〉1,2 + |0, α〉1,2, with and without photon losses. By separating the coherent state into , we derive exact results of the logarithmic negativity EN, which quantifies the degree of entanglement between the two bosonic modes. Without particle losses, EN = 1 for the N00N state; while for the ECS, EN increases from 0 to 1 as |α|2 → ∞. In the presence of photon losses, we find that the ECS with large enough photon number is more robust than that of the N00N state. An optimal ECS is obtained by maximizing EN with respect to |α|2.

A new bipartite coherent-entangled state is introduced in the two-mode Fock space, which exhibits the properties of both a coherent state and an entangled state. The set of coherent-entangled states makes up a complete and partly nonorthogonal representation. A simple experimental scheme to produce the coherent-entangled state using an asymmetric beamsplitter is proposed. Some applications of the coherent-entangled state in quantum optics are also presented.

Quantum discord measures the fraction of the pair-wise mutual information that is locally inaccessible in a multipartite system. Nonzero quantum discord has interesting and significant applications because although non-zero entanglement guarantees the existence of quantum correlation in a bipartite quantum system, zero entanglement does not guarantee the absence of a quantum correlation. On the other hand, many quantum optics systems can be described as deformed quantum oscillators. In this work, we investigate the quantum discord of bipartite entangled nonlinear coherent states, in the context of the so-called f-deformed coherent states algebra. To calculate the quantum discord, we consider quasi- Werner mixed states bases on bipartite entangled f-deformed coherent states. Two explicit analytic expressions are derived for the quantum discord of two different nonlinear deformed coherent states. The first one considers deformed coherent states obtained as eigenstates of the annihilation deformed operator, and the second one is obtained by using a deformed displacement operator. We compare the quantum discord of those states, when the nonlinear deformation function is either associated with the SU(1,1) coherent states in the Gilmore-Perelomov or Barut-Girardello representations, respectively.

We describe a nonlocal method for generating entangled coherent states of a two-mode field wherein the field modes never meet. The proposed method is an extension of an earlier proposal [C. C. Gerry, Phys. Rev. A 59, 4095 (1999)] for the generation of superpositions of coherent states. A single photon injected into a Mach-Zehnder interferometer with cross-Kerr media in both arms coupling with two external fields in coherent states produces entangled coherent states upon detection at one of the output ports. We point out that our proposal can be alternatively viewed as a ``which path'' experiment, and in the case of only one external field, we describe the implementation of a quantum eraser.

Following the approach of Solano et al (2003 Phys. Rev. Lett. 90 027903) we propose a scheme for a generation of a few classes of entangled (nonlinear) coherent states. To achieve this purpose, the interaction of a spatially narrow collection of two-level atoms with a quantized field in a high-Q factor cavity in the presence of a strong-driving classical field is studied. We perform appropriate Hamiltonians describing the atom-field interaction by focusing on two particular forms of intensity-dependent functions which are directly related to su(1, 1) and su(2) Lie algebras. It is shown that the dynamical evolution of the considered systems can generate bipartite, tripartite (nonlinear) and more complicated entangled states corresponding to the mentioned groups depending on the number of atoms in the cavity. In the processes of the abovementioned generation schemes, even and odd nonlinear coherent states are produced. In the end, in a particular circumstance with the two-mode quantized field we can turn easily from Jaynes–Cummings to anti-Jaynes–Cummings interactions which brings us to the maximally entangled number state. Finally, to quantify the degree of entanglement of the produced states, the measures of von Neumann and linear entropies are applied.

A scheme of teleporting a superposition of coherent states |α⟩ and | − α⟩ using a beam splitter and two phase shifters was proposed by van Enk and Hirota (2001 Phys. Rev. A 64 022313). The authors concluded that the probability for successful teleportation is 1/2. In this paper, it is shown that the authors' scheme can be altered slightly so as to obtain an almost perfect teleportation for an appreciable value of |α|2. For |α|2 = 5, the minimum of average fidelity, which is the minimum of the sum of the product of probability of occurrence of any case, and the corresponding fidelity is less than 1 by a quantity ∼10−4. We also discuss the effect of decoherence on teleportation fidelity. We find that if no photons are counted in both final outputs, the minimum assured fidelity is still non-zero except when there is no decoherence and the information is an even coherent state. For non-zero photon counts, minimum assured fidelity decreases with an increase in |α|2 for low noise. For high noise, however, it increases, attains a maximum value and then decreases with |α|2. The average fidelity depends appreciably on the information for low values of |α|2 only.

In this paper, we propose an optical scheme to generate four-mode W-type entangled coherent states (ECSs) in free travelling optical fields by using a single-photon source, coherent state sources, beam splitters, photodetectors and cross-Kerr media. Our scheme consists of a symmetric optical setup in which there are two Mach–Zehnder interferometers in which each arm contains a cross-Kerr medium. We calculate the average fidelity of generated W-type ECSs, the success probability of producing four-mode W-type ECSs and the average fidelity of the generated four-mode W-type ECSs. It is indicated that our protocol is nearly deterministic.

We study entangling and disentangling functions of optical Fourier multiport devices in which input-output relation for the creation and annihilation operators is given by a finite Fourier transform. It is shown that these Fourier multiport devices can act as entanglement converters which can not only create entanglement from an unentangled state at the input but also destroy entanglement in an entangled state at the input. Creation and destruction of two-mode and three-mode entangled coherent states (ECSs) are investigated in detail. The creation and destruction of Bell-type two-mode ECS, GHZ-type and W-type three-mode ECSs are indicated explicitly through using Fourier four-port and six-port devices, respectively.

We describe a one-step, deterministic and scalable scheme for creating macroscopic arbitrary entangled coherent states (ECSs) of separate nitrogen-vacancy center ensembles (NVEs) that couple to a superconducting flux qubit. We discuss how to generate the entangled states between the flux qubit and two NVEs by the resonant driving. Then the ECSs of the NVEs can be obtained by projecting the flux qubit, and the entanglement detection can be realized by transferring the quantum state from the NVEs to the flux qubit. Our numerical simulation shows that even under current experimental parameters the concurrence of the ECSs can approach unity. We emphasize that this method is straightforwardly extendable to the case of many NVEs.

We propose a scheme for generation of three-mode W-type entangled coherent states (ECSs) in free-travelling optical fields by using a single-photon source, coherent state sources, beam splitters, photodetectors, and three-mode cross-Kerr media. The scheme consists of a Mach-Zehnder interferometer (MZI) in which each arm contains a cross-Kerr medium. We calculate the success probability of the generated W-type ECSs, and the total success probability is unity under the ideal conditions.

The effect of channel decoherence on entangled coherent states: A theoretical analysis

The quantum Fisher information (QFI) of certain multipartite entangled quantum states is larger than what is reachable by separable states, providing a metrological advantage. Are these nonclassical correlations strong enough to potentially violate a Bell inequality? Here, we present evidence from two examples. First, we discuss a Bell inequality designed for spin-squeezed states which is violated only by quantum states with a large QFI. Second, we relax a well-known lower bound on the QFI to find the Mermin Bell inequality as a special case. However, a fully general link between QFI and Bell correlations is still open.