We consider the question of whether it is possible to convert the entanglement between spatially separated modes of massive particles into a form that would allow the experimental observation of nonlocal quantum correlations. In the simplest setups analogous to optics experiments, that conversion is prohibited by fundamental conservation laws. However, we show that using auxiliary particles, that goal can be achieved with varying levels of success, depending on the nature and number of auxiliary particles used. In particular, we find that an auxiliary Bose-Einstein condensate allows the conversion arbitrarily many times with a small error that depends only on the initial state of the condensate.
We consider the question of whether it is possible to convert the entanglement between spatially separated modes of massive particles into a form that would allow the experimental observation of nonlocal quantum correlations. In the simplest setups analogous to optics experiments, that conversion is prohibited by fundamental conservation laws. However, we show that using auxiliary particles, that goal can be achieved with varying levels of success, depending on the nature and number of auxiliary particles used. In particular, we find that an auxiliary Bose-Einstein condensate allows the conversion arbitrarily many times with a small error that depends only on the initial state of the condensate.
We consider the question of whether it is possible to use the entanglement between spatially separated modes of massive particles to observe nonlocal quantum correlations. Mode entanglement can be obtained using a single particle, indicating that it requires careful consideration before concluding whether experimental observation---e.g., violation of Bell inequalities---is possible or not. In the simplest setups analogous to optics experiments, that observation is prohibited by fundamental conservation laws. However, we show that using auxiliary particles, mode entanglement can be converted into forms that allow the observation of quantum nonlocality. The probability of successful conversion depends on the nature and number of auxiliary particles used. In particular, we find that an auxiliary Bose-Einstein condensate allows the conversion arbitrarily many times with a small error that depends only on the initial state of the condensate.
Experiments violating Bell inequalities have formed our belief that the world at its smallest is genuinely nonlocal. While many nonlocality experiments use the first quantized picture, the physics of fields of indistinguishable particles is captured most conveniently by second quantization. This implies the possibility of nonlocal correlations, such as entanglement, between modes of the field. In this paper, we propose an experimental scheme that tests the theoretically predicted entanglement between modes in space occupied by massive bosons. A successful test of this scheme would not only indicate that mode entanglement is as genuine as particle entanglement, despite the particle number superselection rule, but would also prove that this superselection rule can be overcome.
Two spatially separated observers, Alice and Bob, share a bipartite two-qubit entangled state and perform measurements to witness entanglement. After their measurements, they pass their qubits to a subsequent pair of observers who try to perform the same task independently, and so on. Here we ask: what is the maximum number of such pairs that can perform this task successfully? It has previously been conjectured that not more than one pair of observers can detect Clauser-Horne-Shimony-Holt ``Bell-nonlocal'' correlations in this setup. We prove that, on the contrary, entanglement can be witnessed by arbitrarily many pairs of observers. The dissimilar nature between entanglement and Bell-nonlocal correlations is therefore uncovered in a rather radical way, when considering sequentially acting pairs of observers. We prove this statement to be true when the initial shared state is any pure entangled state, a class of Bell-nonlocal mixed states, or a class of Bell-local entangled states.
These two accompanying papers treat two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems where the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. We focus on the meaning of entanglement, the quantum paradoxes associated with entangled states, and important tests that allow an experimentalist to determine whether a quantum state - in particular, one for massive bosons is entangled. Our tests for entanglement fully utilise the symmetrisation principle and super-selection rules for bosonic massive particles. These papers provides detailed arguments necessary for the conclusions of a recent paper which presented results for rigorously demonstrating the entanglement of a two-mode Bose-Einstein condensate (BEC). In the first paper (I), we discuss the meaning of entanglement for systems of identical particles. For such systems, the relevant quantum density operators must satisfy the symmetrisation principle and global and local super-selection rules prohibiting states in which there are coherences between differing particle numbers. We fully justify these requirements. In the second quantisation approach, both the system and sub-systems are modes (or sets of modes) rather than particles, particles being associated with different mode occupancies. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement for two-mode bosonic systems. Starting from paper (I) we determine which tests are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, and recent key experiments are analysed.
The predictions of quantum mechanics often differ from our everyday experiences. In the classical case, the state of a system can be predicted precisely, assuming the exact knowledge of all parameters. This is not possible in quantum mechanics. For example, the spin of one particle can be simultaneously in two different states, until a measurement defines the actual state. If multiple particles are in such superposition states, their spins can be coupled with each other such that the measurement of one particle changes the physical state of the other particles. This coupling effect is a fundamental part of quantum mechanics and is called entanglement. It is a central requirement for applications in the fields of quantum communication, quantum cryptography and quantum computing. One possibility to create entanglement are spin-changing collisions in a spinor Bose-Einstein condensate. These collisions are described by a nonlinear process, which creates non-classical correlations between all atoms of the Bose-Einstein condensate. Within this thesis, 87Rb spinor Bose-Einstein condensates are used to demonstrate Einstein-Podolsky-Rosen entanglement by the generation and analysis of a two-mode entangled state in spin space [A1]. These states are applied in an interferometric measurement with a resolution of 2.05+0.34 -0.37 dB beyond the standard quantum limit [A2]. Depending on the dynamics, the process can also be analogous to the dynamical Casimir effect [A3]. The entanglement between the particles in these publications is strongly connected with their indistinguishability. It is thus possible to make the particles distinguishable again and recover the entanglement between the now separated particles? This question is approached by cutting an entangled state of indistinguishable particles into two spatially separated atomic modes - creating a two-mode entangled state in real space. As shown in Ref. [A4], the entanglement between the two separated clouds of a Bose-Einstein condensate can be measured directly. In the future, the created state can be employed for a test of quantum nonlocality.
Abstract These two accompanying papers are concerned with two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems in which the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. There are many aspects of entanglement that can be studied. This two-part review focuses on the meaning of entanglement, the quantum paradoxes associated with entangled states, and the important tests that allow an experimentalist to determine whether a quantum state—in particular, one for massive bosons is entangled. An overall outcome of the review is to distinguish criteria (and hence experiments) for entanglement that fully utilize the symmetrization principle and the super-selection rules that can be applied to bosonic massive particles. In the first paper (I), the background is given for the meaning of entanglement in the context of systems of identical particles. For such systems, the requirement is that the relevant quantum density operators must satisfy the symmetrization principle and that global and local super-selection rules prohibit states in which there are coherences between differing particle numbers. The justification for these requirements is fully discussed. In the second quantization approach that is used, both the system and the sub-systems are modes (or sets of modes) rather than particles, particles being associated with different occupancies of the modes. The definition of entangled states is based on first defining the non-entangled states—after specifying which modes constitute the sub-systems. This work mainly focuses on the two mode entanglement for massive bosons, but is put in the context of tests of local hidden variable theories, where one may not be able to make the above restrictions. The review provides the detailed arguments necessary for the conclusions of a recent paper, where the question of how to rigorously demonstrate the entanglement of a two-mode Bose–Einstein condensate (BEC) has been examined. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement that have been proposed for two-mode bosonic systems. We apply the approach of review (I) to determine which tests, and which modifications of the tests, are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, a theory for the required two-mode interferometry is presented, and key experiments to date are analyzed.