Entanglement-Enhanced Classical Communication on a Noisy Quantum Channel
Charles H. Bennett, Christopher A. Fuchs, John A. Smolin
备注：
10 pages, LaTeX, to appear in Proceedings of the 3rd International Workshop on Quantum Communication and Measurement; See also http://vesta.physics.ucla.edu/~smolin/
We consider the problem of trying to send a single classical bit through a noisy quantum channel when two transmissions through the channel are available as a resource. Classically, two transmissions add nothing to the receiver's capability of inferring the bit. In the quantum world, however, one has the possible further advantage of entangling the two transmissions. We demonstrate that, for certain noisy channels, such entangled transmissions enhance the receiver's capability of a correct inference.
The optimization of entanglement evolution for two-mode Gaussian pure states under one-side Gaussian map is studied. Even there isn't complete information about the one-side Gaussian noisy channel, one can still maximize the entanglement distribution by testing the channel with only two specific states.
The entanglement evolution of two qubits under local, single- and two- sided noisy channels is investigated. It is found that for all pure initial states, the entanglement under a one-sided noisy channel is completely determined by the maximal trace distance which is the main element to construct the measure of non-Markovianity. For the two-sided noisy channel case, when the qubits are initially prepared in a general class of states, no matter pure or mixed, the entanglement can be expressed as the products of initial entanglement and the channels' action on the maximally entangled state.
We use the noisy entanglement-assisted classical capacity formula [arXiv:1609.08592] to create a coherent-attack security framework for Gaussian two-way quantum key distribution protocols in the asymptotic region.
We report the experimental realization of entanglement localization which restores polarization entanglement completely redirected after a linear coupling with incoherent and noisy surrounding photon. The method, based only on measurements of the surrounding photon after the coupling and on post-selection, can localize the entanglement back to original systems for any linear coupling.
We give a proof that entanglement purification, even with noisy apparatus, is sufficient to disentangle an eavesdropper (Eve) from the communication channel. Our proof applies to all possible attacks (individual and coherent). Due to the quantum nature of the entanglement purification protocol, it is also possible to use the obtained quantum channel for secure transmission of quantum information.
We present a protocol in which two or more parties can share multipartite entanglement over noisy quantum channels. The protocol is based on the entanglement purification presented by Shor and Preskill [Phys. Rev. Lett. 85, 441 (2000)] and the quantum teleportation via an isotropic state. We show that a nearly perfect purification implies a nearly perfect sharing of multipartite entanglement between two parties so that the protocol can assure a faithful sharing of multipartite entanglement with Shor and Preskill's proof on the entanglement purification.
Two qubits in pure entangled states going through separate paths and interacting with their own individual environments will gradually lose their entanglement. Here we show that the entanglement change of a two-qubit state due to amplitude damping noises can be recovered by entanglement swapping. Some initial states can be asymptotically purified into maximally entangled states by iteratively using our protocol.
In this paper, we study entanglement dynamics of a two-qubit extended Werner-like state locally interacting with independent noisy channels, i.e., amplitude damping, phase damping and depolarizing channels. We show that the purity of initial entangled state has direct impacts on the entanglement robustness in each noisy channel. That is, if the initial entangled state is prepared in mixed instead of pure form, the state may exhibit entanglement sudden death (ESD) and/or be decreased for the critical probability at which the entanglement disappear.
We discuss the estimation of channel parameters for a noisy quantum channel - the so-called Pauli channel - using finite resources. It turns out that prior entanglement considerably enhances the fidelity of the estimation when we compare it to an estimation scheme based on separable quantum states.
Partial teleportation of entanglement is to teleport one particle of an entangled pair through a quantum channel. This is conceptually equivalent to quantum swapping. We consider the partial teleportation of entanglement in the noisy environment, employing the Werner-state representation of the noisy channel for the simplicity of calculation. To have the insight of the many-body teleportation, we introduce the measure of correlation information and study the transfer of the correlation information and entanglement. We find that the fidelity gets smaller as the initial-state is entangled more f…
We study entanglement dynamics of pure three-qubit Greenberger-Horne-Zeilinger-type (GHZ-type) entangled states when one, two or three qubits being subjected to general local noise. Employing a lower bound for three-qubit concurrence as an entanglement measure, we show that for some many-sided noisy channels the entanglement dynamics can be complete described by the evolution of the entangled states in single-sided channels.
