Local update recovery seeks to maintain quantum information by applying local
correction maps alternating with and compensating for the action of noise.
Motivated by recent constructions based on quantum LDPC codes in the
finite-dimensional setting, we establish an analytic upper bound on the
fault-tolerance threshold for concatenated GKP-stabilizer codes with local
update recovery. Our bound applies to noise channels that are tensor products
of one-mode beamsplitters with arbitrary environment states, capturing, in
particular, photon loss occurring independently in each mode. It shows that for
loss rates above a threshold given explicitly as a function of the locality of
the recovery maps, encoded information is lost at an exponential rate. This
extends an early result by Razborov from discrete to continuous variable (CV)
quantum systems.
To prove our result, we study a metric on bosonic states akin to the
Wasserstein distance between two CV density functions, which we call the
bosonic Wasserstein distance. It can be thought of as a CV extension of a
quantum Wasserstein distance of order 1 recently introduced by De Palma et al.
in the context of qudit systems, in the sense that it captures the notion of
locality in a CV setting. We establish several basic properties, including a
relation to the trace distance and diameter bounds for states with finite
average photon number. We then study its contraction properties under quantum
channels, including tensorization, locality and strict contraction under
beamsplitter-type noise channels. Due to the simplicity of its formulation, and
the established wide applicability of its finite-dimensional counterpart, we
believe that the bosonic Wasserstein distance will become a versatile tool in
the study of CV quantum systems.Comment: 30 pages, 2 figure