Abstract The no-cloning theorem leads to information-theoretic security in various quantum cryptographic protocols. However, this security typically derives from a possibly weaker property that classical information encoded in certain quantum states cannot be broadcast. To formally capture this property, we introduce the study of ‘orthogonality broadcasting.’ When attempting to broadcast the orthogonality of two different qubit bases, we establish that the power of classical and quantum communication is equivalent. However, quantum communication is shown to be strictly more powerful for broadcasting orthogonality in higher dimensions. We then relate orthogonality broadcasting to quantum position verification and provide a new method for establishing error bounds in the no pre-shared entanglement model that can address protocols previous methods could not. Our key technical contribution is an uncertainty relation that uses the geometric relation of the states that undergo broadcasting rather than the non-commutative aspect of the final measurements.Abstract The no-cloning theorem leads to information-theoretic security in various quantum cryptographic protocols. However, this security typically derives from a possibly weaker property that classical information encoded in certain quantum states cannot be broadcast. To formally capture this property, we introduce the study of ‘orthogonality broadcasting.’ When attempting to broadcast the orthogonality of two different qubit bases, we establish that the power of classical and quantum communication is equivalent. However, quantum communication is shown to be strictly more powerful for broadcasting orthogonality in higher dimensions. We then relate orthogonality broadcasting to quantum position verification and provide a new method for establishing error bounds in the no pre-shared entanglement model that can address protocols previous methods could not. Our key technical contribution is an uncertainty relation that uses the geometric relation of the states that undergo broadcasting rather than the non-commutative aspect of the final measurements.

arXiv:2311.00677New Journal of PhysicsMay 1, 20251 citation

Orthogonal representations: Quantum physics

László Lovász

Colloquium Publications - American Mathematical Society/Colloquium PublicationsAug 28, 20190 citations

Preserver problems for the logics associated to Hilbert spaces and related Grassmannians

Mark Pankov

We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$ , i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$ ) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$ and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.We consider the standard quantum logic ${\mathcal L}(H)$ associated to a complex Hilbert space $H$, i.e. the lattice of closed subspaces of $H$ together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic ${\mathcal L}(H)$) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of ${\mathcal L}(H)$ and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.

arXiv:1702.03157Feb 10, 20170 citations

From Classical to Quantum

Newton C. A. da Costa, Francisco Antônio Dória

Synthese Library/Synthese libraryAug 9, 20210 citations

Exactly solvable generalized quantum harmonic oscillators related with the classical orthogonal polynomials

Zehra Çayiç

Jul 1, 20160 citations

We consider the standard quantum logic

{\mathcal L}(H)

associated to a complex Hilbert space

H

, i.e. the lattice of closed subspaces of

H

together with the orthogonal complementation. The orthogonality and compatibility relations are defined for any logic. In the standard quantum logic, they have a simple interpretation in terms of operator theory. For example, two closed subspaces (propositions in the logic

{\mathcal L}(H)

) are compatible if and only if the projections on these subspaces commute. We present both classical and more resent results on transformations of

{\mathcal L}(H)

and the associated Grassmannians which preserve the orthogonality or compatibility relation. The first result in this direction was classical Wigner's theorem.