Mutually unbiased bases

In quantum information theory, a set of bases in Hilbert space Cd are are said to be mutually unbiased to mean, that, if a system is prepared in an eigen state of one of the bases, then all outcomes of the measurement with respect to the other basis are predicted to occur with an equal probability inexorably equal to 1/d. The notion of mutually unbiased bases was first introduced by Schwinger in 1960, and the first person to consider applications of mutually unbiased bases was Ivanovic in the problem of quantum state determination. MUBs and their existence problem, is now known to have several closely related problems and equivalent avatars in several other branches of mathematics and quantum sciences, such as SIC-POVMs, finite projective / affine planes, complex Hadamard matrices etc. [see: Related Problems and Constructions sections in the below for details] MUBs are important for quantum key distribution, more specifically in secure quantum key exchange. Mutually unbiased bases are used in many protocols since the outcome is random when a measurement is made in a basis unbiased to that in which the state was prepared. When two remote parties share two non-orthogonal quantum states, attempts by an eavesdropper to distinguish between these by measurements will affect the system and this can be detected. While many quantum cryptography protocols have relied on 1-qubit technologies, employing higher-dimensional states, such as qutrits, allows for better security against eavesdropping. This motivates the study of mutually unbiased bases in higher-dimensional spaces. Other uses of mutually unbiased bases include quantum state reconstruction, quantum error correction codes, detection of quantum entanglement, and the so-called "mean king's problem". A pair of orthonormal bases and in Hilbert space Cd are said to be mutually unbiased, if and only if the square of the magnitude of the inner product between any basis states and equals the inverse of the dimension d: These bases are unbiased in the following sense: if a system is prepared in a state belonging to one of the bases, then all outcomes of the measurement with respect to the other basis are predicted to occur with equal probability.