correctable error quantum error correction

#Correctable error quantum error correction generator#
#Correctable error quantum error correction code#

#Correctable error quantum error correction code#

In particular, this implies more general code algebras willīe obtained. States that are invariant under the transformations U † g U h. He can only completely recover those properties of the quantum Is not sure whether Alice’s classical message was g or h, then U † g U h for each pair of classical states g, h thatīecame indistinguishable under the noisy channel.

#Correctable error quantum error correction generator#

Therefore, the commutant of the conserved algebra has a generator That classically, because of the noise, we can no longerĭistinguish between the state | h ⟩ and the state | h ′ ⟩. We are only interested in the nonzero generators that is, thoseįor which p g h ≠ 0 and p g h ′ ≠ 0.

It follows that the commutant of the correctable algebra, which isį † g h l F g ′ h ′ l ′ = δ g g ′ √ p g h p g h ′ U † h U h ′. Thus, the error operators F g h l satisfyį g h l ∝ √ p g h | g ⟩ ⟨ h | | l ⟩ ⊗ U l = δ h l √ p g h | g ⟩ ⊗ U h. The overall channel is given by composing E with theĬlassical channel. Then oneĬould ask, what information can be recovered by Bob once he is in Must send to Bob with transition probabilities p g h. Is applied on the classical bits | g ⟩, g ∈ S, that Alice Now suppose that a noisy stochastic channel In the standard teleportation protocol, S is the set of single Unitary operators in L ( H ) for some Hilbert If ρ is a density matrix we can always write ρ ↦ E ( ρ ) = ∑ a E a ρ E † a where g ∈ S of Noise models in quantum information are described by channels, which are (in the Schrödinger picture) trace-preserving (TP) and completely positive (CP) linear maps E on mixed states, ρ, which are operators acting on a Hilbert space H. We also discuss illustrative examples andĬonsider applications to quantum teleportation and informationįlow in quantum interactions. We present results that establish testable conditions forĬorrectability in OAQEC. Correspondingly we refer to the new theory as “operator algebra quantum error correction” (OAQEC). As a convenience, we shall simply refer to such operator algebras as “algebras”. Hence our codes take the form of operator algebras that are closed under Hermitian conjugation that is, finite dimensional C ∗-algebras Dav96. In this context it is natural to consider algebras of observables vN55. This can be alleviated by specifically selecting only some observables to be conserved. To this end, we change the focus from that of states to that of observables: conservation of a state by a given noise model implies the conservation of all of its observables, and is therefore a rather strong requirement. In this paper, we introduce a natural generalization of this theory.