The recently developed Kraus representation for bosonic Gaussian channels is employed to study analytically the robustness of non-Gaussian entanglement against evolution under noisy attenuator and amplifier environments, and compare it with the robustness of Gaussian entanglement. Our results show that some non-Gaussian states with one ebit of entanglement are more robust than all Gaussian states, even the ones with arbitrarily large entanglement, a conclusion of direct consequence to the recent conjecture by Allegra et al. [PRL, 105, 100503 (2010)].
We give a security proof of quantum cryptography based entirely on entanglement purification. Our proof applies to all possible attacks (individual and coherent). It implies the security of cryptographic keys distributed with the help of entanglement-based quantum repeaters. We prove the security of the obtained quantum channel which may not only be used for quantum key distribution, but also for secure, albeit noisy, transmission of quantum information.
We analyze the Pauli-channel estimation with mixed nonseparable states. It turns out that within a specific range entanglement can serve as a nonclassical resource. However, this range is rather small, that is entanglement is not very robust for this application. We further show that Werner states yield the best result of all Bell diagonal states with the same amount of entanglement.
A discussion on the robustness of continuous-variable entangled states under noisy environment is briefly given in direct relation to Phys. Rev. Lett. 105, 100503 (2010).
Entanglement distillation is a fundamental task in quantum information processing. It not only extracts entanglement out of corrupted systems but also leads to protecting systems of interest against intervention with environment. In this work, we consider a realistic scenario of entanglement distillation where noisy quantum operations are applied. In particular, the two-way distillation protocol that tolerates the highest error rate is considered. We show that among all types of noise there are only four equivalence classes according to the distillability condition. Since the four classes are …
We describe a phase transition for long-range entanglement in a three-dimensional cluster state affected by noise. The partially decohered state is modeled by the thermal state of a suitable Hamiltonian. We find that the temperature at which the entanglement length changes from infinite to finite is nonzero. We give an upper and lower bound to this transition temperature.
We investigate the possibility and conditions to factorize the entanglement evolution of a multiqubit system passing through multi-sided noisy channels. By means of a lower bound of concurrence (LBC) as entanglement measure, we derive an explicit formula of LBC evolution of the N-qubit generalized Greenberger-Horne-Zeilinger (GGHZ) state under some typical noisy channels, based on which two kinds of factorizing conditions for the LBC evolution are presented. In this case, the time-dependent LBC can be determined by a product of initial LBC of the system and the LBC evolution of a maximally ent…
We derive the general formula for the capacity of a noiseless quantum channel assisted by an arbitrary amount of noisy entanglement. In this capacity formula, the ratio of the quantum mutual information and the von Neumann entropy of the sender's share of the noisy entanglement plays the role of mutual information in the completely classical case. A consequence of our results is that bound entangled states cannot increase the capacity of a noiseless quantum channel.
