The influence of losses in the interferometric generation and the transmission of continuous-variable entangled light is studied, with special emphasis on Gaussian states. Based on the theory of quantum-state transformation at absorbing dielectric devices, the amount of entanglement is quantified by means of the relative-entropy measure. Upper bounds of entanglement and the distance to the set of separable Gaussian states are calculated. Compared with the distance measure, the bounds can substantially overestimate the entanglement. In particular, they do not show the drastic decrease of entanglement with increasing mean photon number, as does the distance measure.
Security of passive entanglement-based key distribution protocols
Network Innovation Laboratories, NTT, Inc.（NTT网络创新实验室）；NTT Research Center for Theoretical Quantum Information, NTT, Inc.（NTT理论量子信息研究中心）；Basic Research Laboratories, NTT, Inc.（NTT基础研究所）
In general, generation of entangled photon pairs and also the random choice of measurement basis can be implemented with passive optical devices in entanglement based quantum key distribution (QKD) system. However, auxiliary random numbers are still mandatory for conducting the privacy amplification procedure. In this paper, we propose a fully passive entanglement based QKD scheme in which certified random bits for privacy amplification are extracted from detection events of measurement-basis-mismatched entangled photon pairs. The privacy of these random bits can be certified by estimating the…
The distillability of bipartite entangled state as seen by moving observers has been investigated. It is found that the same initial entanglement for a state parameter $α$ and its "normalized partner" $\sqrt{1-α^2}$ will be degraded as seen by moving observer. It is shown that in the ultra relativistic limit, the state does not have distillable entanglement for any $α$.
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 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 new link between non-classicality of quantum states and …
Relations between photon scattering, entanglement and multi-mode detection are investigated. We first establish a general framework in which one- and two-photon elastic scattering processes can be discussed, then we focus on the study of the intrinsic entanglement degradation caused by a multi-mode detection. We show that any multi-mode scattered state cannot maximally violate the Bell-CHSH inequality because of the momentum spread. The results presented here have general validity and can be applied to both deterministic and random scattering processes.
The entanglement survival time is defined as the maximum time a system which is evolving under the action of local Markovian, homogenous in time noise, is capable to preserve the entanglement it had at the beginning of the temporal evolution. In this paper we study how this quantity is affected by the interplay between the coherent preserving and dissipative contributions of the corresponding dynamical generator. We report the presence of a counterintuitive, non-monotonic behaviour in such functional, capable of inducing sudden death of entanglement in models which, in the absence of unitary d…
We study the effect of decoherence on a qubit-qutrit system under the influence of global, local and multilocal decoherence in non-inertial frames. We show that the entanglement sudden death can be avoided in non-inertial frames in the presence of amplitude damping, depolarizing and phase damping channels. However, degradation of entanglement is seen due to Unruh effect. It is shown that for lower level of decoherence, the depolarizing channel degrades the entanglement more heavily as compared to the amplitude damping and phase damping channels. However, for higher values of decoherence parame…
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 multi-mode Gaussian state with the help of passive optical elements, thereby establishing a general connection between squeezing and the entanglement that is attainable by non-squeezing 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 attaina…
Although entropy is a necessary and sufficient quantity to characterize the order of work content for equal energetic (EE) states in the asymptotic limit, for the finite quantum systems, the relation is not so linear and requires detail investigation. Toward this, we have considered a resource theoretic framework taking the energy preserving operations (EPO) as free, to compare the amount of extractable work from two different quantum states. Under EPO, majorization becomes a necessary criterion for state transformation. It is also shown that the passive state energy is a concave function and …
The π-tangle is used to study the behavior of entanglement of a nonmaximal tripartite state of both Dirac and scalar fields in accelerated frame. For Dirac fields, the degree of degradation with acceleration of both one-tangle of accelerated observer and π-tangle, for the same initial entanglement, is different by just interchanging the values of probability amplitudes. A fraction of both one-tangles and the π-tangle always survives for any choice of acceleration and the degree of initial entanglement. For scalar field, the one-tangle of accelerated observer depends on the choice of values of …
The influence of losses in the transmission of continuous-variable entangled light through linear devices such as optical fibers is studied, with special emphasis on Gaussian states. Upper bounds on entanglement and the distance to the set of separable Gaussian states are calculated. Compared with the distance measure, the bounds can substantially overestimate the entanglement and thus do not show the drastic decrease of entanglement with increasing mean photon number, as does the distance measure. In particular, it shows that losses give rise to entanglement saturation, which principally limi…
We address the propagation of twin-beam of radiation through Gaussian phase-sensitive channels, i.e. noisy channels with squeezed fluctuations. We find that squeezing the environment always reduces the survival time of entanglement in comparison to the case of simple dissipation and thermal noise. We also show that the survival time is further reduced if the squeezing phase of the fluctuations is different from the twin-beam phase.
