First, Mutually Unbiased Bases (MUB’s), the primary topic of the paper, represent a fundamental concept in quantum mechanics and information, and relate to the minimum sets of experiments one needs to extract the maximum amount of information from a quantum state.

“The paper describes a new methodology for attacking a very challenging mathematical problem, which originates naturally in the context of quantum mechanics”

Second, MUB’s are critical to several quantum cryptographic protocols and systems, including the much-studied Quantum Key Distribution (QKD) system; in particular,

one can provide increased security and bit rates by using states corresponding to MUB’s of multi-dimensional quantum systems.

Third, the mathematical problem of determining a set of MUB’s of maximum cardinality for any d-dimensional quantum system is a challenging one, with connections to many well- known problems in combinatorics and algebra. For example, the MUB problem is closely related to the problem of determining arrangements of lines in the Grassmannian spaces so that they are as far apart as possible.

Our paper provides a fresh perspective on this challenging and relevant problem. In particular, we provide a construction of a set of MUB’s of maximum size when the dimension is a power of prime. While the existence of such MUB’s was already reported in the literature, we provide a very different construction and also make connections to maximal commuting bases of orthogonal unitary matrices, and other related problems.

This fresh perspective, along with the open problem of determining sets of MUB’s of maximum size for dimensions that are not powers of prime numbers, has fueled a lot of interest in this field following the publication of our paper. For example, even for a six-dimensional system, the maximum size of a set of MUB’s is not known.

The paper describes a new methodology for attacking a very challenging mathematical problem, which originates naturally in the context of quantum mechanics. Other than providing novel and explicit constructions for MUB observables (operators), the paper raises hopes that one might be able to obtain an answer to the question of determining MUB sets of maximum size when the dimension is not a power of prime.

Fundamental to quantum mechanics and quantum information theory is the question: How much can one learn about a given ensemble of quantum states, and how efficiently can one distinguish among different ensembles, corresponding to different quantum states?

Since learning about an unknown quantum state necessarily implies a change or disturbance in the observed state, a related question concerns the associated tradeoff: how much information one gets vs. how much the state itself is disturbed. MUB’s are intrinsically related to such fundamental questions.

For example, consider the scenario in which one is randomly presented with quantum states drawn from two bases that are mutually unbiased. One can show that if one can accurately determine the states from one of the given bases, then one must disturb the states of the other basis considerably. Thus, if a sender randomly sends states chosen from two MUB’s, then an eavesdropper trying to accurately learn which state was sent would get caught: if the sender can accurately predict the states belonging to one of the bases, then the sender would disturb the states from the other bases, which can then be detected by the receiver at the other end.

This result forms the basis of the quantum key distribution (QKD) paradigm, which specifies a secure protocol for sharing secret keys over a quantum (e.g., optical) channel. By studying constructions of sets of MUB’s for different dimensional systems, the paper furthers the understanding of such states, and also opens up the possibility of utilizing high-dimensional quantum states for information transmission and processing purposes.

This work evolved naturally from our group’s interest in quantum computing and quantum information processing. At the time, we were exploring topics such as the design of protocols for various cryptographic tasks, as well as developing tools to analyze the security aspects of given protocols.

Nothing that is remarkable, other than the conjecture that physical systems seem to be somehow "aware" of prime numbers. For quantum systems with dimensions that are powers of a prime one can find sets of MUB’s with maximum size. While there is no proof, the conjecture is that for general composite dimensional systems—for example, when the systems have dimension 6—such maximum size MUB sets do not exist. If such a conjecture is indeed true, it might have interesting philosophical implications.