We investigate the probabilistic cloning and purification of quantum states. The performance of these probabilistic operations is quantified by the average fidelity between the ideal and actual output states. We provide a simple formula for the maximal achievable average fidelity and we explictly show how to construct a probabilistic operation that achieves this fidelity. We illustrate our method on several examples such as the phase covariant cloning of qubits, cloning of coherent states, and purification of qubits transmitted via depolarizing channel and amplitude damping channel. Our examples reveal that the probabilistic cloner may yield higher fidelity than the best deterministic cloner even when the states that should be cloned are linearly dependent and are drawn from a continuous set.
We introduce a new decomposition of the multiqubit states of the form $ρ^{\otimes N}$ and employ it to construct the optimal single qubit purification procedure. The same decomposition allows us to study optimal quantum cloning and state estimation of mixed states.
We construct efficient quantum logic network for probabilistic cloning the quantum states used in implemented tasks for which cloning provides some enhancement in performance.
In probabilistic cloning with two auxiliary systems, we consider and compare three different protocols for the success probabilities of cloning. We show that, in certain circumstances, it may increase the success probability to add an auxiliary system to the probabilistic cloning machine having one auxiliary system, but we always can find another cloning machine with one auxiliary system having the same success probability as that with two auxiliary systems.
We present an example of quantum computational tasks whose performance is enhanced if we distribute quantum information using quantum cloning. Furthermore we give achievable efficiencies for probabilistic cloning the quantum states used in implemented tasks for which cloning provides some enhancement in performance.
A basic feasible probabilistic purification of unknown noisy coherent states, outgoing from different state preparations with unknown mean number of thermal photons, is proposed. The scheme is based only on a linear-optical network with an avalanche photo-diode or heterodyne (homodyne) detection used to post-select a successful processing. The suggested probabilistic method can produce an output state with a lower noise than both quantum deterministic and classical probabilistic distillation method. The purification applied in the state preparation can increase classical capacity of communicat…
The method of quantum cloning is divided into two main categories: approximate and probabilistic quantum cloning. The former method is used to approximate an unknown quantum state deterministically, and the latter can be used to faithfully copy the state probabilistically. So far, many approximate cloning machines have been experimentally demonstrated, but probabilistic cloning remains an experimental challenge, as it requires more complicated networks and a higher level of precision control. In this work, we designed an efficient quantum network with a limited amount of resources, and perform…
We show that non-local resources cannot be used for probabilistic signalling even if one can produce exact clones with the help of a probabilistic quantum cloning machine (PQCM). We show that PQCM cannot help to distinguish two statistical mixtures at a remote location. Thus quantum theory rules out the possibility of sending superluminal signals not only deterministically but also probabilistically. We give a bound on the success probability of producing multiple clones in an entangled system.
From Ref. [Phys. Rev. Lett. 80(1998)4999] one knows that the quantum states secretly chosen from a certain set can be probabilistically cloned with positive cloning efficiencies if and only if all the states in the set are linearly independent. In this paper, we focus on the probabilistic quantum cloning (PQC) of linearly dependent states with nonnegative cloning efficiencies. We show that a linearly independent subset of the linearly dependent quantum states can be probabilistically cloned if and only if any one state in the subset can not be expressed as the linear superposition of the other…
We show that the condition of no faster-than-light signalling restricts the number of quantum states that can be cloned in a given Hilbert space. This condition leads to the constraints on a probabilistic quantum cloning machine (PQCM) recently found by Duan and Guo.
The manipulation of quantum coherence is one of the principal issues in the resource theory of coherence, with two critical topics being the purification and enhancement of coherence. Here, we present two no-go theorems for the deterministic purification of coherence and the probabilistic enhancement of coherence, respectively. Specifically, we prove that a quantum state cannot be deterministically purified if it can be expressed as a convex combination of an incoherent state and a coherent state. Besides, we give an easy-to-verified sufficient and necessary condition to determine whether a st…
We discuss the usefulness of quantum cloning and present examples of quantum computation tasks for which cloning offers an advantage which cannot be matched by any approach that does not resort to it. In these quantum computations, we need to distribute quantum information contained in states about which we have some partial information. To perform quantum computations, we use state-dependent probabilistic quantum cloning procedure to distribute quantum information in the middle of a quantum computation.
