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Lecture notes in computer science

Cryptography from One-Way Communication: On Completeness of Finite Channels

Shweta Agrawal, Yuval Ishai, Eyal Kushilevitz Z, Varun Narayanan, Manoj Prabhakaran, Vinod M. Prabhakaran, Alon Rosen

Published Jan 1, 2020• 7 citations
Publisher
DOI
10.1007/978-3-030-64840-4_22

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Complete information orders of quantum channels: equivalence and undecidability

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How do you decide which quantum channel keeps more information? There is more than one way to ask. You can compare how well two states remain distinguishable, how much information an ensemble retains, or how sensitively an output responds to a small change. These are different mathematical questions. This paper asks when they give the same ordering of quantum channels. The comparison is complete: an input may share correlations with a separate, finite quantum reference, and the reference can be as large as needed. That extra system makes the question richer than looking at the channel output alone. For finite-dimensional channels, the manuscript claims an equivalence among complete comparisons expressed through relative entropy, Holevo information, BKM and SLD Fisher information, and measured chi-squared divergence. It also studies the complementary channel—the part of a Stinespring description that goes to the environment—and claims that the information order reverses under genuine complementation. The second part asks whether these comparisons can be decided by an algorithm. The paper claims coRE-completeness for a class of strictly positive rational-complex measurements, and that no computable bound on reference size suffices for the stated SLD comparison. It also claims fixed-additive-accuracy inapproximability for a bounded entropy defect. These are claims about precisely specified mathematical decision problems, not a claim that every practical channel comparison is impossible. The companion paper, From complete channel order to capacity and one-way distillation, explores consequences for communication and distillation using the order results developed here. See the manuscript page for a further introduction and citation details.

•Oct 14, 2026•1 citation
We give a proof that entanglement purification, even with noisy apparatus, is sufficient to disentangle an eavesdropper (Eve) from the communication channel. In the security regime, the purification process factorizes the overall initial state into a tensor-product state of Alice and Bob, on one side, and Eve on the other side, thus establishing a completely private, albeit noisy, quantum communication channel between Alice and Bob. The security regime is found to coincide for all practical purposes with the purification regime of a two-way recurrence protocol. This makes two-way entanglement purification protocols, which constitute an important element in the quantum repeater, an efficient tool for secure long-distance quantum cryptography.
When does a difference in information become a way to communicate? Imagine two quantum channels as competing routes through noise. One might preserve a distinction that the other loses. That difference is interesting on its own; the question here is whether it can be turned into something operational: a channel that can carry private or quantum information at a positive rate. This paper presents a construction intended to make that step. Starting from a failure of complete quantum-channel information order, it constructs a channel claimed to have strictly positive ordinary private and quantum capacity. “Complete” matters because the comparison allows an input to be correlated with a finite quantum reference: information may be visible in the joint system even when it is hidden from the channel output alone. The construction accounts for the resources it uses, including transmission of the reference and uses of an activation helper. Its conclusion concerns strict positivity: a rate greater than zero, however small. It does not claim a uniform positive gap or fixed-error capacity inapproximability. The paper also develops reductions connecting the capacity questions to one-way distillable entanglement and secret key. The state constructions are designed to retain any prescribed lower bound on two-way distillation. On this basis, the manuscript claims exact-zero undecidability classifications for finite rational-complex memoryless channels and corresponding classifications for the one-way quantities. The information-order equivalences and undecidability premises used in these conclusions are developed in the companion paper, Complete information orders of quantum channels: equivalence and undecidability. A further introduction and citation details are available on the manuscript page.
How do you decide which quantum channel keeps more information? There is more than one way to ask. You can compare how well two states remain distinguishable, how much information an ensemble retains, or how sensitively an output responds to a small change. These are different mathematical questions. This paper asks when they give the same ordering of quantum channels. The comparison is complete: an input may share correlations with a separate, finite quantum reference, and the reference can be as large as needed. That extra system makes the question richer than looking at the channel output alone. For finite-dimensional channels, the manuscript claims an equivalence among complete comparisons expressed through relative entropy, Holevo information, BKM and SLD Fisher information, and measured chi-squared divergence. It also studies the complementary channel—the part of a Stinespring description that goes to the environment—and claims that the information order reverses under genuine complementation. The second part asks whether these comparisons can be decided by an algorithm. The paper claims coRE-completeness for a class of strictly positive rational-complex measurements, and that no computable bound on reference size suffices for the stated SLD comparison. It also claims fixed-additive-accuracy inapproximability for a bounded entropy defect. These are claims about precisely specified mathematical decision problems, not a claim that every practical channel comparison is impossible. The companion paper, From complete channel order to capacity and one-way distillation, explores consequences for communication and distillation using the order results developed here. See the manuscript page for a further introduction and citation details.