Experimental hierarchy of the nonclassicality of single-qubit states via potentials for entanglement, steering, and Bell nonlocality.

Entanglement potentials are a promising way to quantify the nonclassicality of single-mode states. They are defined by the amount of entanglement (expressed by, e.g., the Wootters concurrence) obtained after mixing the examined single-mode state with a purely classical state; such as the vacuum or a coherent state. We generalize the idea of entanglement potentials to other quantum correlations: the EPR steering and Bell nonlocality, thus enabling us to study mutual hierarchies of these nonclassicality potentials. Instead of the usual vacuum and one-photon superposition states, we experimentally test this concept using specially tailored polarization-encoded single-photon states. One polarization encodes a given nonclassical single-mode state, while the other serves as the vacuum place-holder. This technique proves to be experimentally more convenient in comparison to the vacuum and a one-photon superposition as it does not require the vacuum detection.

Experimental hierarchy of two-qubit quantum correlations without state tomography.

A Werner state, which is the singlet Bell state affected by white noise, is a prototype example of states, which can reveal a hierarchy of quantum entanglement, steering, and Bell nonlocality by controlling the amount of noise. However, experimental demonstrations of this hierarchy in a sufficient and necessary way (i.e., by applying measures or universal witnesses of these quantum correlations) have been mainly based on full quantum state tomography, corresponding to measuring at least 15 real parameters of two-qubit states. Here we report an experimental demonstration of this hierarchy by measuring only six elements of a correlation matrix depending on linear combinations of two-qubit Stokes parameters. We show that our experimental setup can also reveal the hierarchy of these quantum correlations of generalized Werner states, which are any two-qubit pure states affected by white noise.

Quantification of Quantum Correlations in Two-Beam Gaussian States Using Photon-Number Measurements.

Identification, and subsequent quantification of quantum correlations, is critical for understanding, controlling, and engineering quantum devices and processes. We derive and implement a general method to quantify various forms of quantum correlations using solely the experimental intensity moments up to the fourth order. This is possible as these moments allow for an exact determination of the global and marginal impurities of two-beam Gaussian fields. This leads to the determination of steering, tight lower and upper bounds for the negativity, and the Kullback-Leibler divergence used as a quantifier of state nonseparability. The principal squeezing variances are determined as well using the intensity moments. The approach is demonstrated on the experimental twin beams with increasing intensity and the squeezed super-Gaussian beams composed of photon pairs. Our method is readily applicable to multibeam Gaussian fields to characterize their quantum correlations.

Experimental kernel-based quantum machine learning in finite feature space.

We implement an all-optical setup demonstrating kernel-based quantum machine learning for two-dimensional classification problems. In this hybrid approach, kernel evaluations are outsourced to projective measurements on suitably designed quantum states encoding the training data, while the model training is processed on a classical computer. Our two-photon proposal encodes data points in a discrete, eight-dimensional feature Hilbert space. In order to maximize the application range of the deployable kernels, we optimize feature maps towards the resulting kernels' ability to separate points, i.e., their "resolution," under the constraint of finite, fixed Hilbert space dimension. Implementing these kernels, our setup delivers viable decision boundaries for standard nonlinear supervised classification tasks in feature space. We demonstrate such kernel-based quantum machine learning using specialized multiphoton quantum optical circuits. The deployed kernel exhibits exponentially better scaling in the required number of qubits than a direct generalization of kernels described in the literature.

Author Correction: Implementation of an efficient linear-optical quantum router.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Experimental hybrid quantum-classical reinforcement learning by boson sampling: how to train a quantum cloner.

We report on experimental implementation of a machine-learned quantum gate driven by a classical control. The gate learns optimal phase-covariant cloning in a reinforcement learning scenario having fidelity of the clones as reward. In our experiment, the gate learns to achieve nearly optimal cloning fidelity allowed for this particular class of states. This makes it a proof of present-day feasibility and practical applicability of the hybrid machine learning approach combining quantum information processing with classical control. The quantum information processing performed by the setup is equivalent to boson sampling, which, in complex systems, is predicted to manifest quantum supremacy over classical simulation of linear-optical setups.

Experimentally attacking quantum money schemes based on quantum retrieval games.

