How much entanglement is needed for quantum error correction?
arxiv(2024)
摘要
It is commonly believed that logical states of quantum error-correcting codes
have to be highly entangled such that codes capable of correcting more errors
require more entanglement to encode a qubit. Here we show that this belief may
or may not be true depending on a particular code. To this end, we characterize
a tradeoff between the code distance d quantifying the number of correctable
errors, and geometric entanglement of logical states quantifying their maximal
overlap with product states or more general "topologically trivial" states. The
maximum overlap is shown to be exponentially small in d for three families of
codes: (1) low-density parity check (LDPC) codes with commuting check
operators, (2) stabilizer codes, and (3) codes with a constant encoding rate.
Equivalently, the geometric entanglement of any logical state of these codes
grows at least linearly with d. On the opposite side, we also show that this
distance-entanglement tradeoff does not hold in general. For any constant d
and k (number of logical qubits), we show there exists a family of codes such
that the geometric entanglement of some logical states approaches zero in the
limit of large code length.
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