Quantum error correction for non-maximally entangled states. (arXiv:1709.04301v3 [quant-ph] UPDATED)

Quantum states have high affinity for errors and hence error correction is of
utmost importance to realise a quantum computer. Laflamme showed that 5 qubits
are necessary to correct a single error on a qubit. In a Pauli error model,
four different types of errors can occur on a qubit. Maximally entangled states
are orthogonal to each other and hence can be uniquely distinguished by a
measurement in the Bell basis. Thus a measurement in Bell basis and a unitary
transformation is sufficient to correct error in Bell states. However, such a
measurement is not possible for non-maximally entangled states. In this work we
show that the 16 possible errors for a non-maximally entangled two qubit system
map to only 8 distinct error states. Hence, it is possible to correct the error
without perfect knowledge of the type of error. Furthermore, we show that the
possible errors can be grouped in such a way that all 4 errors can occur on one
qubit, whereas only bit flip error can occur on the second qubit. As a
consequence, instead of 10, only 8 qubits are sufficient to correct a single
error. We propose an 8-qubit error correcting code to correct a single error in
a non-maximally entangled state. We further argue that for an $n$-qubit
non-maximally entangled state of the form $a|0>^{n} + b|1>^{n}$, it is always
possible to correct a single error with fewer than $5n$ qubits, in fact only
$3n+2$ qubits suffice.

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