Probabilistic quantum cloning and identifying machines can be constructed via unitary-reduction processes [Duan and Guo, Phys. Rev. Lett. 80, 4999 (1998)]. Given the cloning (identifying) probabilities, we derive an explicit representation of the unitary evolution and corresponding Hamiltonian to realize probabilistic cloning (identification). The logic networks are obtained by decomposing the unitary representation into universal quantum logic operations. The robustness of the networks is also discussed. Our method is suitable for a $k$-partite system, such as quantum computer, and may be gen…
We give a proof of impossibility of probabilistic exact $1\to 2$ cloning of any three different states of a qubit. The simplicity of the proof is due to the use of a surprising result of remote state preparation [M.-Yong Ye, Y.-Sheng Zhang and G.-Can Guo, quant-ph/0307027 (2003)]. The result is extented to higher dimentional cases for special ensemble of states.
It has been recently shown that probabilistic protocols based on postselection boost the performances of phase estimation and the replication of quantum clocks. Here we demonstrate that the improvements in these two tasks have to match exactly in the macroscopic limit where the number of clones grows to infinity, preserving the equivalence between asymptotic cloning and estimation for arbitrary values of the success probability. Remarkably, the cloning fidelity depends critically on the number of rationally independent eigenvalues of the clock Hamiltonian. We also prove that probabilistic metr…
We construct a probabilistic quantum cloning machine by a general unitary-reduction operation. With a postselection of the measurement results, the machine yields faithful copies of the input states. It is shown that the states secretly chosen from a certain set $\$=\left\{\left| Ψ_1\right> ,\left| Ψ_2\right> ,... ,\left| Ψ_n\right> \right\} $ can be probabilistically cloned if and only if $% \left| Ψ_1\right>$, $\left| Ψ_2\right>$, $... ,$ and $\left| Ψ_n\right>$ are linearly-independent. We derive the best possible cloning efficiencies. Probabilistic cloning has close connection with the pro…
It is known that to imperfectly clone a phase-set of states uncorrelatedly is impossible due to the linearity and the hermitian-preservation of quantum operations deterministically shown by D'Ariano et al. In this paper we address uncorrelated cloning in probabilistic cases. We show that there exists a linear and hermitian-preserving probabilistic map to imperfectly clone a phase-set of states uncorrelatedly. It is pointed out that the existence of such a map is due to the difference between non-unit-trace output operators and normalized (unit-trace) output operators inherent to probabilistic …
We propose a probabilistic quantum cloning scheme using Greenberger-Horne-Zeilinger states, Bell basis measurements, single-qubit unitary operations and generalized measurements, all of which are within the reach of current technology. Compared to another possible scheme via Tele-CNOT gate [D. Gottesman and I. L. Chuang, Nature 402, 390 (1999)], the present scheme may be used in experiment to clone the states of one particle to those of two different particles with higher probability and less GHZ resources.
We present a probabilistic cloning scheme operating independently of any phase reference. The scheme is based solely on a phase-randomized displacement and photon counting, omitting the need for non-classical resources and non-linear materials. In an experimental implementation, we employ the scheme to clone coherent states from a phase covariant alphabet and demonstrate that the cloner is capable of outperforming the hitherto best-performing deterministic scheme. An analysis of the covariances between the output states shows that uncorrelated clones can be approached asymptotically. An intrig…
Recently, the nonlocal optimal probabilistic cloning (NLOPC) of two non-orthogonal qubit states has been proposed [Phys. Rev. A 86, 052332 (2012)] by means of an experimental setup based on a pair of twin photons in a maximally entangled state. Here we study the performance of the NLOPC protocol when implemented via a partially entangled state. We show that the errors introduced by the use of partial entanglement can be undone by applying a quantum state discrimination process. Since quantum state discrimination is a probabilistic process the correction succeeds with a certain probability but …
We prove generic versions of the no-cloning and no-broadcasting theorems, applicable to essentially {\em any} non-classical finite-dimensional probabilistic model that satisfies a no-signaling criterion. This includes quantum theory as well as models supporting ``super-quantum'' correlations that violate the Bell inequalities to a larger extent than quantum theory. The proof of our no-broadcasting theorem is significantly more natural and more self-contained than others we have seen: we show that a set of states is broadcastable if, and only if, it is contained in a simplex whose vertices are …
We investigate general probabilistic theories in which every mixed state has a purification, unique up to reversible channels on the purifying system. We show that the purification principle is equivalent to the existence of a reversible realization of every physical process, namely that every physical process can be regarded as arising from a reversible interaction of the system with an environment, which is eventually discarded. From the purification principle we also construct an isomorphism between transformations and bipartite states that possesses all structural properties of the Choi-Ja…
