![]() We know that max U ′ | T r ( X ρ U ′ ) | = T r ( | X ρ | ). Next, note that | g ρ ( A, U ′ ) | = | T r ( X ρ U ′ ) | for some operator X ρ. First, using the strong concavity of the fidelity, it can be shown that the max over U in Eq. ( 4) can as well be taken over the convex set of operators A with operator norm ∥ A ∥ ≤ 1. Hence we just have to show that we can exchange the maximizations over U and U ′ in Eq. ( 4). If we reflect the picture with respect to a vertical axis through the middle, Hermitian conjugating each operator, and exchange the wire labels E ′ and B, and E and B ′, we see that we also have max R ′ F ( ˆ N, R ′ ˆ M ) = max U ′ min ρ max U | g ρ ( U, U ′ ) |, where now U ′ is the unitary defining R ′ while U comes from Uhlmann’s expression for the fidelity. The state | 0 ⟩ in the picture is arbitrary, and | ψ ρ ⟩ can be any purification of ρ. The wires labeled B and B ′ represent the target systems for N and M respectively, and E and E ′ are the respective “environments”. Hence the picture represents a complex number. Where the left half circles represent input states, while the right half circles are states which are scalar multiplied with the corresponding outputs. If instead of tracing out the environment after the unitary interaction, we trace out the target system B, we obtain a channel ˆ N which is said to be complementary to N: From the isometry V, one obtains the channel elements E i of N ( ρ ) = ∑ i E i ρ E † i simply by writing the partial trace explicitly in terms of a basis | i ⟩ of E. This isometry V is not unique, but unique up to a further local unitary map on the environment, eventually followed by an embedding into a larger environment. What matters is the isometry V defined by V | ϕ ⟩ : = U ( | ϕ ⟩ ⊗ | ψ ⟩ ) so that It does not matter which state | ψ ⟩ we use since the difference can be absorbed in the unitary. We will make essential use of the fact that a general quantum operation, or channel N, can always be viewed as resulting from a unitary interaction U with an “environment” E whose initial state | ψ ⟩ is known and which is later discarded (traced out). ( 2008b) Bény ( 2009) which are based on the diamond-norm distance rather than the fidelity. The present results are also strictly stronger than those of Ref. One advantage of this generality is that our results apply directly to the more general schemes of subsystem, or operator quantum error correction Kribs et al. Moreover, we prove our result in a very general context namely for the approximation of any channel, not necessarily the identity map on the code. Here we obtain both sufficient and necessary conditions which are a direct generalization of the KL conditions. Sufficient conditions for approximate correctability under the worst-case entanglement fidelity were proposed in Ref. In contrast, most previous work has considered input-dependent fidelities Barnum and Knill ( 2002) Schumacher and Westmoreland ( 2002) Tyson ( 2009) Buscemi ( 2008). Minimization over all inputs is essential if one is interested in guaranteeing a given fidelity when the state to be corrected is not known, as in the case of quantum computing. The entanglement fidelity Schumacher ( 1996) has been shown to be the pertinent fidelity measure in both quantum communication and computation scenarios since it estimates not only how well the state of the system under correction is preserved but also how its entanglement with auxiliary systems is maintained. In this work we focus on the entanglement fidelity minimized over all input states (also known as worst-case entanglement fidelity). Two practical consequences are that the error rate in a logical circuit is well quantified by the average gate fidelity at the logical level and that essentially optimal recovery operators can be determined by independently optimizing the logical fidelity of the effective noise per syndrome.The analysis of approximate error correction depends on the figure of merit used to compare the states after correction to the input states. Here, we prove that encoding a system in a stabilizer code and measuring error syndromes decoheres errors, that is, causes coherent errors to converge toward probabilistic Pauli errors, even when no recovery operations are applied. Consequently, the effective logical noise due to physically realistic coherent errors is relatively unknown. Typical studies of quantum error correction assume probabilistic Pauli noise, largely because it is relatively easy to analyze and simulate. ![]()
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