A central concept in quantum information processing is genuine multipartite entanglement (GME), a type of correlation beyond biseparability, that is, correlations that cannot be explained by statistical mixtures of partially separable states. GME is relevant for characterizing and benchmarking complex quantum systems, and it is an important resource for applications such as quantum communication. Remarkably, it has been found that GME can be activated from multiple copies of biseparable quantum states, which do not possess GME individually. Here, we experimentally demonstrate unambiguous evidence of such GME activation from two copies of a biseparable three-qubit state in a trapped-ion quantum processor. These results not only challenge notions of quantum resources but also highlight the potential of using multiple copies of quantum states to achieve tasks beyond the capabilities of the individual copies.

Multi-copy activation of genuine multipartite entanglement (GME) is a phenomenon whereby multiple copies of biseparable but fully inseparable states can exhibit GME. This was shown to be generically possible in finite dimensions. Here, we extend this analysis to infinite dimensions. We provide examples of GME-activatable non-Gaussian states. For Gaussian states, we apply a necessary biseparability criterion for the covariance matrix and show that it cannot detect GME activation. We further identify fully inseparable Gaussian states that satisfy the criterion but show that multiple and, in some cases, even single copies are GME. Thus, we show that the covariance-matrix biseparability criterion is not sufficient even for Gaussian states.

We derive a set of genuine multi-mode entanglement criteria for second moments of the quadrature operators. The criteria have a common form of the uncertainty relation between sums of variances of position and momentum quadrature combinations. A unique feature of the criteria is that the sums contain the least possible number of variances of at most two-mode combinations. The number of second moments we need to know to apply the criteria thus scales only linearly with the number of modes, as opposed to the quadratic scaling of the already existing criteria. Each criterion is associated with a tree graph, which allowed us to develop a direct method of construction of the criteria based solely on the structure of the underlying tree. The practicality of the proposed criteria is demonstrated by finding a number of examples of Gaussian states of up to six modes, whose genuine multi-mode entanglement is detected by them. The designed criteria are particularly suitable for verification of genuine multipartite entanglement in large multi-mode states or when only a set of two-mode nearest-neighbour marginal covariance matrices of the investigated state is available.

Quantum rotor represents the second simplest model after the harmonic oscillator with profound interdisciplinary but yet unexplored implications. It overreaches its mechanical meaning with significant consequences and promising applications in, e.g., singular optics, super-conductive circuits with Josephson junction or optimal pulse shaping in time frequency domains at ultimate quantum limit. Quantification of complementarity between angular momentum and angular variable is essential for exploitation of this canonical pair in quantum metrology. Whereas the natural measures for position and momentum are variances, the uncertainty associated with angular variable can be linked to moments of inertia of the unit ring about axes passing through its center of mass. This interpretation provides variants for saturable uncertainty relations which can be further used in quantum metrology of the quantum rotor and explains ambiguities in choice of possible uncertainty measures associated with sine and cosine functions of angular variable. Special attention will be payed to uncertainty products which are exactly or approximately minimised by von Mises states, which play the role of squeezed states for quantum rotor and allow optimal detection of quantum states at the ultimate limits.

We establish fundamental upper bounds on the amount of secret key that can be extracted from quantum Gaussian states by using local Gaussian operations, local classical processing, and public communication. For one-way public communication or when two-way public communication is allowed but Alice and Bob first perform destructive local Gaussian measurements, we prove that the key is bounded by the Renyi-2 Gaussian entanglement of formation EGF,2. The saturation of this inequality for pure Gaussian states provides an operational interpretation of the Renyi-2 entropy of entanglement as the secret key rate of pure Gaussian states accessible with Gaussian operations and one-way communication. In the general setting of two-way communication and arbitrary interactive protocols, we argue that 2EGF,2 still serves as an upper bound on the extractable key. We conjecture that the factor of 2 is spurious, suggesting that EGF,2 coincides with the secret key rate of Gaussian states under Gaussian measurements and two-way public communication. We use these results to prove a gap between the secret key rates obtainable with arbitrary versus Gaussian operations. This gap is observed for states produced by sending one half of a two-mode squeezed vacuum through a pure loss channel, in the regime of sufficiently low squeezing or sufficiently high transmissivity. Finally, for a wide class of Gaussian states, including all two-mode states, we prove a recently proposed conjecture on the equality between EGF,2 and the Gaussian intrinsic entanglement. The unified entanglement quantifier emerging from such an equality is then endowed with a direct operational interpretation as the value of a quantum teleportation game.