We present a completely localized solution to the problem of entanglement degradation in non-inertial frames. A two mode squeezed state is considered from the viewpoint of two observers, Alice (inertial) and Rob (accelerated), and a model of localized projective detection is used to study the amount of entanglement that they are able to extract from the initial state. The Unruh vacuum noise plays only a minor role in the degradation process. The dominant source of degradation is a mode mismatch between the mode of the squeezed state Rob observes and the mode he is able to detect from his accel…
Statistical Characterization of Entanglement Degradation Under Markovian Noise in Composite Quantum Systems
Scuola Normale Superiore（高等师范学院）；Center for Life Nano- & Neuro-Science, Italian Institute of Technology (IIT)（意大利理工学院生命纳米与神经科学中心）；Department of Physics, Harvard University（哈佛大学物理系）；Department of Mathematics, University of Bologna（博洛尼亚大学数学系）；NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR（NEST，高等师范学院和CNR纳米科学研究所）
The entanglement between two modes of the free scalar and Dirac fields as seen by two relatively accelerated observers has been investigated. It is found that the same initial entanglement for an initial state parameter $α$ and its "normalized partner" $\sqrt{1-α^{2}}$ will be degraded by the Unruh effect along two different trajectories except for the maximally entangled state, which just shows the inequivalence of the quantization for a free field in the Minkowski and Rindler coordinates. In the infinite acceleration limit the state doesn't have the distillable entanglement for any $α$ for t…
Quantum key distribution (QKD) based on entangled photon pairs holds the potential for repeater-based quantum networks connecting clients over long distance. We demonstrate long-distance entanglement distribution by means of polarisation-entangled photon pairs through two successive deployed 96 km-long telecommunications fibres in the same submarine cable. One photon of each pair was detected directly after the source, while the other travelled the fibre cable in both directions for a total distance of 192 km and attenuation of 48 dB. The observed two-photon Bell state exhibited a fidelity 85%…
A cost-effective QKD transmitter is evaluated in a 16km reach, 2:16-split PON and yields 5.10-7secure bits/pulse. Co-existence with 20 down-and 1 upstream channel is possible at low QBER degradation of 0.93% and 1.1%.
Applying the recently developed formalism of quantum-state transformation at absorbing dielectric four-port devices [L.~Knöll, S.~Scheel, E.~Schmidt, D.-G.~Welsch, and A.V.~Chizhov, Phys. Rev. A {\bf 59}, 4716 (1999)], we calculate the quantum state of the outgoing modes of a two-mode squeezed vacuum transmitted through optical fibers of given extinction coefficients. Using the Peres--Horodecki separability criterion for continuous variable systems [R.~Simon, Phys. Rev. Lett. {\bf 84}, 2726 (2000)], we compute the maximal length of transmission of a two-mode squeezed vacuum through an absorbin…
Entanglement between two free bosonic modes can be determined via detection of each mode by different observers and then observing the correlations between their measurements. We show that such entanglement is degraded as a function of time if one observer begins in a state of inertial motion but ends in a state of uniform acceleration while the other remains inertial. At late times we recover previously established results for observers in relative uniform acceleration.
We investigate the degree to which entanglement survives when a correlated pair of two-state systems are copied using either local or non-local processes. We show how the copying process degrades the entanglement, due to a residual correlation between the copied output and the copying machine (itself made of qubits).