The concept of quantum money (QM) was proposed by Wiesner in the 1970s. Its main advantage is that every attempt to copy QM unavoidably leads to imperfect counterfeits. In the Wiesner's protocol, quantum banknotes need to be delivered to the issuing bank for verification. Thus, QM requires quantum communication which range is limited by noise and losses. Recently, Bozzio et al. (2018) have demonstrated experimentally how to replace challenging quantum verification with a classical channel and a quantum retrieval game (QRG). This brings QM significantly closer to practical realisation, but still thorough analysis of the revised scheme QM is required before it can be considered secure. We address this problem by presenting a proof-of-concept attack on QRG-based QM schemes, where we show that even imperfect quantum cloning can, under some circumstances, provide enough information to break a QRG-based QM scheme.

Demonstration of Controlled Quantum Teleportation for Discrete Variables on Linear Optical Devices.

We report an experimental implementation of tripartite controlled quantum teleportation on quantum optical devices. The protocol is performed through bi- and tripartite entangled channels of discrete variables and qubits encoded in the polarization of individual photons. The experimental results demonstrate successful controlled quantum teleportation with a fidelity around 83%, well above the classical limit. By realizing the controlled quantum teleportation through a biseparable state, we show that tripartite entanglement is not a necessary resource for controlled quantum teleportation, and the controller's capability to allow or prohibit the teleportation cannot be considered to be a manifestation of tripartite entanglement. These results open new possibilities for further application of controlled quantum teleportation by lowering the teleportation channel's requirements.

Experimental Measurement of the Hilbert-Schmidt Distance between Two-Qubit States as a Means for Reducing the Complexity of Machine Learning.

We report on the experimental measurement of the Hilbert-Schmidt distance between two two-qubit states by many-particle interference. We demonstrate that our three-step method for measuring distances in the Hilbert space is far less complex than reconstructing density matrices and that it can be applied in quantum-enhanced machine learning to reduce the complexity of calculating Euclidean distances between multidimensional points, which can be especially interesting for near term quantum technologies and quantum artificial intelligence research. Our results are also a novel example of applying mixed states in quantum information processing. Usually working with mixed states is undesired, but here it gives the possibility of encoding extra information as the degree of coherence between the given two dimensions of the density matrix.

Implementation of an efficient linear-optical quantum router.

For several decades, scientists have been aware of significant benefits allowing quantum information processing technologies to surpass their classical counterparts. Recent technological development allows these benefits to be tested experimentally and in some cases also implemented in practical devices. So far the majority of experimental quantum networks was limited to peer-to-peer communications between two parties. Practical implementation of quantum communications networks, however, needs to address the problem of scalability to serve large numbers of users. Similarly to classical computer networks, their quantum counterparts would require routing protocols to direct the signal from its source to destination. Devices implementing these routing protocols are called quantum routers and have recently been subject of an intense research. In this paper, we report on experimental implementation of a linear-optical quantum router. Our device allows single-photon polarization-encoded qubits to be routed coherently into two spatial output modes depending on the state of two identical control qubits. The polarization qubit state of the routed photon is maintained during the routing operation. The success probability of our scheme can be increased up to 25% making it the most efficient linear-optical quantum router developed to this date.

Experimental temporal quantum steering.

Temporal steering is a form of temporal correlation between the initial and final state of a quantum system. It is a temporal analogue of the famous Einstein-Podolsky-Rosen (spatial) steering. We demonstrate, by measuring the photon polarization, that temporal steering allows two parties to verify if they have been interacting with the same particle, even if they have no information about what happened with the particle in between the measurements. This is the first experimental study of temporal steering. We also performed experimental tests, based on the violation of temporal steering inequalities, of the security of two quantum key distribution protocols against individual attacks. Thus, these results can lead to applications for secure quantum communications and quantum engineering.

Priority Choice Experimental Two-Qubit Tomography: Measuring One by One All Elements of Density Matrices.