Reaching ultimate performance of quantum technologies requires the use of detection at quantum limits and access to all resources of the underlying physical system. We establish a full quantum analogy between the pair of angular momentum and exponential angular variable, and the structure of canonically conjugate position and momentum. This includes the notion of optimal simultaneous measurement of the angular momentum and angular variable, the identification of Einstein-Podolsky-Rosen-like variables and states, and, finally, a phase -space representation of quantum states. Our construction is based on close interconnection of the three concepts and may serve as a template for the treatment of other observables. This theory also provides a test bed for implementation of quantum technologies combining discrete and continuous quantum variables.

Reaching ultimate performance of quantum technologies requires the use of detection at quantum limits and access to all resources of the underlying physical system. We establish a full quantum analogy between the pair of angular momentum and exponential angular variable, and the structure of canonically conjugate position and momentum. This includes the notion of optimal simultaneous measurement of the angular momentum and angular variable, the identification of Einstein-Podolsky-Rosen-like variables and states, and finally a phase-space representation of quantum states. Our construction is based on close interconnection of the three concepts and may serve as a template for the treatment of other observables. This theory also provides a new testbed for implementation of quantum technologies combining discrete and continuous quantum variables.

Genuine multipartite entanglement underlies correlation experiments corroborating quantum mechanics and it is an expedient empowering many quantum technologies. One of many counterintuitive facets of genuine multipartite entanglement is its ability to exhibit an emergent character. That is, one can infer its presence in some multipartite states merely from a set of its separable marginals. Here we show that the effect can also be found in the context of Gaussian states of bosonic systems. Specifically, we construct examples of multimode Gaussian states carrying genuine multipartite entanglement which can be verified solely from separable nearest -neighbor two-mode marginals. The key tool of our construction is an entanglement witness acting only on some two-mode reductions of the global covariance matrix, which we find by a numerical solution of a semidefinite program. We also propose an experimental scheme for preparation of the simplest three-mode state, which requires interference of three correlatively displaced squeezed beams on two beam splitters. Besides revealing the concept of emergent genuine multipartite entanglement in the Gaussian scenario and bringing it closer to experimentally testable form, our results pave the way to effective diagnostics methods of global properties of multipartite states without complete tomography.

Oblivious transfer is an important primitive in modern cryptography. Applications include secure multiparty computation, oblivious sampling, e-voting, and signatures. Information-theoretically secure perfect 1-out-of 2 oblivious transfer is impossible to achieve. Imperfect variants, where both participants’ ability to cheat is still limited, are possible using quantum means while remaining classically impossible. Precisely what security parameters are attainable remains unknown. We introduce a theoretical framework for studying semirandom quantum oblivious transfer, which is shown to be equivalent to regular oblivious transfer in terms of cheating probabilities. We then use it to derive bounds on cheating. We also present a protocol with lower cheating probabilities than previous schemes, together with its optical realization. We show that a lower bound of 2/3 on the minimum achievable cheating probability can be directly derived for semirandom protocols using a different method and definition of cheating than used previously. The lower bound increases from 2/3 to approximately 0.749 if the states output by the protocol are pure and symmetric. The oblivious transfer scheme we present uses unambiguous state elimination measurements and can be implemented with the same technological requirements as standard quantum cryptography. In particular, it does not require honest participants to prepare or measure entangled states. The cheating probabilities are 3/4 and approximately 0.729 for sender and receiver, respectively, which is lower than in existing protocols. Using a photonic testbed, we have implemented the protocol with honest parties, as well as optimal cheating strategies. Because of the asymmetry of the receiver’s and sender’s cheating probabilities, the protocol can be combined with a “trivial” protocol to achieve an overall protocol with lower average cheating probabilities of approximately 0.74 for both sender and receiver. This demonstrates that, interestingly, protocols where the final output states are pure and symmetric are not optimal in terms of average cheating probability.

Genuine multipartite entanglement underlies correlation experiments corroborating quantum mechanics and it is an expedient empowering many quantum technologies. One of many counterintuitive facets of genuine multipartite entanglement is its ability to exhibit an emergent character, that is, one can infer its presence in some multipartite states merely from a set of its separable marginals. Here, we show that the effect can be found also in the context of Gaussian states of bosonic systems. Specifically, we construct examples of multimode Gaussian states carrying genuine multipartite entanglement which can be verified solely from separable nearest-neighbour two-mode marginals. The key tool of our construction is a genuine multipartite entanglement witness acting only on some two-mode reductions of the global covariance matrix, which we find by a numerical solution of a semi-definite programme. We also propose an experimental scheme for preparation of the simplest three-mode state, which requires interference of three correlatively displaced squeezed beams on two beam splitters. Besides revealing the concept of emergent genuine multipartite entanglement in the Gaussian scenario and bringing it closer to experimentally testable form, our results pave the way to effective diagnostics methods of global properties of multipartite states without complete tomography.

