Abstract
In this article we consider flagged extensions of convex combination of quantum channels, and find general sufficient conditions for the degradability of the flagged extension. An immediate application is a bound on the quantum Q and private P capacities of any channel being a mixture of a unitary map and another channel, with the probability associated to the unitary component being larger than 1/2. We then specialize our sufficient conditions to flagged Pauli channels, obtaining a family of upper bounds on quantum and private capacities of Pauli channels. In particular, we establish new state-of-the-art upper bounds on the quantum and private capacities of the depolarizing channel, BB84 channel and generalized amplitude damping channel. Moreover, the flagged construction can be naturally applied to tensor powers of channels with less restricting degradability conditions, suggesting that better upper bounds could be found by considering a larger number of channel uses.
Highlights
Protecting quantum states against noise is a fundamental requirement for harnessing the power of quantum computers and technologies
It can be expressed in terms of an entropic functional, the coherent information Ic, which can be computed as a maximization over quantum states used as inputs of the communication line
A striking feature of the problem is the potential super-additivity of the coherent information [SS96; DSS98; SS07; FW08; SY08; SSY11; Cub+15; LLS18a; BL21; SG21; Sid[21]; Sid[20]; NPJ20; Yu+20], which which hinders the direct evaluation of the quantum capacity imposing an infinite number of optimizations on Hilbert spaces of dimension that grows exponentially in n
Summary
Protecting quantum states against noise is a fundamental requirement for harnessing the power of quantum computers and technologies. The quantum capacity Q of a channel is the maximal amount of qubits which can be transmitted reliably, per use of the channel. It can be expressed in terms of an entropic functional, the coherent information Ic, which can be computed as a maximization over quantum states used as inputs of the communication line. A striking feature of the problem is the potential super-additivity of the coherent information [SS96; DSS98; SS07; FW08; SY08; SSY11; Cub+15; LLS18a; BL21; SG21; Sid[21]; Sid[20]; NPJ20; Yu+20], which which hinders the direct evaluation of the quantum capacity imposing an infinite number of optimizations on Hilbert spaces of dimension that grows exponentially in n. The existence of an algorithmically feasible evaluation of the quantum capacity remains as one of the most important open problems in quantum Shannon theory [Hol[19]; Wil17], while finding computable upper or lower bounds on the quantum capacity constitutes important progress
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