We study entanglement degradation of two non-interacting qubits subject to independent baths with broadband spectra typical of solid state nanodevices. We obtain the analytic form of the concurrence in the presence of adiabatic noise for classes of entangled initial states presently achievable in experiments. We find that adiabatic (low frequency) noise affects entanglement reduction analogously to pure dephasing noise. Due to quantum (high frequency) noise, entanglement is totally lost in a state-dependent finite time. The possibility to implement on-chip both local and entangling operations …
We study the entanglement of families of Unruh modes in the Bell states $|Φ^\pm\rangle =1/\sqrt{2}(|00\rangle\pm|11\rangle)$ and $|Ψ^\pm\rangle=1/\sqrt{2}(|01\rangle\pm|10\rangle)$ shared by two accelerated observers and find fundamental differences in the robustness of entanglement against acceleration for these states. States $Ψ^\pm$ are entangled for all finite accelerations, whereas, due to the Unruh effect, states $Φ^\pm$ lose their entanglement for finite accelerations. This is true for Bell states of two bosonic modes, as well as for Bell states of a bosonic and a fermionic mode. Furthe…
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…
We incorporate active and passive quantum error-correcting techniques to protect a set of optical information modes of a continuous-variable quantum information system. Our method uses ancilla modes, entangled modes, and gauge modes (modes in a mixed state) to help correct errors on a set of information modes. A linear-optical encoding circuit consisting of offline squeezers, passive optical devices, feedforward control, conditional modulation, and homodyne measurements performs the encoding. The result is that we extend the entanglement-assisted operator stabilizer formalism for discrete vari…
Quantum number-path entanglement is a resource for super-sensitive quantum metrology and in particular provides for sub-shotnoise or even Heisenberg-limited sensitivity. However, such number-path entanglement has thought to have been resource intensive to create in the first place --- typically requiring either very strong nonlinearities, or nondeterministic preparation schemes with feed-forward, which are difficult to implement. Very recently, arising from the study of quantum random walks with multi-photon walkers, as well as the study of the computational complexity of passive linear optica…
In this short paper we compute the various bipartite quantum correlations in the presence of the $(4+n)$-dimensional Schwarzschild black hole. In particular, we focus on the $n$-dependence of various bosonic bipartite entanglements. For the case between Alice and Rob, where the former is free falling observer and the latter is at the near-horizon region, the quantum correlation is degraded compared to the case in the absence of the black hole. The degradation rate increases with decreasing $n$. We also compute the physically inaccessible correlations. It is found that there is no creation of q…
We analyze the entanglement degradation provoked by the Hawking effect in a bipartite system Alice-Rob when Rob is in the proximities of a Schwarzschild black hole while Alice is free falling into it. We will obtain the limit in which the tools imported from the Unruh entanglement degradation phenomenon can be used properly, keeping control on the approximation. As a result, we will be able to determine the degree of entanglement as a function of the distance of Rob to the event horizon, the mass of the black hole, and the frequency of Rob's entangled modes. By means of this analysis we will s…
We analyse the entanglement and the non-locality of a (1+1)-dimensional massless Dirac field confined to a cavity on a worldtube that consists of inertial and uniformly accelerated segments, for small accelerations but arbitrarily long travel times. The correlations between the accelerated field modes and the modes in an inertial reference cavity are periodic in the durations of the individual trajectory segments, and degradation of the correlations can be entirely avoided by fine-tuning the individual or relative durations of the segments. Analytic results for selected trajectories are presen…
We propose a scheme to investigate whether non-uniform motion degrades entanglement of a relativistic quantum field that is localised both in space and in time. For a Dirichlet scalar field in a cavity in Minkowski space, in small but freely-adjustable acceleration of finite but arbitrarily long duration, degradation of observable magnitude occurs for massless transverse quanta of optical wavelength at Earth gravity acceleration and for kaon mass quanta already at microgravity acceleration. We outline a space-based experiment for observing the effect and its gravitational counterpart.
The entanglement of different classes of initially entangled qubit pair is investigated in the presence of short laser pulses of rectangular and exponential shapes with either one or both qubits are excited. For the rectangular pulse, the detuning parameter protects the entanglement from degradation and controls its upper and lower bounds. We show that the upper bounds of entanglement decrease if both particles are excited, but the lower bounds are much better if only one particle is excited. The phenomena of entanglement degradation, sudden- death and -birth are shown for small initial time. …
We show that the multipulse application can suppress the degradation of the quantum entanglement with focusing on the concurrence, the degree of entanglement. By evaluating the time evolution of concurrence with a linearly interacting spin-boson model under pulse application, we find that the effectiveness of the multipulse control depends on the non-Markovian nature of the reservoir.
We investigate how entangled inertial Unruh-DeWitt detectors are affected by interaction with a quantum field using a nonperturbative method. Inertial detectors in a $(3+1)$-dimensional Minkowski spacetime with instantaneous switching ($δ$-switching) experience degradation of their initial entanglement as their coupling strength with a scalar field increases. Somewhat surprisingly, initially separable or weakly entangled detectors can extract mutual information from the vacuum. We also find that entanglement degradation is not reduced if communication via the field is possible; rather this onl…
We study the entanglement dynamics between two cavities when one of them is harmonically shaken in the context of quantum information theory. We find four different regimes depending on the frequency of the motion and the spectrum of the moving cavity. If the moving cavity is three dimensional only two modes inside get coupled and the entanglement can either degrade asymptotically with time or oscillate depending on the driving. On the other hand, if the cavity has an equidistant spectrum the entanglement can either vanish asymptotically if it is driven with its fundamental frequency or have a…
We present a procedure for confirming the generation of long strings of photons in an entangled (cluster) state that does not rely on complete state tomography and that works even at low collection efficiencies. The scheme has the added advantage of being passive - it does not require switching of optical elements to perform measurements in different bases for instance.