Oblivious transfer (OT) is a cryptographic primitive which is universal for multiparty computation. Unfortunately, perfect information-theoretically secure (ITS) quantum oblivious transfer is impossible. Imperfect information-theoretically secure quantum oblivious transfer is possible, but the smallest possible cheating probabilities are not known. We present an imperfect information-theoretically secure quantum oblivious transfer protocol with no restrictions on dishonest parties, and its experimental implementation. The cheating probabilities are 0.75 and 0.729 for sender and receiver respectively, which is lower than in existing protocols. Using a photonic test-bed, we have implemented the protocol with honest parties, as well as optimal cheating strategies.

We construct a classical cryptographic analog of multipartite entanglement which can be verified solely from separable marginals. That is a set of marginal classical probability distributions carrying no secret correlations, which are compatible only with a joint distribution containing secret correlations. This demonstrates that the emergent correlation properties do not exist only in the quantum world. Viewed from a different perspective, our result reveals another possible application of the quantum marginal problems in that they can help us to design and solve interesting classical marginal problems.

We prove experimentally the predicted existence of a three-qubit quantum state with genuine multipartite entanglement which can be certified solely from its separable two-qubit reduced density matrices.The qubits are encoded into different degrees of freedom of a pair of correlated photons and the state is prepared by letting the photons to propagate through a linear optical circuit.The presence of genuine multipartite entanglement is verified by finding numerically a fully decomposable entanglement witness acting nontrivially only on the reductions of the global state.Our result confirms viability of detection of emerging global properties of composite quantum systems from their parts which lack the properties.

The distribution of quantum correlations among remote users is a key procedure underlying many quantum information technologies. Einstein-Podolsky-Rosen steering, which is one kind of such a correlation stronger than entanglement, has been identified as a resource for secure quantum networks. We show that this resource can be established between two and even more distant parties by transmission of a system being separable from all the parties. For two-user scenarios, we design a protocol allowing one to distribute one-way Gaussian steering which can be used subsequently for one-sided device-independent (1sDI) quantum key distribution. Further, we extend the protocol to three-user scenarios to distribute richer steerability properties including one-to-multimode steering and collective steering which can be used for 1sDI quantum secret sharing. All the proposed protocols can be implemented with squeezed states, beam splitters, and displacements, and thus they can be readily realized experimentally. Our findings reveal that not only entanglement but even steering can be distributed via communication of a separable system. Viewed from a different perspective, the present protocols also demonstrate that one can switch multipartite states between different steerability classes by operations on parts of the states.

We develop a recently proposed theory of a quantifier of bipartite Gaussian entanglement called Gaussian intrinsic entanglement (GIE) [L. Mista, Jr. and R. Tatham, Phys. Rev. Lett. 117, 240505 (2016)]. Gaussian intrinsic entanglement provides a compromise between computable and physically meaningful entanglement quantifiers and so far it has been calculated for two-mode Gaussian states including all symmetric partialminimumuncertainty states, weakly mixed asymmetric squeezed thermal states with partial minimum uncertainty, and weakly mixed symmetric squeezed thermal states. We improve the method of derivation of GIE and show that all previously derived formulas for GIE of weakly mixed states in fact hold for states with higher mixedness. In addition, we derive analytical formulas for GIE for several other classes of two-mode Gaussian states with partial minimum uncertainty. Finally, we show that, like for all previously known states, also for all currently considered states the GIE is equal to Gaussian Renyi-2 entanglement of formation. This finding strengthens a conjecture about the equivalence of GIE and Gaussian Renyi-2 entanglement of formation for all bipartite Gaussian states.