The quantum teleportation with noisy EPR state is discussed. Using an optimal decomposition technique, we compute the concurrence, entanglement of formation and Groverian measure for various noisy EPR resources. It is shown analytically that all entanglement measures reduce to zero when $\bar{F} \leq 2/3$, where $\bar{F}$ is an average fidelity between Alice and Bob. This fact indicates that the entanglement is a genuine physical resource for the teleportation process. This fact gives valuable clues on the optimal decomposition for higher-qubit mixed states. As an example, the optimal decompos…
The faithful distribution of entanglement over noisy channels is a vital prerequisite for many quantum technological applications. Quantum information can be encoded in different degrees of freedom (DOF) of photons, where each encoding comes with its own advantages and disadvantages with respect to noise resilience and practicality in manipulation. In this work, we experimentally implement a deterministic entanglement purification protocol that allows us to faithfully distribute entanglement in one DOF over a noisy channel and then remotely transfer it to another DOF for manipulation. In parti…
Quantum metrology and quantum information necessitate a profound study of suitable states. Attenuations induced by free-space communication links or fluctuations in the generation of such states limit the quantum enhancement in possible applications. For this reason we investigate quantum features of mixtures of so-called N00N states propagating in atmospheric channels. First, we show that noisy N00N states can still yield a phase resolution beyond classical limitations. Second, we identify entanglement of noisy N00N states after propagation in fluctuating loss channels. To do so, we apply the…
Quantum metrology overcomes standard precision limits and plays a central role in science and technology. Practically it is vulnerable to imperfections such as decoherence. Here, we demonstrate quantum metrology for noisy channels such that entanglement with ancillary qubits enhances the quantum Fisher information for phase estimation but not otherwise. Our photonic experiment covers a range of noise for various types of channels, including for two randomly alternating channels such that assisted entanglement fails for each noisy channel individually. We have simulated noisy channels by implem…
Gradual loss of the entanglement of a twin beam containing around 25 photon pairs with the increasing external noise is experimentally investigated. The entanglement is quantified by the non-classicality depths and the non-classicality counting parameters related to several non-classicality criteria. The reduction of intensity moments of the analyzed multi-mode twin beams to single-mode ones allows to determine the negativity as another quantifier of the entanglement. Both the raw photocount histograms and the reconstructed photon-number distributions are analyzed in parallel.
Prior entanglement between sender and receiver, which exactly doubles the classical capacity of a noiseless quantum channel, can increase the classical capacity of some noisy quantum channels by an arbitrarily large constant factor depending on the channel, relative to the best known classical capacity achievable without entanglement. The enhancement factor is greatest for very noisy channels, with positive classical capacity but zero quantum capacity. Although such quantum channels can be simulated classically, no violation of causality is implied, because the simulation requires at least as …
The dynamics of entanglement between two photons with one of them passing through noisy quantum channels is characterized. It is described by a simple factorization law which was first theoretically proposed by Konrad {\it et al.} [Nature Phys., 4, 99 (2008)]. Quantum state tomography process is employed to reconstruct the reduced density matrixes of the final states and the corresponding concurrences are calculated. Good fittings between experimental results and theoretical predictions are found, which imply the validity of the general factorization law in the characterization of entanglement…
We present an experimental scheme based on spontaneous parametric down-conversion to produce multiple photon pairs in maximally entangled polarization states using an arrangement of two type-I nonlinear crystals. By introducing correlated polarization noise in the paths of the generated photons we prepare mixed entangled states whose properties illustrate fundamental results obtained recently in quantum information theory, in particular those concerning bound entanglement and privacy.
Known entanglement purification protocols for mixed states use collective measurements on several copies of the state in order to increase the entanglement of some of them. We address the question of whether it is possible to purify the entanglement of a state by processing each copy separately. While this is possible for pure states, we show that this is impossible, in general, for mixed states. The importance of this result both conceptually and for experimental realization of purification is discussed. We also give explicit invariants of an entangled state of two qubits under local actions …
Nonclassicality, entanglement, as well as dimensionality of a noisy twin beam are determined using characteristic function of the beam written in the Fock basis. One-to-one correspondence between the negativity quantifying entanglement and the nonclassicality depth is revealed. Twin beams, which are either entangled or nonclassical (independent of their entanglement), are observed only for the limited degrees of noise that degrades their quantumness. Dimensionality of the twin beam quantified by the participation ratio is compared with the dimensionality of entanglement determined from the neg…
A quantum internet aims at harnessing networked quantum technologies, namely by distributing bipartite entanglement between distant nodes. However, multipartite entanglement between the nodes may empower the quantum internet for additional or better applications for communications, sensing, and computation. In this work, we present an algorithm for generating multipartite entanglement between different nodes of a quantum network with noisy quantum repeaters and imperfect quantum memories, where the links are entangled pairs. Our algorithm is optimal for GHZ states with 3 qubits, maximising sim…
