This thesis investigates the entanglement of distinguishable and indistinguishable particles, introducing a new error model for Hardy's test, experimentally verified using superconducting qubits. We address challenges in implementing quantum protocols based on this test and propose potential solutions and present two performance measures for qubits in superconducting quantum computers. We demonstrate that if quantum particles can create hyper-hybrid entangled states and achieve unit fidelity quantum teleportation, arbitrary state cloning is possible. This leads to two no-go theorems: hyper-hybrid entangled states cannot be formed with distinguishable particles, and unit fidelity quantum teleportation is unattainable with indistinguishable particles. These results establish unique correlations for each particle type, creating a clear distinction between the two domains. We also show that hyper-hybrid entangled states can be formed with indistinguishable fermions and generalize this for both fermions and bosons. We develop a generalized DoF trace-out rule applicable to single or multiple degrees of freedom for both types of particles. This framework allows us to derive expressions for teleportation fidelity and singlet fraction, establishing an upper bound for the generalized singlet fraction. We present an optical circuit that generates entanglement in distinguishable particles. Using our trace-out rule, we show that for two indistinguishable particles with multiple DoFs, the monogamy of entanglement can be maximally violated. We assert that indistinguishability is essential for this violation in qubit systems. For three indistinguishable particles, we confirm that monogamy is upheld using squared concurrence. Finally, we propose a novel entanglement swapping protocol involving two indistinguishable particles, enhancing quantum networks and quantum repeaters.
This thesis investigates the entanglement of distinguishable and indistinguishable particles, introducing a new error model for Hardy's test, experimentally verified using superconducting qubits. We address challenges in implementing quantum protocols based on this test and propose potential solutions and present two performance measures for qubits in superconducting quantum computers. We demonstrate that if quantum particles can create hyper-hybrid entangled states and achieve unit fidelity quantum teleportation, arbitrary state cloning is possible. This leads to two no-go theorems: hyper-hybrid entangled states cannot be formed with distinguishable particles, and unit fidelity quantum teleportation is unattainable with indistinguishable particles. These results establish unique correlations for each particle type, creating a clear distinction between the two domains. We also show that hyper-hybrid entangled states can be formed with indistinguishable fermions and generalize this for both fermions and bosons. We develop a generalized DoF trace-out rule applicable to single or multiple degrees of freedom for both types of particles. This framework allows us to derive expressions for teleportation fidelity and singlet fraction, establishing an upper bound for the generalized singlet fraction. We present an optical circuit that generates entanglement in distinguishable particles. Using our trace-out rule, we show that for two indistinguishable particles with multiple DoFs, the monogamy of entanglement can be maximally violated. We assert that indistinguishability is essential for this violation in qubit systems. For three indistinguishable particles, we confirm that monogamy is upheld using squared concurrence. Finally, we propose a novel entanglement swapping protocol involving two indistinguishable particles, enhancing quantum networks and quantum repeaters.
Hyper-hybrid entanglement for two indistinguishable bosons has been recently proposed by Li et al. [Y. Li, M. Gessner, W. Li, and A. Smerzi, Phys. Rev. Lett. 120, 050404 (2018)]. In the current paper, we show that this entanglement exists for two indistinguishable fermions also. Next, we establish two no-go results: no hyper-hybrid entanglement for two distinguishable particles, and no unit fidelity quantum teleportation using indistinguishable particles. If either of these is possible, then the no-signaling principle would be violated. While several earlier works have attempted extending many results on distinguishable particles to indistinguishable ones, and vice versa, the above two no-go results establish a nontrivial separation between the two domains. Finally, we propose an efficient entanglement swapping using only two indistinguishable particles, whereas a minimum number of either three distinguishable or four indistinguishable particles is necessary for existing protocols.
Quantum teleportation efficiently transfers quantum information between distant locations by utilizing a pre-established composite system. Assessing the effectiveness of teleportation hinges on its fidelity, representing the similarity between input and output states. This fidelity, in turn, relies on a singlet fraction, quantifying the resemblance of the composite system to maximally entangled states. The relation between teleportation fidelity and singlet fraction given by Horodecki et al. [Phys. Rev. A 60, 1888 (1999)] does not hold for distinguishable particles with multiple degrees of freedom or indistinguishable particles with single or multiple degrees of freedom. In this paper, we propose generalized expressions for teleportation fidelity and singlet fraction and derive their relations, applicable to both distinguishable and indistinguishable particles with single or multiple degrees of freedom. We derive an upper bound for the generalized singlet fraction for distinguishable particles using the monogamy of singlet fraction by Kay et al. [Phys. Rev. Lett. 103, 050501 (2009)]. We also show how our relation helps characterize different types of composite states in terms of their distinguishability, separability, the presence of maximally entangled structure, and the number of degrees of freedom. We complement our theory with two practical illustrations. First, we demonstrate two counter-intuitive values of generalized singlet fraction using our optical circuit and the circuit of Li et al. [Phys. Rev. Lett. 120, 050404 (2018)]. Finally, we show that using an additional degree of freedom as an ancilla instead of a particle can be advantageous in quantum cryptographic protocols.