Einstein-Podolsky-Rosen (EPR) steering is an asymmetric form of correlations which is intermediate between quantum entanglement and Bell nonlocality, and can be exploited as a resource for quantum communication with one untrusted party. In particular, steering of continuous-variable Gaussian states has been extensively studied theoretically and experimentally, as a fundamental manifestation of the EPR paradox. While most of these studies focused on quadrature measurements for steering detection, two recent works revealed that there exist Gaussian states which are only steerable by suitable non-Gaussian measurements. In this paper we perform a systematic investigation of EPR steering of bipartite Gaussian states by pseudospin measurements, complementing and extending previous findings. We first derive the density-matrix elements of two-mode squeezed thermal Gaussian states in the Fock basis, which may be of independent interest. We then use such a representation to investigate steering of these states as detected by a simple nonlinear criterion, based on second moments of the correlation matrix constructed from pseudospin operators. This analysis reveals previously unexplored regimes where non-Gaussian measurements are shown to be more effective than Gaussian ones to witness steering of Gaussian states in the presence of local noise. We further consider an alternative set of pseudospin observables, whose expectation value can be expressed more compactly in terms of Wigner functions for all two-mode Gaussian states. However, according to the adopted criterion, these observables are found to be always less sensitive than conventional Gaussian observables for steering detection. Finally, we investigate continuous-variable Werner states, which are non-Gaussian mixtures of Gaussian states, and find that pseudospin measurements are always more effective than Gaussian ones to reveal their steerability. Our results provide useful insights on the role of non-Gaussian measurements in characterizing quantum correlations of Gaussian and non-Gaussian states of continuous-variable quantum systems.

We introduce a cryptographically motivated quantifier of entanglement in bipartite Gaussian systems called Gaussian intrinsic entanglement (GIE). The GIE is defined as the optimized mutual information of a Gaussian distribution of outcomes of measurements on parts of a system, conditioned on the outcomes of a measurement on a purifying subsystem. We show that GIE vanishes only on separable states and exhibits monotonicity under Gaussian local trace-preserving operations and classical communication. In the two-mode case, we compute GIE for all pure states as well as for several important classes of symmetric and asymmetric mixed states. Surprisingly, in all of these cases, GIE is equal to Gaussian Rényi-2 entanglement. As GIE is operationally associated with the secret-key agreement protocol and can be computed for several important classes of states, it offers a compromise between computable and physically meaningful entanglement quantifiers.

L.M. acknowledges Project No. P205/12/0694 of Czech Science Foundation (GACR). N.K. is grateful for the support provided by the A. von Humboldt Foundation. N.Q. and N.K. acknowledge the support from the Scottish Universities Physics Alliance (SUPA) and the Engineering and Physical Sciences Research Council (EPSRC). The project was supported within the framework of the BMBF grant “QuORep” and in the framework of the International Max Planck Partnership (IMPP) with Scottish Universities.

A beam splitter is a basic linear optical element appearing in many optics experiments and is frequently used as a continuous-variable entangler transforming a pair of input modes from a separable Gaussian state into an entangled state. However, a beam splitter is a passive operation that can create entanglement from Gaussian states only under certain conditions. One such condition is that the input light is suitably squeezed. We demonstrate, experimentally, that a beam splitter can create entanglement even from modes which do not possess such a squeezing provided that they are correlated to, but not entangled with, a third mode. Specifically, we show that a beam splitter can create three-mode entanglement by acting on two modes of a three-mode fully separable Gaussian state without entangling the two modes themselves. This beam splitter property is a key mechanism behind the performance of the protocol for entanglement distribution by separable states. Moreover, the property also finds application in collaborative quantum dense coding in which decoding of transmitted information is assisted by interference with a mode of the collaborating party.

Intrinsic entanglement (IE) is a quantity which aims at quantifying bipartite entanglement carried by a quantum state as an optimal amount of the intrinsic information that can be extracted from the state by measurement. We investigate in detail the properties of a Gaussian version of IE, the so-called Gaussian intrinsic entanglement (GIE). We show explicitly how GIE simplifies to the mutual information of a distribution of outcomes of measurements on a conditional state obtained by a measurement on a purifying subsystem of the analyzed state, which is first minimized over all measurements on the purifying subsystem and then maximized over all measurements on the conditional state. By constructing for any separable Gaussian state a purification and a measurement on the purifying subsystem which projects the purification onto a product state, we prove that GIE vanishes on all Gaussian separable states. Via realization of quantum operations by teleportation, we further show that GIE is non-increasing under Gaussian local trace-preserving operations and classical communication. For pure Gaussian states and a reduction of the continuous-variable GHZ state, we calculate GIE analytically and we show that it is always equal to the Gaussian Rényi-2 entanglement. We also extend the analysis of IE to a non-Gaussian case by deriving an analytical lower bound on IE for a particular form of the non-Gaussian continuous-variable Werner state. Our results indicate that mapping of entanglement onto intrinsic information is capable of transmitting also quantitative properties of entanglement and that this property can be used for introduction of a quantifier of Gaussian entanglement which is a compromise between computable and physically meaningful entanglement quantifiers.