In standard optical tomographic methods, the off-diagonal elements of a density matrix ρ are measured indirectly. Thus, the reconstruction of ρ, even if it is based on linear inversion, typically magnifies small errors in the experimental data. Recently, an optimal tomography solution measuring all the elements of ρ one-by-one without error magnification has been theoretically proposed. We implemented this method for two-qubit polarization states. For comparison, we also experimentally implemented other well-known tomographic protocols, either based solely on local measurements (of, e.g., the Pauli operators and James-Kwiat-Munro-White projectors) or with mutually unbiased bases requiring both local and global measurements. We reconstructed seventeen separable, partially and maximally entangled two-qubit polarization states. Our experiments show that our method has the highest stability against errors in comparison to other quantum tomographies. In particular, we demonstrate that each optimally-reconstructed state is embedded in an uncertainty circle of the smallest radius, both in terms of trace distance and disturbance. We explain how to experimentally estimate uncertainty radii for all the implemented tomographies and show that, for each reconstructed state, the relevant uncertainty circles intersect indicating the approximate location of the corresponding physical density matrix.

Experimental implementation of optimal linear-optical controlled-unitary gates.

We show that it is possible to reduce the number of two-qubit gates needed for the construction of an arbitrary controlled-unitary transformation by up to 2 times using a tunable controlled-phase gate. On the platform of linear optics, where two-qubit gates can only be achieved probabilistically, our method significantly reduces the amount of components and increases success probability of a two-qubit gate. The experimental implementation of our technique presented in this Letter for a controlled single-qubit unitary gate demonstrates that only one tunable controlled-phase gate is needed instead of two standard controlled-not gates. Thus, not only do we increase the success probability by about 1 order of magnitude (with the same resources), but also avoid the need for conducting quantum nondemolition measurement otherwise required to join two probabilistic gates. Subsequently, we generalize our method to a higher order, showing that n-times controlled gates can be optimized by replacing blocks of controlled-not gates with tunable controlled-phase gates.

Experimental eavesdropping based on optimal quantum cloning.

The security of quantum cryptography is guaranteed by the no-cloning theorem, which implies that an eavesdropper copying transmitted qubits in unknown states causes their disturbance. Nevertheless, in real cryptographic systems some level of disturbance has to be allowed to cover, e.g., transmission losses. An eavesdropper can attack such systems by replacing a noisy channel by a better one and by performing approximate cloning of transmitted qubits which disturb them but below the noise level assumed by legitimate users. We experimentally demonstrate such symmetric individual eavesdropping on the quantum key distribution protocols of Bennett and Brassard (BB84) and the trine-state spherical code of Renes (R04) with two-level probes prepared using a recently developed photonic multifunctional quantum cloner [Lemr et al., Phys. Rev. A 85, 050307(R) (2012)]. We demonstrated that our optimal cloning device with high-success rate makes the eavesdropping possible by hiding it in usual transmission losses. We believe that this experiment can stimulate the quest for other operational applications of quantum cloning.

Experimental realization of linear-optical partial swap gates.

We present a linear-optical implementation of a class of two-qubit partial SWAP gates for polarization states of photons. Different gate operations, including the SWAP and entangling sqrt[SWAP], can be obtained by changing a classical control parameter, namely, the path difference in the interferometer. Reconstruction of output states, full quantum process tomography, and an evaluation of entanglement of formation prove very good performance of the gates.

Fiber-optics implementation of an asymmetric phase-covariant quantum cloner.

We present the experimental realization of optimal symmetric and asymmetric phase-covariant 1-->2 cloning of qubit states using fiber optics. The state of each qubit is encoded into a single photon which can propagate through two optical fibers. The operation of our device is based on one- and two-photon interference. We have demonstrated the creation of two copies for a wide range of qubit states from the equator of the Bloch sphere. The measured fidelities of both copies are close to the theoretical values and they surpass the theoretical maximum obtainable with the universal cloner.

How quantum correlations enhance prediction of complementary measurements.

If there are correlations between two qubits, then the results of the measurement on one of them can help to predict measurement results on the other one. It is an interesting question as to what can be predicted about the results of two complementary projective measurements on the first qubit. To quantify these predictions the complementary knowledge excesses are used. A nontrivial constraint restricting them is derived. For any mixed state and for arbitrary measurements the knowledge excesses are bounded by a factor depending only on the maximal violation of Bell's inequalities. This result is experimentally verified on two-photon Werner states prepared by means of spontaneous parametric down-conversion.