The conditional entropy power inequality for quantum additive noise channels

Abstract

We prove the quantum conditional entropy power inequality for quantum additive noise channels. This inequality lower bounds the quantum conditional entropy of the output of an additive noise channel in terms of the quantum conditional entropies of the input state and the noise when they are conditionally independent given the memory. We also show that this conditional entropy power inequality is optimal in the sense that we can achieve equality asymptotically by choosing a suitable sequence of Gaussian input states. We apply the conditional entropy power inequality to find an array of information-theoretic inequalities for conditional entropies which are the analogs of inequalities which have already been established in the unconditioned setting. Furthermore, we give a simple proof of the convergence rate of the quantum Ornstein-Uhlenbeck semigroup based on entropy power inequalities. arXiv:1803.00470v2 [quant-ph]

Linear growth of the entanglement entropy for quadratic Hamiltonians and arbitrary initial states

“…The main idea of the proof of ( 35) is perturbing the state with the quantum heat semigroup. The same idea has been crucial in the proofs of several quantum versions of the Entropy Power Inequality [43][44][45][46][47][48][49][50][51][52][53][54], of which (35) can be considered a generalization. Let Ḡ achieve the maximum in (35) (if the maximum is not achieved, the result can be obtained with a limiting argument).…”

The squashed entanglement of the noiseless quantum Gaussian attenuator and amplifier

“…A conditional version of the quantum Entropy Power Inequality has been proved, where all the entropies are conditioned on an external quantum system [26][27][28] . In this paper, we exploit its version for the two-mode squeezing operation:…”

“…Lower bounds to the squashed entanglement are notoriously difficult to prove, since the optimization in (1) over all the possible extensions of the quantum state is almost never analytically treatable. We overcome this difficulty with the quantum conditional Entropy Power Inequality [26][27][28][29] , which holds for any conditioning quantum system.…”

Upper bounds on the private capacity for bosonic Gaussian channels

“…The inequality states that the output-entropy of a bosonic Gaussian channel, such as a beam-splitter (or amplifier), is always increased under two independent input bosonic Gaussian states. Also, the qEPI has been proved several ways with some applications [15][16][17][18] and extended to the conditional cases in discrete and Gaussian regimes [19][20][21][22][23]. The power of quantum entropy power inequalities is that those are only carrying the information about von Neumann entropy without details of the quantum state itself.…”

“…The power of quantum entropy power inequalities is that those are only carrying the information about von Neumann entropy without details of the quantum state itself. Recently, it has been known that qEPIs have many applications for obtaining upper bounds of the classical capacity [22,24] as well as the quantum capacity [25] on bosonic Gaussian channels with general Gaussian noises beyond the thermal-noise case. The general noise means that it can be possible to take the environment as in the form of a squeezed (or even non-Gaussian) quantum state [24].…”