From finite projective geometry to quantum phase enciphering (Discrete Math of MUBs) H. Rosu, M. Planat, M. Saniga (IPICyT-Mx, LPMO-Fr, Astronomical Inst.-Sk)

Abstract Scope: To review the derivation of mutually unbiased bases in arbitrary Hilbert space dimension. To study: Their relationship to Fourier transforms, Galois fields and rings, generalized Hopf fibrations and projective geometry. Some possible applications: state tomography secure exchange of quantum keys

Introduction Technical problems in quantum information theory already connect such distinct disciplines as number theory abstract algebra projective geometry For a partial list of open problems related to the development of quantum computing technologies, see

MUBs: Definition and Basic Facts Two different orthonormal bases A and B of a d-dimensional Hilbert space are called mutually unbiased if and only if for all a A and all b B. An aggregate of mutually unbiased bases is a set of orthonormal bases which are pairwise mutually unbiased. It has been found that the maximum number of such bases cannot be greater than d + 1 in d-dimensional Hilbert space [Wootters & Fields, 1989). It is also known that this limit is reached if d is a power of a prime. Yet, a still unanswered question is if there are non-prime-power values of d for which this bound is attained. Based on numerical calculations it is generally agreed [Zauner, 2003] that the maximum number of such bases is 1 + the lowest factor min( ) in the prime number decomposition of d = П i

MUBs and Finite Projective Planes (J. Opt. B: Quant. Semiclass. Opt., in press, Saniga,Planat,Rosu) CONJECTURE Whether or not there exists a set of d+1 MUBs in a d-dimensional Hilbert space if d not a power of a prime is intimately linked with the question of the existence of projective planes whose order is not a power of prime

MUBs: Applications The main application of MUBs pertains to secure quantum key exchange (quantum cryptography). This is because any attempt by an eavesdropper (say Eve) to distinguish between two non- orthogonal quantum states shared by two remote parties (say Alice and Bob) will occur at the price of introducing a disturbance to the signal, thus revealing the attack, and allowing to reject the corrupted quantum data. Modern protocols, e.g., the original BB84 protocol, use only 1- qubit technologies → dimension d = 2, usually the polarisation states of the photon. But the security against eavesdropping increases when all the three bases of qubits are used, or by using qudits, or entanglement-based protocols.

MUBs vs SIC-POVMS Quantum state recovery and secure quantum key distribution can also be achieved using positive operator valued measures (POVMs) which are symmetric informationally complete (SIC-POVMs) [Renes et al, JMP 2004]. These are sets of d 2 normalized vectors a and b such that | | = 1/(d + 1) 1/2 when a ≠ b. Unlike the MUBs the SIC-POVM’s could exist in all finite dimensions. Recently, SIC-POVMs have been constructed in dimension d = 6 by Grassl.

MUBs and Quantum Fourier Transforms There is a useful relationship between MUBs and QFTs. Consider a basis with indices n in the ring Z d of integers modulo d. The dual basis defined by the quantum Fourier transform is

Particular case: qubits d=2, ω=-1 Note that the two othogonal bases and are mutually unbiased. The third base, mutually unbiased to them is obtained from H (Hadamard matrix) by the action of a π/2 rotation S

2 d MUBs as eigenvectors of Pauli matrices B 0, B 1 and B 2 are also the eigenvectors of Pauli spin matrices respectively.

Generalization of Pauli matrices for prime dimension p For prime dimension p there is a natural generalization of σ x and σ z that are called shift and clock operators, respectively It can be shown that eigenvectors of the unitary operators generate the set of the d+1 MUBs in dimension p.

Problem: Could MUBs be obtained in any d and any field of numbers by Fourier transform as in d=2 ? Write the quantum Fourier transform such that the exponent ω now acts on a finite (Galois) field G=GF(p m ) with characteristic p and d=p m elements. Denote and ● the two operations in the field, corresponding to + and · in the field of real numbers. Then, the GF Fourier transform reads

Cont’d (1) Given any two polynomials k and n in G then there is a uniquely determined pair a and b in G such that where deg b < deg a, so that the exponent in the GF quantum Fourier transform could be written in the form

Cont’d (2) The last formula is valid for the case of prime dimension d=p when E is an integer. Otherwise, it has to be replaced by the trace of GF(p m ) down to GF(p) defined as follows and therefore

Cont’d (3) In a field of odd characteristic p the latter formula provides a set of d bases of index a for the base and the index b for the vector in the base, mutually unbiased to each other and to the computational base B 0. Formula was obtained by Wootters & Fields and also by Klappenecker & Rotteler

Cont’d (4) The same formula provides an interesting relation between the MUBs and quantum phase operators It is known that the Fourier basis can be derived as the eigenvectors of a quantum phase operator

Cont’d (5) Using the properties of the field trace, one can show that each base of index a can be associated to a quantum phase operator