![MUB bases plot](https://upload.wikimedia.org/wikipedia/commons/6/6c/MUB_bases_plot.svg)
In
quantum information
Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both th ...
theory, mutually unbiased bases in
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
C
''d'' are two
orthonormal bases
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, ...
and
such that the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
of the
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of the
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
between any basis states
and
equals the
inverse of the
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
''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
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events.
In other words, measurement is a process of determining how large or small a physical quantity is as compared ...
with respect to the other basis are predicted to occur with equal probability.
Overview
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.
Another area where mutually unbiased bases can be applied is
quantum key distribution
Quantum key distribution (QKD) is a secure communication method which implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared random secret key known only to them, which can then be ...
, more specifically in secure quantum key exchange.
[M. Planat et al, A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements, http://hal.ccsd.cnrs.fr/docs/00/07/99/18/PDF/MUB_FP.pdf.] 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
In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system, ...
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
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
, and the so-called "mean king's problem".
Existence problem
Let
denote the maximum number of mutually unbiased bases in the ''d''-dimensional Hilbert space C
''d''. It is an open question
how many mutually unbiased bases,
, one can find in C
''d'', for arbitrary ''d''.
In general, if
:
is the
prime-power factorization of ''d'', where
: