Generalized Bell Measurements and Equiangular Lines

Abstract

This thesis is ultimately concerned with a natural and optimal generalization of the familiar Bell state measurement on four basis states. Measurements in quantum mechanics are interactive processes, meaning the observer actively changes the state they extract information from. Different measurements differ in the degree of the disturbance induced, and in the amount and type of information gained. For example, some measurements result in well-defined post-measurement states while others provide only outcome statistics. The variety of measurements each have unique features enabling all sorts of fine-tuned and tailored manipulations of quantum states.

Another central feature of the Bell state measurement which the generalized measurements inherit is that the measurements are constructed of maximally entangled states. Mathematically, this introduces an interesting restriction of the construction of equiangular lines to the subset of maximally entangled states. Quantum mechanically, this opens the door to unknown yet exciting possibilities for symmetric joint measurements as SIC-POVMs have done for measurements over single qudits.

In this thesis, equiangular sets of the simplest case of bipartite-qubit states are considered and interpreted as generalized Bell measurements. The construction of this generalization borrows techniques from frame theory and builds on the tradition of equiangular lines in real and complex dimensions, so a treatment of the relevant mathematics is presented first. Then a central point of distinction between the generalized Bell measurements and the Bell state measurement is the difference between projective measurements and positive-operator valued ones, so a discussion of measurements in quantum mechanics and entanglement comes next. Finally, explicit constructions are given and studied in the setting of quantum state discrimination. It is found that the generalized five-state measurement is comparatively more non-local, or less distinguishable, than the Bell
state measurement even with access to two copies. Furthermore, 5-state separable sets and 6-state maximally entangled sets are constructed. Various properties including uniqueness and optimality are also discussed.

Popular Abstract

The Bell state measurement is a measurement on the spins of two qubits whose outcome is one of four maximally entangled states. This measurement is distinguished from other measurements in quantum mechanics principally because the Bell state measurement is that of the correlations of two potentially distant qubits. Whereas measurements of spin or angular momentum are measurements of properties intrinsic to the measured particle, the outcome of the Bell measurement on a maximally entangled state determines the spins of two qubits to be either parallel or anti-parallel and the state to be either symmetric or anti-symmetric with respect to the exchanging of the particles. In other words, the Bell measurement is truly a measurement of nonlocal correlations between particles. This measurement is essential in various quantum information protocols including teleportation, dense coding, and entanglement swapping.

It is known that in quantum mechanics, not all measurements are created equal. All non-trivial measurements effect perturbations on the state, but some gently probe the state while others effectively destroy all the state information upon measurement and provide well-defined post-measurement states. The latter are the source of the popularly recognized “instantaneous collapse” property of quantum states upon observation. The Bell state measurement belongs to this class of measurements. Later, it was realized that by borrowing structures of remarkable symmetry from discrete mathematics, less-destructive generalized measurements of single particles can be constructed which have enabled optimally efficient and unbiased quantum state tomography, more robust entanglement detection, more secure and efficient quantum key distribution, etc. Could such symmetric measurements be analogously constructed for the nonlocal measurements described earlier? If so, what quantum information tasks stand to benefit from the generalized Bell measurements? These are the questions addressed in this thesis.

• author: Wei, Amanda
• supervisor: Armin Tavakoli
• organization: Department of Physics; Mathematical Physics
• course: FYSM33 20232
• year: 2024
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: POVM, quantum measurements, equiangular lines, Bell measurement
• language: English
• id: 9170960
• date added to LUP: 2024-08-12 09:42:22
• date last changed: 2024-08-12 09:42:22