Phase Transition in the Quantum Capacity of Quantum Channels

• URL: http://arxiv.org/abs/2408.05733v3
• Date: Wed, 9 Oct 2024 18:58:20 GMT
• Title: Phase Transition in the Quantum Capacity of Quantum Channels
• Authors: Shayan Roofeh, Vahid Karimipour,
• Abstract summary: determining the capacities of quantum channels is one of the fundamental problems of quantum information theory. We prove that every quantum channel $Lambda$ in arbitrary dimension, when contaminated by white noise in the form $Lambda_x(rho)=(1-x)Lambda(rho)+xtextTr(rho) fracId$, completely loses its capacity of transmiting quantum states when $xgeq frac12$.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Determining the capacities of quantum channels is one of the fundamental problems of quantum information theory. This problem is extremely challenging and technically difficult, allowing only lower and upper bounds to be calculated for certain types of channels. In this paper, we prove that every quantum channel $\Lambda$ in arbitrary dimension, when contaminated by white noise in the form $\Lambda_x(\rho)=(1-x)\Lambda(\rho)+x\text{Tr}(\rho) \frac{I}{d}$, completely loses its capacity of transmiting quantum states when $x\geq \frac{1}{2}$, no matter what type of encoding and decoding is used. In other words, the quantum capacity of the channel vanishes in this region. To show this, we prove that the quantum depolarizing channel in $d$ dimension, defined as $\mathcal{D}_x (\rho):=(1-x)\rho+\frac{x}{d}I_d$ is anti-degradable when $x\geq \frac{1}{2}$. Although there are indirect proofs of this property, we explicitly find the quantum channel ${\cal N}_x$, which anti-degrades this channel. Furthermore, in this way, we find a new channel, namely ${\cal D}^c_x$ which is degradable in the same region. Being degradable, means that this new channel enjoys additive property for its quantum capacity.

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