Compression for quantum population coding

Abstract

Abstract-We study the compression of n quantum systems, each prepared in the same state belonging to a given parametric family of quantum states. For a family of states with f independent parameters, we devise an asymptotically faithful protocol that requires a hybrid memory of size (f /2) log n, including both quantum and classical bits. Our construction uses a quantum version of local asymptotic normality and, as an intermediate step, solves the problem of compressing displaced thermal states of n identically prepared modes. In both cases, we show that (f /2) log n is the minimum amount of memory needed to achieve asymptotic faithfulness. In addition, we analyze how much of the memory needs to be quantum. We find that the ratio between quantum and classical bits can be made arbitrarily small, but cannot reach zero: unless all the quantum states in the family commute, no protocol using only classical bits can be faithful, even if it uses an arbitrarily large number of classical bits.

Compression for Qubit Clocks

“…The memory cost is (1/2) log n qubits in the leading order, with the same amount of ancillary classical bits if the purity of the clock state is undetermined. The cost matches the general statement in [15], which says that (1/2) logn (qu)bits are needed per degree of freedom. Compared to [15], the protocols here are constructed using a completely different approach from the protocols, tailor-made for qubit clock states: The second order term of its memory cost is O(log log n) in contrast to x log n (for any positive x) as in [15].…”

“…The cost matches the general statement in [15], which says that (1/2) logn (qu)bits are needed per degree of freedom. Compared to [15], the protocols here are constructed using a completely different approach from the protocols, tailor-made for qubit clock states: The second order term of its memory cost is O(log log n) in contrast to x log n (for any positive x) as in [15]. The error of the protocols here also vanishes faster than that of the protocols in [15] as n grows large.…”

Communication Cost of Quantum Processes

“…This task is very similar to the simulation task considered in this paper, except that the parameter t is now invisible. The counterpart of this task for states is the task of compressing multicopy states, recently studied both theoretically [53], [54], [55], [56], [46] and experimentally [57]. The task of channel compression is more involved since the input of the channel is not necessary in the many-copy form, and these compression protocols for states cannot be applied directly.…”

Attaining the Ultimate Precision Limit in Quantum State Estimation

“…The calculations can be found in Appendix F. The quantum version of local asymptotic normality has been derived in several different forms [17,16,39] with applications in quantum statistics [40,12], benchmarks [41] and data compression [42]. Here we use the version of [17], which states that n identical copies of a qudit state can be locally approximated by a c-q Gaussian state in the large n limit.…”

“…(35) RLD quantum Fisher information at t 0 J t 0 Eq. (42) general method is illustrated through examples. The conclusions are drawn in Section 11.…”