Quantum Error Correction Inapplicable to Really Entangled States
draft-ohta-qec-inapplicable-00

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Author Masataka Ohta 
Last updated 2020-10-30
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INTERNET DRAFT                                                   M. Ohta
draft-ohta-qec-inapplicable-00.txt         Tokyo Institute of Technology
Intended status: Informational                          October 30, 2020
Expires: May 3, 2021

    Quantum Error Correction Inapplicable to Really Entangled States

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Abstract

   Though quantum error correction assumes localized error model of Shor
   that errors on a qubit are caused by interaction with its local
   environment, enabling essentially classical error correction for
   unentangled states, the model is applied to entangled states
   improperly without involving local environment states in the
   entanglement.

   That is, when an entangled state (Q) is represented as superposition
   of unentangled terms (Qi) as Q=Q1+Q2+...+Qn, local environment states
   around qubits are, in general, different term by term. Q will be,
   with term-specific error operators (Ei), E1*Q1+E2*Q2+...+En*Qn, not,
   with a common error operator (E) assumed by Shor, E*(Q1+Q2+...+Qn).

M. Ohta                  Expires on May 3, 2021                 [Page 1]
INTERNET DRAFT QEC Inapplicable to Really Entangled States  October 2020

   A complication is that Shor's error model is a little quantum,
   allowing for two different local environment states around a qubit.
   As such, quantum error correction is applicable to some trivially
   entangled states including states used by Shor code but not to really
   entangled states.

1. Introduction

   An assumption of noise model for quantum error correction by Shor [1]
   is "The critical assumption here is that decoherence only affects one
   qubit of our superposition, while the other qubits remain unchanged.
   It is not clear how reasonable this assumption is physically, but it
   corresponds to the assumption in classical information theory of the
   independence of noise.", which means a qubit suffers from error as a
   result of interaction with local environment around the qubit but no
   interaction occurs with other qubits or local environment of other
   qubits.  Though some extension to consider certain interaction
   between a qubit and other qubits or environment of other qubits is
   possible, some locality is still assumed.

   The error model is directly applicable to unentangled, that is,
   essentially classical, states, resulting in localized errors,
   corrections of which are essentially classical error correction.

   However, it is unreasonable to expect such localized errors for
   entangled states, because the states themselves do not have locality.
   Actually, with a 2 qubit entangled state: |00>+|11>, if the first
   qubit coherently interacts with its environment to be |0>, the entire
   state becomes |00>, which means the second qubit is also affected,
   Though the case is trivial enough to be explained by Shor's error
   model as superposition of identity (no error) and sign flip (|0> and
   |1> become |0> and -|1>, correspondingly) error:
   |00>=((|00>+|11>)+(|00>-|11>))/2, such an explanation dose not deny
   lack of locality of errors on entangled states.

   As Shor overlooked the fact that when qubit states are entangled,
   their environment states are, in general, also entangled, errors on
   really entangled states are highly non-local to which quantum error
   correction is not applicable.

   That is, when an entangled state (Q) is represented as superposition
   of (minimum number of) unentangled terms (Qi) as Q=Q1+Q2+...+Qn,
   local environment states around a qubit are, in general, involved in
   the entanglement and different term by term, resulting in different
   error operators (Ei). As a result, Q will be disturbed by noise to be
   E1*Q1+E2*Q2+...+En*Qn, whereas, Shor thought a common error operator
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