We investigate the creation of squeezing via operations subject to noise and losses and ask for the optimal use of such devices when supplemented by noiseless passive operations. Both single and repeated uses of the device are optimized analytically and it is proven that in the latter case the squeezing converges exponentially fast to its asymptotic optimum, which we determine explicitly. For the case of multiple iterations we show that the optimum can be achieved with fixed intermediate passive operations. Finally, we relate the results to the generation of entanglement and derive the maximal two-mode entanglement achievable within the considered scenario.
We investigate the creation of squeezing via operations subject to noise and losses and ask for the optimal use of such devices when supplemented by noiseless passive operations. Both single and repeated uses of the device are optimized analytically and it is proven that in the latter case the squeezing converges exponentially fast to its asymptotic optimum, which we determine explicitly. For the case of multiple iterations we show that the optimum can be achieved with fixed intermediate passive operations. Finally, we relate the results to the generation of entanglement and derive the maximal two-mode entanglement achievable within the considered scenario.
We investigate the creation of squeezing via operations subject to noise and losses and ask for the optimal use of such devices when supplemented by noiseless passive operations. Both, single and repeated uses of the device are optimized and it is proven that in the latter case the squeezing converges exponentially fast to its asymptotic optimum, which we determine explicitly. For the case of multiple iterations we show that the optimum can be achieved with fixed intermediate passive operations. Finally, we relate the results to the generation of entanglement and derive the maximal two-mode entanglement achievable within the considered scenario.
Loss and noise limit the use of light to distribute entanglement, transmit quantum information, and establish secret keys. Gaussian states have a compact description in terms of means and covariance matrices, but their entanglement still requires global optimization. We develop a framework for computing Gaussian-restricted versions of the relative entropy of entanglement and Rains entanglement. An integral representation reveals a convex geometry under matrix geometric means and reduces Gaussian Rains entanglement to relative entropy minimized over Gaussian states with positive partial transpose. We obtain unique optimal reference states and additivity for independent Gaussian input states. Our convex-majorization algorithm has an explicit geometric convergence rate, uniform in the faithful regularization parameter and valid for pure input states. Controlled numerical integration and approximate convex subproblem solutions give a bit-complexity bound that is polynomial in accuracy bits for each fixed rational input covariance matrix, with explicit dependence on covariance size. These calculations bound extractable entanglement and, through the relative entropy of entanglement, extractable secret key. Teleportation simulation, energy allocation, and a convex optimization of passive-basis loss cores give bounds on quantum and private communication under an input-energy constraint, allowing arbitrary two-way classically assisted protocols. Two- and three-mode examples compare the bounds with achievable rates, including matching pure-loss benchmarks. Finally, we separate the Gaussian-restricted quantities from their unrestricted counterparts and give a photon-number truncation method with controlled error for computing the latter.
Single-photon-level non-Gaussian operations—photon addition, photon subtraction, and their coherent superposition—are powerful tools with which to increase entanglement in continuous-variable optical states. Although such operations are typically not deterministic, there may be other advantages such as noiseless manipulation. Therefore, to fully account for the efficacy of a particular non-Gaussian operation in a practical scenario, we develop figures of merit which trade off the advantages of such protocols against their success probability. Specifically, we define 'entanglement enhancement rate' as the increase in entanglement per trial of a generic non-Gaussian operation on a two-mode squeezed vacuum (TMSV) state. We consider states generated by photon-subtraction, photon-addition and a coherent superposition of subtraction and addition. We compare each strategy when applied to one or both modes of a TMSV state, and also in the presence of channel losses prior to the operation. In many cases, additional properties are analytically calculable, including excess noise arising from the operation and the fidelity of the resulting states to particular Gaussian and non-Gaussian states. Finally, by incorporating loss, we derive optimal interaction parameters for each non-Gaussian operation which maximize the effectiveness of the particular protocol under investigation.
We investigate the entangling capability of passive optical elements, both qualitatively and quantitatively. We present a general necessary and sufficient condition for the possibility of creating distillable entanglement in an arbitrary multimode Gaussian state with the help of passive optical elements, thereby establishing a general connection between squeezing and the entanglement that is attainable by nonsqueezing operations. Special attention is devoted to general two-mode Gaussian states, for which we provide the optimal entangling procedure, present an explicit formula for the attainable degree of entanglement, and discuss several practically important special cases.
We investigate the deterministic generation and distribution of entanglement in large quantum networks by driving distant qubits with the output fields of a nondegenerate parametric amplifier. In this setting, the amplifier produces a continuous Gaussian two-mode squeezed state, which acts as a quantum-correlated reservoir for the qubits and relaxes them into a highly entangled steady state. Here we are interested in the maximal amount of entanglement and the optimal entanglement generation rates that can be achieved with this scheme under realistic conditions taking, in particular, the finite amplifier bandwidth, waveguide losses, and propagation delays into account. By combining exact numerical simulations of the full network with approximate analytic results, we predict the optimal working point for the amplifier and the corresponding qubit-qubit entanglement under various conditions. Our findings show that this passive conversion of Gaussian into discrete-variable entanglement offers a robust and experimentally very attractive approach for operating large optical, microwave, or hybrid quantum networks, for which efficient parametric amplifiers are currently developed.
We quantify the maximum amount of entanglement of formation (EoF) that can be achieved by continuous-variable states under passive operations, which we refer to as the EoF potential. Focusing, in particular, on two-mode Gaussian states we derive analytical expressions for the EoF potential for specific classes of states. For more general states, we demonstrate that this quantity can be upper bounded by the minimum amount of squeezing needed to synthesize the Gaussian modes, a quantity called squeezing of formation. Our work, thus, provides a link between nonclassicality of quantum states and the nonclassicality of correlations.