It has been observed by numerous authors that a quantum system being entangled with another one limits its possible entanglement with a third system: this has been dubbed the "monogamous nature of entanglement". In this paper we present a simple identity which captures the trade-off between entanglement and classical correlation, which can be used to derive rigorous monogamy relations.We also prove various other trade-offs of a monogamy nature for other entanglement measures and secret and total correlation measures.

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The exact eigenspectra and eigenstates of spin-1 and spin-2 vectorial Bose-Einstein condensates (BECs) are found, and their response to a weak magnetic field is studied and compared with their mean-field counterparts. Whereas mean-field theory predicts the vanishing population of the zero magnetic-quantum-number component of a spin-1 antiferromagnetic BEC, the component is found to become populated as the magnetic field decreases. The spin-2 BEC exhibits an even richer magnetic response due to quantum correlation between 3 bosons. PACS numbers: 03.75. Fi, 05.30.Jp Bose-Einstein condensates (BECs) of alkali-metal atoms have internal degrees of freedom due to the hyperfine spin of the atoms. These degrees of freedom are frozen in a magnetic trap, but an optical trap liberates them to allow BEC to be in a superposition of magnetic sublevels [1]. BEC is therefore described by a vectorial rather than scalar order parameter. A new feature in this BEC system as compared to superfluid helium-three is the fact that its response to an external magnetic field is dominated by electronic rather than nuclear spin, and hence the response is much stronger than that of superfluid helium-three. This opens up possibilities of manipulating the magnetism of superfluid vapors. Observation of spin domains by an MIT group [2] offers an excellent example of such manipulations. While the experiments reported so far achieved only the spin-1 vectorial BEC, the spin-2 BEC also appears feasible by using the F = 2 multiplet of bosons such as 23 Na, 87 Rb, or 85 Rb. The mean-field theory (MFT) for describing a vectorial BEC was developed independently by Ohmi and Machida [3] and by Ho [4] by generalizing the Gross-Pitaevskii equation under the restriction of gauge and spin-rotation symmetry; they also used it to predict various spin textures and topological excitations. Law et al.[5] utilized techniques developed in quantum optics [6,7] to study many-body states of spin-1 BEC in the absence of external fields, and found that spin-exchange collisions lead to rather complicated dynamical behavior of BEC that MFT fails to capture. In this Letter, we study magnetic response of spin-1 and spin-2 BECs by explicitly constructing exact eigenspectra and eigenstates, and compare the results with their mean-field counterparts.We first consider a system of spin-1 bosons interacting via s-wave scattering. The second-quantized Hamiltonian of the bosons subject to a uniform magnetic field B and in a confining potential U (r) is given bŷwhere M is the mass of the bosons,Ψ α describes their field operator with magnetic quantum number α = −1, 0, 1, andc 0 andc 1 are related to scattering lengths a 0 and a 2 of two colliding bosons with total angular momentum 0 and 2 byc 0 = 4πhHere and in the rest of this Letter, it is assumed that repeated indices are to be summed, and that the total number N of bosons in the system is fixed. We further assume that the external magnetic field is weak and |c 1 | ≪ c 0 so that the coordinate wave function φ(r) is independe...

Quantum cryptography exploits the fundamental laws of quantum mechanics to provide a secure way to exchange private information. Such an exchange requires a common random bit sequence, called a key, to be shared secretly between the sender and the receiver. The basic idea behind quantum key distribution (QKD) has widely been understood as the property that any attempt to distinguish encoded quantum states causes a disturbance in the signal. As a result, implementation of a QKD protocol involves an estimation of the experimental parameters influenced by the eavesdropper's intervention, which is achieved by randomly sampling the signal. If the estimation of many parameters with high precision is required, the portion of the signal that is sacrificed increases, thus decreasing the efficiency of the protocol. Here we propose a QKD protocol based on an entirely different principle. The sender encodes a bit sequence onto non-orthogonal quantum states and the receiver randomly dictates how a single bit should be calculated from the sequence. The eavesdropper, who is unable to learn the whole of the sequence, cannot guess the bit value correctly. An achievable rate of secure key distribution is calculated by considering complementary choices between quantum measurements of two conjugate observables. We found that a practical implementation using a laser pulse train achieves a key rate comparable to a decoy-state QKD protocol, an often-used technique for lasers. It also has a better tolerance of bit errors and of finite-sized-key effects. We anticipate that this finding will give new insight into how the probabilistic nature of quantum mechanics can be related to secure communication, and will facilitate the simple and efficient use of conventional lasers for QKD.

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We implemented a joint weak measurement of the trajectories of two photons in a photonic version of Hardy's experiment. The joint weak measurement has been performed via an entangled meter state in polarization degrees of freedom of the two photons. Unlike Hardy's original argument in which the contradiction is inferred by retrodiction, our experiment reveals its paradoxical nature as preposterous values actually read out from the meter. Such a direct observation of a paradox gives us new insights into the spooky action of quantum mechanics.

The ground states of Bose-Einstein condensates of spin-2 bosons are classified into three distinct (ferromagnetic, "antiferromagnetic", and cyclic) phases depending on the s-wave scattering lengths of binary collisions for total-spin 0, 2, and 4 channels. Many-body spin correlations and magnetic response of the condensate in each of these phases are studied in a mesoscopic regime, while lowlying excitation spectra are investigated in the thermodynamic regime. In the mesoscopic regime, where the system is so tightly confined that the spatial degrees of freedom are frozen, the exact, many-body ground state for each phase is found to be expressed in terms of the creation operators of pair or trio bosons having spin correlations. These pairwise and trio-wise units are shown to bring about some unique features of spin-2 BECs such as a huge jump in magnetization from minimum to maximum possible values and the robustness of the minimum-magnetization state against an applied magnetic field. In the thermodynamic regime, where the system is spatially uniform, low-lying excitation spectra in the presence of magnetic field are obtained analytically using the Bogoliubov approximation. In the ferromagnetic phase, the excitation spectrum consists of one Goldstone mode and four single-particle modes. In the antiferromagnetic phase, where spin-singlet "pairs" undergo Bose-Einstein condensation, the spectrum consists of two Goldstone modes and three massive ones, all of which become massless when magnetic field vanishes. In the cyclic phase, where boson "trios" condense into a spin-singlet state, the spectrum is characterized by two Goldstone modes, one singleparticle mode having a magnetic-field-independent energy gap, and a gapless single-particle mode that becomes massless in the absence of magnetic field.

An experimental scheme for concentrating entanglement in partially entangled photon pairs is proposed. In this scheme, two separated parties obtain one maximally entangled photon pair from previously shared two partially entangled photon pairs by local operations and classical communication. A practical realization of the proposed scheme is discussed, which uses imperfect photon detectors and spontaneous parametric down-conversion as a photon source. This scheme also works for purifying a class of mixed states.