Quantum Permutation Matrices

Avval Amil, Shashank Gupta

Cryptanalysis increases the level of confidence in cryptographic algorithms. We analyze the security of a symmetric cryptographic algorithm - quantum permutation pad (QPP) [8]. We found the instances of ciphertext the same as plaintext even after the action of QPP with the probability 1/N when the entire set of permutation matrices of dimension N is used and with the probability 1/N^m when an incomplete set of m permutation matrices of dimension N are used. We visually show such instances in a cipher image created by QPP of 256 permutation matrices of different dimensions. For any practical usage of QPP, we recommend a set of 256 permutation matrices of a dimension more or equal to 2048.

J.P. McCarthy

Jian Yan, Lin Chen

Permutation matrices are an old and widely used family of matrices in mathematics and physics. In this paper, we classify bipartite permutation matrices under local permutation equivalence when the equivalence is in the form of the Kronecker product of two permutation matrices. We classify all [Formula: see text] bipartite permutation matrices of Schmidt rank up to three. We also study the equivalence for general bipartite permutation gates in terms of Schmidt rank two. We apply our result to study the entangling power of two-qubit permutation gates. Under the local unitary equivalence, we show that the five distinct two-qubit gates under local permutation equivalence can be reduced to exactly three gates. Our results present a physical understanding of the permutation group from the point of view of quantum information and experiments.

Dong Wang, Hanwu Chen, Wanning Zhu +1 more

Gemma De las Cuevas, Tom Drescher, Tim Netzer

Quantum permutation matrices and quantum magic squares are generalizations of permutation matrices and magic squares, where the entries are no longer numbers but elements from arbitrary (non-commutative) algebras. The famous Birkhoff–von Neumann theorem characterizes magic squares as convex combinations of permutation matrices. In the non-commutative case, the corresponding question is as follows: Does every quantum magic square belong to the matrix convex hull of quantum permutation matrices? That is, does every quantum magic square dilate to a quantum permutation matrix? Here, we show that this is false even in the simplest non-commutative case. We also classify the quantum magic squares that dilate to a quantum permutation matrix with commuting entries and prove a quantitative lower bound on the diameter of this set. Finally, we conclude that not all Arveson extreme points of the free spectrahedron of quantum magic squares are quantum permutation matrices.