We address the degradation of continuous variable (CV) entanglement in a noisy channel focusing on the set of photon-number entangled states. We exploit several separability criteria and compare the resulting separation times with the value of non-Gaussianity at any time, thus showing that in the low-temperature regime: i) non-Gaussianity is a bound for the relative entropy of entanglement and ii) Simon' criterion provides a reliable estimate of the separation time also for nonGaussian states. We provide several evidences supporting the conjecture that Gaussian entanglement is the most robust …
Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of pre-assigned accuracy can be obtained straightforwardly. Our method also works for POVM measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional…
Entangled quantum states, such as N00N states, are of major importance for quantum technologies due to their quantum-enhanced performance. At the same time, their quantum correlations are relatively vulnerable when they are subjected to imperfections. Therefore, it is crucial to determine under which circumstances their distinct quantum features can be exploited. In this paper, we study the entanglement property of noisy N00N states. This class of states is a generalization of N00N states including various attenuation effects, such as mixing, constant or fluctuating losses, and dephasing. To v…
The extremality of Gaussian states is exploited to show that Gaussian states are the most robust, among all possible bipartite continuous-variable states at fixed energy, against disentanglement due to noisy evolutions in Markovian Gaussian channels involving dissipation and thermal hopping. This proves a conjecture raised recently in [M. Allegra, P. Giorda, and M. G. A. Paris, Phys. Rev. Lett. {\bf 105}, 100503 (2010)], providing a rigorous validation of the conclusions of that work. The problem of identifying continuous variable states with maximum resilience to entanglement damping in more …
Whether noisy quantum devices without error correction can provide quantum advantage over classical computers is a critical issue of current quantum computation. In this work, the random quantum circuits, which are used as the paradigm model to demonstrate quantum advantage, are simulated with depolarizing noise on experiment relevant two-dimensional architecture. With comprehensive numerical simulation and theoretical analysis, we find that the maximum achievable operator entanglement entropy, which indicates maximal simulation cost, has area law scaling with the system size for constant nois…
We consider properties of states of many qubits, which arise after sending certain entangled states via various noisy channels (white noise, coloured noise, local depolarization, dephasing and amplitude damping). Entanglement of these states is studied and their ability to violate certain classes of Bell inequalities. States which violate them allow for higher than classical efficiency of solving related distributed computational tasks with constrained communication. This is a direct property of such states -- not requiring their further modification via stochastic local operations and classic…
We show how, in the presence of quantum noises generated by non-commuting Lindblad operators, a maximally entangled Bell state may suffer entanglement sudden death (ESD). Similarly, ESD may occur when a Bell state is under the action of a quantum operation generated by a Hamiltonian in the presence of a quantum noise, provided that the Hamiltonian does not commute with the corresponding Lindblad operator. Practically this means noisy quantum operations could cause ESD during the encoding process in quantum communication protocols like special dense coding ($\cal S$). Next, we show how in the p…
Being able to implement effective entanglement distribution in noisy environments is a key step towards practical quantum communication, and long-term efforts have been made on the development of it. Recently, it has been found that the null-result weak measurement (NRWM) can be used to enhance probabilistically the entanglement of a single copy of amplitude-damped entangled state. This paper investigates remote distributions of bipartite and multipartite entangled states in the amplitudedamping environment by combining NRWMs and entanglement distillation protocols (EDPs). We show that the NRW…
We present a formal quantum mechanical analysis of the communication protocol of Prevedel {\it et al.}\ [Phys. Rev. Lett. \textbf{106}, 110505 (2011)], in which entanglement shared by sender and receiver is used to enhance, beyond that achievable via the optimal classical strategy, the probability of successful transmission of a bit through a particular noisy classical channel ${\cal N}$. We provide a full analysis of this protocol when the shared entanglement resides in a two-qubit system. Our analysis shows the measurement choices specified by the protocol yield the maximum possible enhancem…
We study entanglement dynamics of the three-qubit system which is initially prepared in pure Greenberger-Horne- Zeilinger (GHZ) or W state and transmitted through one of the Pauli channels $σ_z, \, σ_x, \, σ_y$ or the depolarizing channel. With the help of the lower bound for three-qubit concurrence we show that the W state preserves more entanglement than the GHZ state in transmission through the Pauli channel $σ_z$. For the Pauli channels $σ_x, \, σ_y$ and the depolarizing channel, however, the entanglement of the GHZ state is more resistant against decoherence than the W-type entanglement. …
Two separated observers, by applying local operations to a supply of not-too-impure entangled states ({\em e.g.} singlets shared through a noisy channel), can prepare a smaller number of entangled pairs of arbitrarily high purity ({\em e.g.} near-perfect singlets). These can then be used to faithfully teleport unknown quantum states from one observer to the other, thereby achieving faithful transfrom one observer to the other, thereby achieving faithful transmission of quantum information through a noisy channel. We give upper and lower bounds on the yield $D(M)$ of pure singlets ($\ket{Ψ^-}$)…
Some properties of multi-qubit systems interacting with noisy environment is discussed. The amount of the survival entanglement is quantified for the GHZ and W-states. It is shown that the entanglement decay depends on the noise type ( correlated or non-correlated), number of interacted qubits with the environment and the initial state which passes through this noisy environment. In general, the GHZ is more fragile than the W-state. The phenomena of entanglement sudden death appears only for non-correlated noise.
