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Physical Review A

Estimation of a generalized amplitude-damping channel

Akio Fujiwara

Published Jul 29, 2004• 66 citations
Publisher

Abstract

The problem of finding the optimal strategy for estimating a generalized amplitude damping channel ${\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ by means of the extension $\mathrm{id}\ensuremath{\bigotimes}{\ensuremath{\Gamma}}_{\ensuremath{\eta}}^{(p)}$ is addressed. We first evaluate the quantum Fisher information of output states based on the symmetric logarithmic derivative and specify all pure-state inputs that maximize the quantum Fisher information. We next investigate the ${\ensuremath{\nabla}}^{e}$ autoparallelity of output state manifolds and characterize the condition for the existence of an efficient estimator. A comparison of these results concludes that, while there is no uniformly optimal input for all $p$ and $\ensuremath{\eta}$, a maximally entangled input is an admissible one under a nonasymptotic setting.

DOI
10.1103/physreva.70.012317

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Determining a noisy quantum channel's classical capacity and entanglement cost generally requires regularization over many channel uses. We remove both regularizations for every qubit-to-qubit channel admitting a pure output. For each such channel, Holevo information, channel entanglement of formation, and parallel entanglement cost are additive with those of any finite-dimensional partner channel. This class includes all qubit-to-qubit channels of Kraus rank at most two. For the amplitude damping channel with damping probability ppp, the unassisted classical capacity equals the known single-use Holevo information, attained by a binary pure-state ensemble with collective decoding, and the entanglement cost is h2((1+p)/2)h_2((1+\sqrt p)/2)h2​((1+p​)/2) ebits per use. The common mechanism is a support criterion for strong superadditivity of entanglement of formation: one marginal has no support on the sector in which both local systems are orthogonal to fixed distinguished vectors. We prove this criterion in arbitrary finite dimensions using a triangular block-matrix entropy inequality and decompositions preserving two expectations. For amplitude damping, we also derive an exact finite-block Holevo deficit, identify the unique optimal average input for p<1p<1p<1, and construct a binary Kraus representation attaining the uniform formation bound.
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Due to the anisotropy of quantum lossy channels one must choose optimal bases of input states for best estimating them. In this paper, we obtain that the equal probability Schrödinger cat states are optimal for estimating a single lossy channel and they are also the optimal bases of input states for estimating composite lossy channels. On the other hand, by using the symmetric logarithmic derivative (SLD) Fisher information of output states exported from the lossy channels we obtain that if we take the equal probability Schrödinger cat states as the bases of input states the maximally entangled inputs are not optimal, however if the bases of the input states are not the equal probability Schrödinger cat states the maximally entangled input states may be optimal for the estimating composite lossy channel.
We present a quantum parameter estimation theory for a generalized Pauli channel Γ θ : ( d ) → ( d ), where the parameter θ is regarded as a coordinate system of the probability simplex d 2 −1 . We show that for each degree n of extension (id ⊗ Γ θ ) ⊗ n : (( d ⊗ d ) ⊗ n ) → (( d ⊗ d ) ⊗ n ), the SLD Fisher information matrix for the output states takes the maximum when the input state is an n -tensor product of a maximally entangled state τ ME ∊ ( d ⊗ d ). We further prove that for the corresponding quantum Cramér–Rao inequality, there is an efficient estimator if and only if the parameter θ is ∇ m -affine in d 2 −1 . These results rely on the fact that the family {id ⊗ Γ θ (τ ME )} θ of output states can be identified with d 2 −1 in the sense of quantum information geometry. This fact further allows us to investigate submodels of generalized Pauli channels in a unified manner.