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arXiv:2101.00230 · Communications in Physics

High-dimensional Private Quantum Channels and Regular Polytopes

Junseo Lee, Kabgyun Jeong

Published May 10, 2021• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.15625/0868-3166/15762

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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.
Abstract At present, generalizations of quantum communication protocols from qubits to systems with higher-dimensional state spaces (qudits) commonly employ mutually unbiased bases (MUB), the construction of which is known for any dimension equal to a prime power. However, in low dimensions, there also exist formally more symmetric state systems described by regular complex polytopes, which are a generalization of the concept of regular polytopes to complex spaces. This work considers the application of a model originally proposed by R. Penrose and based on the geometry of the dodecahedron and two entangled particles with spin 3/2. In a more general case, two arbitrary quantum systems with four basis states (ququarts) can be used instead. It was subsequently shown that this system of 40 states is equivalent to the Witting configuration and is related to the four-dimensional complex polytope described by Coxeter. The paper proposes a quantum key distribution protocol based on contextuality, utilizing this configuration.
The private quantum channel (PQC) maps any quantum state to the maximally mixed state for the discrete as well as the bosonic Gaussian quantum systems, and it has fundamental meaning on the quantum cryptographic tasks and the quantum channel capacity problems. In this paper, we primally introduce a notion of approximate private quantum channel (ε-PQC) on fermionic Gaussian systems (i.e., ε-FPQC), and construct its explicit form of the fermionic (Gaussian) private quantum channel. First of all, we suggest a general structure for ε-FPQC on the fermionic Gaussian systems with respect to the Schatten p-norm class, and then we give an explicit proof of the statement in the trace norm case. In addition, we study that the cardinality of a set of fermionic unitary operators agrees on the ε-FPQC condition in the trace norm case. This result may give birth to intuition on the construction of emerging fermionic Gaussian quantum communication or computing systems.
At present, generalisations of quantum communication protocols from qubits to systems with higher- dimensional state spaces (qudits) commonly employ mutually unbiased bases (MUB), the construction of which is known for any dimension equal to a prime power. However, in low dimensions, there also exist formally more symmetric state systems described by regular complex polytopes, which are a generalisation of the concept of regular polytopes to complex spaces. This work considers the application of a model originally proposed by R. Penrose and based on the geometry of the dodecahedron and two entangled particles with spin 3/2. In a more general case, two arbitrary quantum systems with four basis states (ququarts) can be used instead. It was subsequently shown that this system of 40 states is equivalent to the Witting con guration and is related to the four-dimensional complex polytope described by Coxeter. The paper proposes a quantum key distribution protocol based on contextuality, utilising this con guration.
The Perfect-Mirsky conjecture states that the set of all possible eigenvalues of an n by n doubly stochastic matrix is the union of certain specific n−1n-1n−1 regular polygons in the complex plane. It is known to be true for n=1,2,3n=1,2,3n=1,2,3 and recently a counterexample was found for n=5n=5n=5. We show that the conjecture is true for n=4n=4n=4. We also modify the original conjecture for n>4n>4n>4 in order to accommodate the n=5 counterexample, as well as provide evidence for our new conjecture. An important class of quantum channels are the Pauli channels; channels whose Kraus operators are all tensor products of Pauli matrices. A private quantum channel is a quantum channel that is non-injective for some set of inputs. Private quantum channels generalize the classical notion of the one-time pad to the quantum setting. PQCs are most useful when the set of inputs they privatize is isomorphic as an algebra to an algebra of qubits. We use the theory of conditional expectations to show that in certain cases, the set of algebras privatized by a channel is the set of algebras quasiorthogonal to the range of the channel. Quasiorthogonality is an idea from operator algebras that has begun to find applications elsewhere in quantum information. Using these ideas, we prove that conditional expectation Pauli channels with commuting Kraus operators can always privatize fraclog(n)2\\frac{\\log(n)}{2}fraclog(n)2 qubits, where nnn is the number of Kraus operators. We also use ideas from Fourier analysis on finite groups to give a method for finding privatized algebras general Pauli channels. Using character theory, we extend this analysis beyond Pauli channels, and show how to find private algebras for qutrit and other channels. Finally, using ideas from Lie theory, we show a connection between quasiorthogonality and the Cartan decomposition of a Lie algebra, giving new methods for finding algebras privatized by a channel. We also use the KAKKAKKAK decomposition of a Lie algebra to begin looking at ways to encode algebras so they are privatized.
We show that every three-dimensional subspace of qutrit-qudit complex or real systems has a distinguishable basis under one-way local projective measurements and classical communication (LPCC). This solves a long-standing open problem proposed in [J. Phys. A, 40, 7937, 2007]. We further construct a three-dimensional space whose locally distinguishable basis is unique and apply the uniqueness property to the task of state transformation. We also construct a three-dimensional locally distinguishable multipartite space assisted with entanglement. On the other hand, we show that four-dimensional indistinguishable bipartite subspace under one-way LPCC exists. Our work offers profound insights and introduces a theoretical tool for understanding the local distinguishability of subspace. As a consequence, every qutrit channel has optimal environment-assisting classical capacity, and the environment-assisted classical capacity of every rank-three channel is at leastlog⁡23\log _{2} 3log2​3. We also show that every two-qutrit state can be converted into a generalized classical state near the quantum-classical boundary by an entanglement-breaking channel.