Two important results of quantum physics are the no-cloning theorem and the monogamy of entanglement. The former forbids the creation of an independent and identical copy of an arbitrary unknown quantum state and the latter restricts the shareability of quantum entanglement among multiple quantum systems. For distinguishable particles, one of these results implies the other. In this Letter, we show that in qubit systems with indistinguishable particles (where each particle cannot be addressed individually) a maximum violation of the monogamy of entanglement is possible by the measures that are monogamous for distinguishable particles. To derive this result, we formulate the degree of freedom trace-out rule for indistinguishable particles corresponding to a spatial location where each degree of freedom might be entangled with the other degrees of freedom. Our result removes the restriction on the shareability of quantum entanglement for indistinguishable particles, without contradicting the no-cloning theorem.
This thesis develops the quantitative study of quantum entanglement in systems of identical particles. Understanding this topic is essential for the construction of quantum information processing devices involving identical particles. A brief overview of necessary concepts and methods, such as the density matrix, the entanglement in pure and mixed states of distinguishable particles, and some common applications of entanglement is given in the introduction. Some competing methods of calculating the entanglement in bipartite pure states of indistinguishable particles are examined. It is shown that only the 'site entropy' measure introduced by Zanardi satisfies all the criteria for a correct entanglement measure. A teleportation protocol which utilizes all the entanglement carried (in both the spin and space degrees of freedom) in a doubly- occupied molecular bonding orbital is presented. The output from an interferometer in a thought experiment described by Omar et al. is studied as an example to see whether entanglement can be separated into space-only, spin-only, and space-spin components. A similar exercise is performed for a doubly-occupied molecular bonding orbital. The relationship between these results and the application of superselection rules (SSRs) to the quantification of useful entanglement is discussed. A numerical method for estimating the entanglement of formation of a mixed state of arbitrary dimension by a conjugate gradient algorithm is described. The results of applying an implementation of the algorithm to both random and isotropic states of 2 qutrits (i.e. two three-dimensional systems) is described. Existing work on calculating entanglement between two sites in various spin systems is outlined. New methods for calculating the entanglement between two sites in various types of degenerate quantum gas - a Fermi gas, a Bose condensate, and a BCS superconductor - are described. The results of numerical studies of the entanglement in a normal metal and a BCS superconductor are reported, both with and without the application of superselection rules for local particle number conservation.
Recently, it has been realized that indistinguishability is a resource for quantum information processing. A new method to represent the indistinguishable particles by Franco et al. (Sci Rep 6:20603, 2016, https://doi.org/10.1038/srep20603 ) and measure the concurrence is developed by Nosrati et al. (npj Quantum Inf 6:39, 2020, https://doi.org/10.1038/s41534-020-0271-7 ). The monogamy property says that quantum entanglement cannot be shared freely between more than two particles. For three distinguishable particles, the monogamy of entanglement was first expressed as an inequality using squared concurrence where each particle has a single degree of freedom (for pure or mixed states). Using multiple degrees of freedom, similar inequality was shown to be held between two distinguishable particles. However, for two indistinguishable particles, where each particle cannot be addressed individually, the monogamy inequality was shown to be violated maximally for a specific state. Thus a question naturally arises: what happens to the monogamy of entanglement in the case of three or more indistinguishable particles? We prove that monogamy holds in this scenario and the inequality becomes equality for all pure indistinguishable states. Further, we provide three major operational meanings of our result. Finally, we present an experimental schematic using photons to observe our result.
There are two traditionally-separated approaches to optical quantum information: the continuous- and discrete-variable strategies respectively linked to the wave-like and particle-like nature of light. This thesis work is focused on the novel hybrid approach aiming to join the capabilities of both strategies into single systems. Using hybrid methods, we report on the use of optical parametric oscillators and superconducting-nanowire single-photon detectors to generate highly non-classical non-Gaussian states such as high-purity single-photons, Schr\"odinger cat states and finally hybrid entanglement of light between continuous- and discrete-variable encoding. The potential of this resource is studied in a number of protocols. We first consider non-locality tests with hybrid entanglement such as Bell inequality violation and experimentally demonstrate the violation of Einstein-Podolsky-Rosen steering inequalities. We finally report on the progress made towards the implementation of a new setup for the demonstration of hybrid quantum teleportation between continuous and discrete-variable encodings. Our demonstrations prove the versatility of this hybrid resource and open the possibility of implementing scalable quantum networks linking systems of dissimilar nature.