Entanglement witnesses form an effective method to locally detect entanglement in the laboratory without having the prior knowledge of the full density matrix. However, separable states can be erroneously indicated as entangled in such detections in the presence of wrong measurements or loss in detectors. Measurement-device-independent entanglement witnesses (MDI-EWs) never detect fake entanglement even under wrong measurements and for a particular kind of lossy detectors. A crucial assumption in the case of faithful detection of entanglement employing MDI-EWs is that the preparation devices p…
Maximally entangled states--a resource for quantum information processing--can only be shared through noiseless quantum channels, whereas in practice channels are noisy. Here we ask: Given a noisy quantum channel, what is the maximum attainable purity (measured by singlet fraction) of shared entanglement for single channel use and local trace preserving operations? We find an exact formula of the maximum singlet fraction attainable for a qubit channel and give an explicit protocol to achieve the optimal value. The protocol distinguishes between unital and nonunital channels and requires no loc…
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…
This paper addresses some general questions of quantum information theory arising from the transmission of quantum entanglement through (possibly noisy) quantum channels. A pure entangled state is prepared of a pair of systems $R$ and $Q$, after which $Q$ is subjected to a dynamical evolution given by the superoperator $\superop^{Q}$. Two interesting quantities can be defined for this process: the entanglement fidelity $F_{e}$ and the entropy production $S_{e}$. It turns out that neither of these quantities depends in any way on the system $R$, but only on the initial state and dynamical evolu…
We show that quantum mechanical entanglement can prevail even in noisy open quantum systems at high temperature and far from thermodynamical equilibrium, despite the deteriorating effect of decoherence. The system consists of a number N of interacting quantum particles, and it can interact and exchange particles with some environment. The effect of decoherence is counteracted by a simple mechanism, where system particles are randomly reset to some standard initial state, e.g. by replacing them with particles from the environment. We present a master equation that describes this process, which …
We address the question as to whether an entangled state that satisfies local realism will give a violation of the same, after entanglement swapping in a suitable scenario. We consider such possibility as a kind of superadditivity in nonclassicality. Importantly, it will indicate that checking for violation of local realism, in the state obtained after entanglement swapping, can be a method for detecting entanglement in the input state of the swapping procedure. We investigate various entanglement swapping schemes, which involve mixed initial states. The strength of violation of local realism …
We present and experimentally demonstrate a communication protocol that employs shared entanglement to reduce errors when sending a bit over a particular noisy classical channel. Specifically, it is shown that, given a single use of this channel, one can transmit a bit with higher success probability when sender and receiver share entanglement compared to the best possible strategy when they do not. The experiment is realized using polarization-entangled photon pairs, whose quantum correlations play a critical role in both the encoding and decoding of the classical message. Experimentally, we …
It is well-known that the violation of a local uncertainty relation can be used as an indicator for the presence of entanglement. Unfortunately, the practical use of these non-linear witnesses has been limited to few special cases in the past. However, new methods for computing uncertainty bounds became available. Here we report on an experimental implementation of uncertainty-based entanglement witnesses, benchmarked in a regime dominated by strong local noise. We combine the new computational method with a local noise tomography in order to design noise-adapted entanglement witnesses. This p…
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…
