Mutually Unbiased Bases

Mutually unbiased bases are special collections of orthonormal coordinate systems for a quantum system such that if a particle is prepared in any state that belongs to one basis, measuring it with respect to another gives outcomes that are completely unpredictable and equally likely. In other words, the probability distribution over the possible results of the second measurement does not depend on which particular state from the first basis was chosen; each outcome occurs with the same chance.

This property matters because it captures a maximal form of quantum incompatibility: knowing something precisely in one basis tells you nothing about measurements in another. That extreme unpredictability is the resource that underpins many protocols where security or information hiding relies on an eavesdropper’s inability to guess measurement results, and it also provides a convenient way to probe all aspects of an unknown quantum state with minimal bias.

You encounter mutually unbiased bases wherever one needs collections of measurements that together reveal the full character of a quantum system. They appear in quantum key distribution schemes such as BB84, where the sender and receiver randomly switch between two unbiased bases to guarantee secrecy. They are also used in quantum state tomography to design measurement sets that efficiently reconstruct an unknown state, and in foundational studies that explore the limits of knowledge about complementary properties.