Mutually unbiased bases

In quantum information theory, mutually unbiased bases in Hilbert space Cd are two orthonormal bases { | e 1 ⟩ , … , | e d ⟩ } {displaystyle {|e_{1} angle ,dots ,|e_{d} angle }} and { | f 1 ⟩ , … , | f d ⟩ } {displaystyle {|f_{1} angle ,dots ,|f_{d} angle }} such that the square of the magnitude of the inner product between any basis states | e j ⟩ {displaystyle |e_{j} angle } and | f k ⟩ {displaystyle |f_{k} angle } equals the inverse of the dimension d: In quantum information theory, mutually unbiased bases in Hilbert space Cd are two orthonormal bases { | e 1 ⟩ , … , | e d ⟩ } {displaystyle {|e_{1} angle ,dots ,|e_{d} angle }} and { | f 1 ⟩ , … , | f d ⟩ } {displaystyle {|f_{1} angle ,dots ,|f_{d} angle }} such that the square of the magnitude of the inner product between any basis states | e j ⟩ {displaystyle |e_{j} angle } and | f k ⟩ {displaystyle |f_{k} angle } 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. 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, 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'. Let M ( d ) {displaystyle {mathfrak {M}}(d)} denote the maximum number of mutually unbiased bases in the d-dimensional Hilbert space Cd. It is an open question how many mutually unbiased bases, M ( d ) {displaystyle {mathfrak {M}}(d)} , one can find in Cd, for arbitrary d.