As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.As the quantum analog of the classical one-time pad, the private quantum channel (PQC) plays a fundamental role in the construction of the maximally mixed state (from any input quantum state), which is very useful for studying secure quantum communications and quantum channel capacity problems. However, the undoubted existence of a relation between the geometric shape of regular polytopes and private quantum channels in the higher dimension has not yet been reported. Recently, it was shown that a one-to-one correspondence exists between single-qubit PQCs and three-dimensional regular polytopes (i.e., regular polyhedra). In this paper, we highlight these connections by exploiting two strategies known as a generalized Gell-Mann matrix and modified quantum Fourier transform. More precisely, we explore the explicit relationship between PQCs over a qutrit system (i.e., a three-level quantum state) and regular 4-polytopes. Finally, we attempt to devise a formula for such connections on higher dimensional cases.

arXiv:2101.00230Communications in PhysicsMay 10, 20211 citation

The exact strong converse exponent for private communication over quantum channels

Hao-Chung Cheng, Christoph Hirche, Marco Tomamichel

We determine the exact strong converse exponent for secret-key transmission and generation over general finite-dimensional quantum wiretap channels, measuring reliability and secrecy jointly by squared fidelity to an ideal secret key. We introduce a novel Renyi private information whose regularization characterizes this exponent. An exact integral representation in terms of ordinary private information gives a uniform continuity bound, showing that the regularized quantity converges to private capacity as the Renyi order approaches one. This establishes for any wiretap channel an exponential fidelity decay at every rate above private capacity.We determine the exact strong converse exponent for secret-key transmission and generation over general finite-dimensional quantum wiretap channels, measuring reliability and secrecy jointly by squared fidelity to an ideal secret key. We introduce a novel Renyi private information whose regularization characterizes this exponent. An exact integral representation in terms of ordinary private information gives a uniform continuity bound, showing that the regularized quantity converges to private capacity as the Renyi order approaches one. This establishes for any wiretap channel an exponential fidelity decay at every rate above private capacity.

quant-pharXiv:2609.38144Sep 30, 20260 citations

Limits and restrictions of private quantum channel

Jan Bouda, Mário Ziman

We study private quantum channels on a single qubit, which encrypt given set of plaintext states $P$ . Specifically, we determine all achievable states $ρ^{(0)}$ (average output of encryption) and for each particular set $P$ we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set $P$ is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional $P$ varies continuously between 1 and 2 bits depending on the state $ρ^{(0)}$ . Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.We study private quantum channels on a single qubit, which encrypt given set of plaintext states $P$. Specifically, we determine all achievable states $ρ^{(0)}$ (average output of encryption) and for each particular set $P$ we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set $P$ is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional $P$ varies continuously between 1 and 2 bits depending on the state $ρ^{(0)}$. Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.

arXiv:quant-ph/0506107Jun 14, 20051 citation

Geometry and Entanglement of Super-Qubit Quantum States

Oktay K. Pashaev, Aygül Koçak

arXiv:2410.04361Springer proceedings in mathematics & statisticsJan 1, 20250 citations

We study private quantum channels on a single qubit, which encrypt given set of plaintext states

P

. Specifically, we determine all achievable states

ρ^{(0)}

(average output of encryption) and for each particular set

P

we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set

P

is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional

P

varies continuously between 1 and 2 bits depending on the state

ρ^{(0)}

. Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.