We consider probabilistic cloning of a state chosen from a mutually nonorthogonal set of pure states, with the help of a party holding supplementary information in the form of pure states. When the number of states is 2, we show that the best efficiency of producing m copies is always achieved by a two-step protocol in which the helping party first attempts to produce m-1 copies from the supplementary state, and if it fails, then the original state is used to produce m copies. On the other hand, when the number of states exceeds two, the best efficiency is not always achieved by such a protoco…
We provide several applications of a previously introduced isomorphism between physical operations acting on two systems and entangled states [1]. We show: (i) how to implement (weakly) non-local two qubit unitary operations with a small amount of entanglement; (ii) that a known, noisy, non-local unitary operation as well as an unknown, noisy, local unitary operation can be purified; (iii) how to perform the tomography of arbitrary, unknown, non-local operations; (iv) that a set of local unitary operations as well as a set of non-local unitary operations can be stored and compressed; (v) how t…
Quantum cloning is an essential operation in quantum information and quantum computing. Similar to the `copy' operation in classical computing, the cloning of flying bits for further processing from the solid-state quantum bits in storage is an operation frequently used in quantum information processing. Here we propose a high-fidelity and controllable quantum cloning scheme between solid bits and flying bits. In order to overcome the obstacles from the no-cloning theorem and the weak phonon-photon interaction, we introduce a hybrid optomechanical system that performs both the probabilistic cl…
Quantum-limited amplifiers increase the amplitude of quantum signals at the price of introducing additional noise. Quantum purification protocols operate in the reverse way, by reducing the noise while attenuating the signal. Here we investigate a scenario that interpolates between these two extremes. We search for the optimal physical process that generates $M$ approximate copies of pure and amplified coherent state, starting from $N$ copies of a noisy coherent state with Gaussian modulation. We prove that the optimal deterministic processes are always Gaussian, whereas non-Gaussianity powers…
Probabilistically creating n perfect clones from m copies for one of N priori known quantum states with minimum failure probability is a long-standing problem. We provide a rigorous proof for the geometric approach to this probabilistic quantum cloning problem when N = 2. Then, we give the general geometric form of the sufficient and necessary condition of probabilistic cloning for N known quantum states. By this general geometric approach, we realize the optimal probabilistic quantum cloning of N known quantum states with priori probabilities. The results are also applicable to the identifica…
We propose an entanglement purification scheme based on material qubits and ancillary coherent multiphoton states. We consider a typical QED scenario where material qubits implemented by two-level atoms fly sequentially through a cavity and interact resonantly with a single mode of the radiation field. We explore the theoretical possibilities of realizing a high-fidelity two-qubit quantum operation necessary for the purification protocol with the help of a postselective balanced homodyne photodetection. We demonstrate that the obtained probabilistic quantum operation can be used as a bilateral…
The no-cloning theorem states that an unknown quantum state cannot be cloned exactly and deterministically due to the linearity of quantum mechanics. Associated with this theorem is the quantitative no-cloning limit that sets an upper bound to the quality of the generated clones. However, this limit can be circumvented by abandoning determinism and using probabilistic methods. Here, we report an experimental demonstration of probabilistic cloning of arbitrary coherent states that clearly surpasses the no-cloning limit. Our scheme is based on a hybrid linear amplifier that combines an ideal det…
In this paper, we study the entanglement and quantum discord of the output modes in the unified $1\rightarrow 2$ state-dependent cloning and probabilistic quantum cloning. The tripartite entanglement among the output modes and the quantum cloning machine is also considered. We find that the roles of the quantum correlations including the bipartite and tripartite entanglement and quantum discord strongly depend on the quantum cloning machines as well as the cloned state. In particular, it is found that this quantum cloning scheme can be realizable even without any quantum correlation.
An optimal universal cloning transformation is derived that produces M copies of an unknown qubit from a pair of orthogonal qubits. For M>6, the corresponding cloning fidelity is higher than that of the optimal copying of a pair of identical qubits. It is shown that this cloning transformation can be implemented probabilistically via parametric down-conversion by feeding the signal and idler modes of a nonlinear crystal with orthogonally polarized photons.
We present a computational circuit which realizes contextually the Tele-UNOT gate and the universal optimal quantum cloning machine (UOQCM). We report the experimental realization of the probabilistic UOQCM with polarization encoded qubits. This is achieved by combining on a symmetric beam-splitter the input qubit with an ancilla in a fully mixed state.