We analyze optimal schemes and also propose some practical schemes for the nonlocal conversion from two shared Bell pairs to four-qubit entangled states in optical quantum networks. In the analysis, we consider two-qubit operations as nonlocal operations and minimize the number of access to ancillary qubits as possible. First, we consider two-qubit unitary operations without using ancillary qubits and derive a necessary and sufficient condition for convertible states. Second, we consider nonlocal optical systems composed of passive linear optics and postselection. For the passive linear optica…
On the equivalence between squeezing and entanglement potential for two-mode Gaussian states
Australian National University（澳大利亚国立大学）；University of California, Los Angeles (UCLA)（加利福尼亚大学洛杉矶分校）；Agency for Science, Technology and Research (A*STAR)（科学技术研究局）；Nanyang Technological University（南洋理工大学）
Entanglement degradation in open quantum systems is reviewed in the Choi-Jamiołkowski representation of linear maps. In addition to physical processes of entanglement dissociation and entanglement annihilation, we consider quantum dynamics transforming arbitrary input states into those that remain positive under partial transpose (PPT-inducing channels). Such evolutions form a convex subset of distillation-prohibiting channels. A relation between the above channels and entanglement-binding ones is clarified. An example of the distillation-prohibiting map $Φ\otimesΦ$ is given, where $Φ$ is not …
We address the interaction of two Gaussian states interferring at a beam splitter and analyze the correlations exhibited by the resulting bipartite system. We demonstrate that nonlocal correlations (entanglement) arise if and only if the fidelity between the two input Gaussian states falls under a threshold value depending only on their purities and on the transmissivity of the beam splitter. Our result clarifies the role of squeezing as a prerequisite for entanglement and provide a tool to optimize the generation of entanglement by passive devices.
We study a limited set of optical circuits for creating near maximal polarisation entanglement {\em without} the usual large vacuum contribution. The optical circuits we consider involve passive interferometers, feed-forward detection, down-converters and squeezers. For input vacuum fields we find that the creation of maximal entanglement using such circuits is impossible when conditioned on two detected auxiliary photons. So far, there have been no experiments with more auxiliary photons. Thus, based on the minimum complexity of the circuits required, if near maximal polarisation entanglement…
We investigate the degradation of quantum entanglement in the Schwarzschild-de Sitter black hole spacetime, by studying the mutual information and the logarithmic negativity for maximally entangled, bipartite states for massless minimal scalar fields. This spacetime is endowed with a black hole as well as a cosmological event horizon, giving rise to particle creation at two different temperatures. We consider two independent descriptions of thermodynamics and particle creation in this background. The first involves thermal equilibrium of an observer with either of the horizons. We show that as…
We characterize theoretically and experimentally the degradation of polarization entanglement in a fiber-optic entanglement distribution system where one of the optical fibers is exposed to the effects of polarization mode dispersion (PMD). We show gradual reduction of entanglement with increasing PMD and find that the highest PMD tolerance is achieved when the bandwidth of the pump used to generate the entangled photons in a $χ^{(3)}$ process is approximately half the bandwidth of the quantum channels.
Several recent experiments have demonstrated the viability of a passive device that can generate spin-entangled currents in two separate leads. However, manipulation and measurement of individual flying qubits in a solid state system has yet to be achieved. This is particularly difficult when a macroscopic number of these indistinguishable qubits are present. In order to access such an entangled current resource, we therefore show how to use it to generate distributed, static entanglement. The spatial separation between the entangled static pair can be much higher than that achieved by only ex…
Entanglement degradation in regular and singular spacetimes
School of Science and Technology, University of Camerino（卡梅里诺大学理学院）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Perugia（意大利国家核物理研究所佩鲁伽分部）；Department of Nanoscale Science and Engineering, University at Albany SUNY（纽约州立大学奥尔巴尼分校纳米科学与工程系）；INAF - Osservatorio Astronomico di Brera（意大利国家天体物理研究所布雷拉天文台）；Al-Farabi Kazakh National University（阿尔法拉比哈萨克国立大学）
We implement an entanglement distribution network based on wavelength-multiplexing and optical switching for quantum communication applications. Using a high-brightness source based on spontaneous parametric down-conversion in periodically-poled lithium niobate waveguides, we generate polarisation entangled photon pairs with a broad spectrum covering the telecom wavelengths around 1550nm. The photon pairs have entanglement fidelities up to 99%, and are distributed via passive wavelength multiplexing in a static multi-user network. We furthermore demonstrate a possible network application in a …
We study steady-state continuous variable entanglement in a three-mode optomechanical system consisting of an active optical cavity (gain) coupled to a passive optical cavity (loss) supporting a mechanical mode. For a driving laser which is blue-detuned, we show that coupling between optical and mechanical modes is enhanced in the unbroken-$\mathcal{PT}$-symmetry regime. We analyze the stability and this shows that steady-state solutions are more stable in the gain and loss systems. We use these stable solutions to generate distant entanglement between the mechanical mode and the optical field…
