Aspects of Quantum Entanglement and Indistinguishability

Abstract

Entanglement of distinguishable and indistinguishable particles under different sce- narios and related properties and results constitute the core component of this thesis. We propose a new error-modeling for Hardy’s test and also perform experimental verification of it in superconducting qubits. Further, we point out the difficulties associated with the practical implementation of quantum protocols based on Hardy’s test and propose possible remedies. We also propose two performance measures for any two qubits of any quantum computer based on superconducting qubits. Next, we prove that if quantum particles (either distinguishable or indistinguish- able) can simultaneously produce and perform hyper-hybrid entangled state and unit fidelity quantum teleportation respectively then using that cloning of any arbitrary quantum state is possible. This theorem results two no-go theorems: (1) hyper-hybrid entangled state is not possible for distinguishable particles and (2) unit fidelity quan- tum teleportation is not possible for indistinguishable particles. These theorems establish that there exists some quantum correlation or application unique to in- distinguishable particles only and yet some unique to distinguishable particles only, giving a separation between the two domains. We also establish that the hyper-hybrid entangled state is possible using two indistinguishable fermions and we generalize it for bosons and fermions.We establish a generalized degree of freedom trace-out rule that covers single or multiple degree of freedom scenarios for both distinguishable and indistinguishable systems. Using this, we propose generalized expressions for teleportation fidelity and singlet fraction and derive their relations, applicable for both distinguishable and in- distinguishable particles with single or multiple degrees of freedom. We also derive an upper bound for the generalized singlet fraction for distinguishable and indistin- guishable particles. We further show how our relation helps to characterize different types of composite states in terms of their distinguishability, separability, presence of maximally entangled structure and the number of degrees of freedom. Finally, we demonstrate an optical circuit to generate entanglement for distinguishable particles each having two degrees of freedom and characterize it using our relation. Further, using generalized degree of freedom trace-out rule, we show that, for two indistinguishable particles each having more than one degree of freedom, the monogamy of entanglement can be violated maximally using the measures that are monogamous for distinguishable particles. This results the following theorem that “In qubit systems, indistinguishability is a necessary criterion for maximum viola- tion of monogamy of entanglement by the same measures that are monogamous for distinguishable particles". For three indistinguishable particles each having multiple degree of freedom, we show that monogamy of entanglement is obeyed using squared concurrence as an entanglement measure. We also establish that the monogamy inequality becomes equality for all pure indistinguishable states, but the inequality remains for mixed indistinguishable states. This can be used as a one-sided test of distinguishability for particles in pure states We show that the cost of adding an ancilla particle can be bypassed by using an additional degrees of freedom and creating multi-degree of freedom entanglement. Next, we show that entangled indistinguishable particles may alter certain important parameters in cryptographic protocols, in particular, we demonstrate how indistin- guishability can change Hardy’s probability Finally we propose a novel entanglement swapping protocol without Bell state measurement using only two indistinguishable particles that will be very useful in quantum networks specially in quantum repeaters.