A general multi-step N->M probabilistic optimal universal cloning protocol is presented together with the experimental realization of the (1 -> 3) and (2 -> 3) machines. Since the present method exploits the bosonic nature of the photons, it can be applied to any particle obeying to the Bose statistics. On a technological perspective, the present protocol is expected to find applications as a novel, multi-qubit symmetrizator device to be used in modern quantum information networks.
We derive the transformation for the optimal universal quantum anti-cloner which produces two anti-parallel outputs for a single input state. The fidelity is shown to be 2/3 which is same as the measurement fidelity. We consider a probabilistic quantum anti-cloner and show quantum states can be anti-cloned exactly with non-zero probability and its efficiency is higher than the efficiency of distinguishing between the two states.
The manipulation of quantum "resources" such as entanglement, coherence and magic states lies at the heart of quantum science and technology, empowering potential advantages over classical methods. In practice, a particularly important kind of manipulation is to "purify" the quantum resources, since they are inevitably contaminated by noise and thus often lose their power or become unreliable for direct usage. Here we prove fundamental limitations on how effectively generic noisy resources can be purified enforced by the laws of quantum mechanics, which universally apply to any reasonable kind…
We present a protocol based on continuous-variable quantum teleportation and Gaussian post- selection that can be used to correct errors introduced by a lossy channel. We first show that the global transformation enacted by the protocol is equivalent to an effective system composed of a noiseless amplification (or attenuation), and an effective quantum channel, which can in theory have no loss and an amount of thermal noise arbitrarily small, hence tending to an identity channel. An application of our protocol is the probabilistic purification of quantum non-Gaussian states using only Gaussian…
We present a probabilistic scheme for generating and purifying maximally-entangled states of two atoms inside an optical cavity via no-photon detection in the output cavity mode, where ideal detectors may not be required. The intermediate mixed states can be continuously "filtered" so as to violate Bell inequalities in a parametrized manner. The scheme relies on an additional strong-driving field that yields unusual dynamics in cavity QED experiments, simultaneously realizing Jaynes-Cummings and anti-Jaynes-Cummings interactions.
Entanglement purification is a very important element for long-distance quantum communication. Different from all the existing entanglement purification protocols (EPPs) in which two parties can only obtain some quantum systems in a mixed entangled state with a higher fidelity probabilistically by consuming quantum resources exponentially, here we present a deterministic EPP with hyperentanglement. Using this protocl, the two parties can, in principle, obtain deterministically maximally entangled pure states in polarization without destroying any less-entangled photon pair, which will improve …
A linear optical probabilistic scheme for the optimal cloning of a pair of orthogonally-polarized photons is devised, based on single- and two-photon interferences. It consists in a partial symmetrization device, realized with a modified unbalanced Mach-Zehnder interferometer, followed by two independent Hong-Ou-Mandel interferometers. This scheme has the advantage that it enables quantum cloning without the need for stimulated amplification in a nonlinear medium. It can also be modified so to make an optical two-qubit partial SWAP gate, thereby providing a potentially useful tool to linear op…
Optimal procedures play an important role in quantum information. It turns out that some naturally occurring processes like emission of light from an atom can realize optimal transformations. Here we study how arbitrary symmetric states of a number of d-level systems can be cloned using a multilevel atomic system. It is shown that optimality is always ensured even though the output number of systems is probabilistic.
we envisage a novel quantum cloning machine, which takes an input state and produces an output state whose success branch can exist in a linear superposition of multiple copies of the input state and the failure branch exist in a superposition of composite state independent of the input state. We prove that unknown non-orthogonal states chosen from a set $\cal S$ can evolve into a linear superposition of multiple clones by a unitary process if and only if the states are linearly independent. We derive a bound on the success probability of the novel cloning machine. We argue that the determinis…
We discuss the recently introduced concept of non-deterministic noiseless linear amplification, demonstrating that such an operation can only be performed perfectly with vanishing probability of success. We show that a weak measurement, which imprints the weak value of an observable of a pre-selected and post-selected system onto a probe system, can be used to approximate probabilistic noiseless amplification. This result may be applied to various tasks in continuous variable quantum information, including entanglement concentration, probabilistic cloning, and in quantum repeaters. We discuss …
The impossibility of superluminal communication is a fundamental principle of physics. Here we show that this principle underpins the performance of several fundamental tasks in quantum information processing and quantum metrology. In particular, we derive tight no-signaling bounds for probabilistic cloning and super-replication that coincide with the corresponding optimal achievable fidelities and rates known. In the context of quantum metrology, we derive the Heisenberg limit from the no-signaling principle for certain scenarios including reference frame alignment and maximum likelihood stat…
We investigate the multiphoton states generated by high-gain optical parametric amplification of a single injected photon, polarization encoded as a ''qubit''. The experiment configuration exploits the optimal phase-covariant cloning in the high gain regime. The interference fringe pattern showing the non local transfer of coherence between the injected qubit and the mesoscopic amplified output field involving up to 4000 photons has been investigated. A probabilistic new method to extract full information about the multiparticle output wavefunction has been implemented.
