{equation, function, exp} {qubit, qubits, gate} {let, theorem, proof} {algorithm, log, probability} equation qubit let algorithm function qubits theorem log exp gate proof probability phys gates set problem order state matrix complexity solution operation lemma function functions computation matrices let {level, atom, field} {theory, mechanics, state} {states, state, optimal} {measurement, state, measurements} level theory states measurement atom mechanics state state field state optimal measurements atomic classical matrix probability atoms physical set stochastic state physics density process transition interpretation unitary information {group, space, representation} {time, decoherence, evolution} {field, particle, equation} {photon, photons, single} group time field photon space decoherence particle photons representation evolution equation single algebra state particles detector wigner environment classical detection representations phys momentum beam basis interaction theory polarization {particle, mechanics, theory} {state, states, entangled} {information, entropy, channel} {operator, operators, space} particle state information operator mechanics states entropy operators theory entangled channel space time entanglement log hamiltonian particles pure state time experiment local states real physical separable channels case {time, wave, function} {state, states, coherent} {energy, state, states} {light, field, probe} time state energy light wave states state field function coherent states probe phys mode phys noise probability phys fig optical initial number ground signal fig modes lattice fig {alice, bob, state} {key, protocol, security} {observables, space, algebra} {force, casimir, field} alice key observables force bob protocol space casimir state security algebra field measurement bit set phys protocol eve state boundary information bob observable energy teleportation information theorem vacuum {classical, space, random} {entanglement, phys, rev} {error, code, errors} {spin, pulse, spins} classical entanglement error spin space phys code pulse random rev errors spins phys entangled codes pulses distribution state correction electron time matrix ancilla magnetic map concurrence state nuclear {state, phys, rev} {trap, ion, state} {temperature, thermal, energy} {energy, gaussian, time} state trap temperature energy phys ion thermal gaussian rev state energy time states ions density nonlinear scheme laser number theory fidelity phys phys linear lett atoms electron phys {cavity, atom, atoms} {wave, scattering, interference} {state, algorithm, problem} {bell, inequality, local} cavity wave state bell atom scattering algorithm inequality atoms interference problem local field potential time phys mode barrier hamiltonian inequalities modes phys adiabatic bells mirror momentum cluster correlations {cos, sin, state} {vol, operators, histories} {phase, path, phys} {time, systems, information} cos vol phase time sin operators path systems state histories phys information angle operator geometric input rotation probability adiabatic space case set phases physical cosh decoherence non output