Quantum capacity of a bosonic dephasing channel

Abstract

We study the LOCC-assisted quantum capacity of bosonic dephasing channel with energy constraint on input states. We start our analysis by focusing on the energy-constrained squashed entanglement of the channel, which is an upper bound for the energy-constrained LOCC-assisted quantum capacity. As computing energy-constrained squashed entanglement of the channel is challenging due to a double optimization (over the set of density matrices and over the isometric extensions of a squashing channel), we first derive an upper bound for it, and then we discuss how tight that bound is for energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel. We prove that the optimal input state is diagonal in the Fock basis. Furthermore, we prove that the optimal squashing channel in the set of weakly-degradable and anti-degradable channels is a weakly-self-complementary channel. Restricting the search for optimal squashing channel to the set of weakly-self-complementary one-mode Gaussian quantum channels, we derive explicit upper and lower bounds for the energy-constrained LOCC-assisted quantum capacity of the bosonic dephasing channel in terms of its quantum capacity with different noise parameters. Thanks to the tightness of these bounds we provide a very good estimation of the LOCC-assisted quantum capacity of bosonic dephasing channel. We also study weakly-self-complementary qubit squashing channels which gives tighter upper bound to the energy-constrained LOCC-assisted quantum capacity of bosonic dephasing channel.

Exact solution for the quantum and private capacities of bosonic dephasing channels

“…The most paradigmatic example of a bosonic dephasing channel is that corresponding to a normal distribution 𝑝 𝛾 (𝜙) = (2𝜋𝛾) −1/2 𝑒 −𝜙 2 /(2𝛾) of the angle 𝜙 over the whole real line. This is the main example studied in [29,30], which is based on a physical model discussed in those works. Here, 𝛾 > 0 denotes the variance of such a distribution: the larger 𝛾, the larger the uncertainty of the rotation angle in (3), and therefore the stronger the dephasing on the input state.…”

“…Many of the aforementioned forms of decoherence can be unified under a single model, known as the bosonic dephasing channel [29,30]. The action of such a channel on the density operator 𝜌 of a single-mode bosonic system is given by…”

“…We note here that while the quantum capacity [29,30] and the assisted quantum capacity [51] of the bosonic dephasing channel N 𝑝 in (3) have been investigated, neither of them has been calculated so far. The determination of the quantum capacity of this channel, in particular, has been an open problem since [29].…”

Discrimination of dephasing channels

“…The dephasing channel we are going to consider is the finite dimensional version of the bosonic channel discussed in [6]. Actually, it is a completely positive and trace preserving map defined on the set of density operators over H as…”

“…The usage of finite channels to get insights about continuous channels would be even more useful when the latter are non-Gaussian. One of the first examples of non-Gaussian continuous variable channel studied in quantum information theory is the dephasing channel [6]. It describes the wash out of coherence properties (off diagonal terms) of a state with respect to the Fock basis.…”