Optimal quantum machines can be implemented by linear projective operations. In the present work a general qubit symmetrization theory is presented by investigating the close links to the qubit purification process and to the programmable teleportation of any generic optimal anti-unitary map. In addition, the contextual realization of the N ->M cloning map and of the teleportation of the N->(M-N) universal NOT gate is analyzed by a novel and very general angular momentum theory. An extended set of experimental realizations by state symmetrization linear optical procedures is reported. These in…
We introduce the study of quantum protocols that probabilistically simulate quantum channels from a sender in the future to a receiver in the past. The maximum probability of simulation is determined by causality and depends on the amount and type (classical or quantum) of information that the channel can transmit. We illustrate this dependence in several examples, including ideal classical and quantum channels, measure-and-prepare channels, partial trace channels, and universal cloning channels. For the simulation of partial trace channels, we consider generalized teleportation protocols that…
Universal quantum state purification under energy-preserving constraints
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；South China University of Technology（华南理工大学）；The University of Hong Kong（香港大学）
Quantum no-cloning, the impossibility of perfectly cloning an arbitrary unknown quantum state, is one of the most fundamental limitations due to the laws of quantum mechanics, which underpin the physical security of quantum key distribution. Quantum physics does allow, however, approximate cloning with either imperfect state fidelity and/or probabilistic success. Whereas approximate quantum cloning of single-particle states has been tested previously, experimental cloning of quantum entanglement -- a highly non-classical correlation -- remained unexplored. Based on a multiphoton linear optics …
Sparrow Quantum（Sparrow量子）；University of Copenhagen（哥本哈根大学）；University of Bristol（布里斯托大学）；The Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所）
We propose a scheme to enhance the fidelity of symmetric quantum cloning machine using a weak measurement. By adjusting the intensity of weak measurement parameter $p$, we obtain the copies with different optimal fidelity. Choosing proper value of $p$, we can obtain the perfect copies for initial qubits. In this paper, we focus on $1-2$ quantum cloning for two nonorthogonal states. Sets containing more than two linear independent states are also discussed briefly. Due to weak measurements being probabilistic, we obtain high fidelity at risk of probability. If the weak measurement successes, we…
No-cloning theorem is fundamental for quantum mechanics and for quantum information science that states an unknown quantum state cannot be cloned perfectly. However, we can try to clone a quantum state approximately with the optimal fidelity, or instead, we can try to clone it perfectly with the largest probability. Thus various quantum cloning machines have been designed for different quantum information protocols. Specifically, quantum cloning machines can be designed to analyze the security of quantum key distribution protocols such as BB84 protocol, six-state protocol, B92 protocol and the…
We show that thresholds for fault-tolerant quantum computation are solely determined by the quality of single-system operations if one allows for d-dimensional systems with $8 \leq d \leq 32$. Each system serves to store one logical qubit and additional auxiliary dimensions are used to create and purify entanglement between systems. Physical, possibly probabilistic two-system operations with error rates up to 2/3 are still tolerable to realize deterministic high quality two-qubit gates on the logical qubits. The achievable error rate is of the same order of magnitude as of the single-system op…
Nonlocal gate operation is based on sharing an ancillary pair of qubits in perfect entanglement. When the ancillary pair are partially entangled, the efficiency of the gate operation drops. Using general transformations, we devise probabilistic nonlocal gates, which perform the nonlocal operation conclusively when the ancillary pair are only partially entangled. We show that a controlled purification protocol can be implemented by the probabilistic nonlocal operation.
No-cloning theorem forbids perfect cloning of an unknown quantum state. A universal quantum cloning machine (UQCM), capable of producing two copies of any input qubit with the optimal fidelity, is of fundamental interest and has applications in quantum information processing. This is enabled by delicately tailored nonclassical correlations between the input qubit and the copying qubits, which distinguish the UQCM from a classical counterpart, but whose experimental demonstrations are still lacking. We here implement the UQCM in a superconducting circuit, and investigate these correlations. The…
We show that in a cloning process, whether deterministic inexact or probabilistic exact, one can take an arbitrary blank state while still using a fixed cloning machine.
