Mutually unbiased bases
In quantum information theory, a set of bases in Hilbert space Cd are said to be mutually unbiased if when a system is prepared in an eigenstate 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 Julian Schwinger in 1960,[1] and the first person to consider applications of mutually unbiased bases was I. D. Ivanovic[2] in the problem of quantum state determination.
[3] MUBs 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.
[3] This motivates the study of mutually unbiased bases in higher-dimensional spaces.
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
equals the inverse of the dimension d:[11] 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.
denote the maximum number of mutually unbiased bases in the d-dimensional Hilbert space Cd.
, one can find in Cd, for arbitrary d. In general, if is the prime-power factorization of d, where then the maximum number of mutually unbiased bases which can be constructed satisfies[11] It follows that if the dimension of a Hilbert space d is an integer power of a prime number, then it is possible to find d + 1 mutually unbiased bases.
Searches for a set of four mutually unbiased bases when d = 6, both by using Hadamard matrices[11] and numerical methods[14][15] have been unsuccessful.
The general belief is that the maximum number of mutually unbiased bases for d = 6 is
[11] The MUBs problem seems similar in nature to the symmetric property of SIC-POVMs.
unbiased bases yields a geometric structure known as a finite projective plane, while a SIC-POVM (in any dimension that is a prime power) yields a finite affine plane, a type of structure whose definition is identical to that of a finite projective plane with the roles of points and lines exchanged.
In this sense, the problems of SIC-POVMs and of mutually unbiased bases are dual to one another.
, the analogy can be taken further: a complete set of mutually unbiased bases can be directly constructed from a SIC-POVM.
be two unitary operators in the Hilbert space Cd such that for some phase factor
[11] The dimension of the Hilbert space is important when generating sets of mutually unbiased bases using Weyl groups.
When d is a prime number, then the usual d + 1 mutually unbiased bases can be generated using Weyl groups.
We label the elements of the computational basis of Cd using the finite field:
For d = 3 these matrices would have the form The problem of finding a set of k+1 mutually unbiased bases therefore corresponds to finding k mutually unbiased complex Hadamard matrices.
[11] An example of a one parameter family of Hadamard matrices in a 4-dimensional Hilbert space is There is an alternative characterization of mutually unbiased bases that considers them in terms of uncertainty relations.
Entropic uncertainty relations are often preferable[24] to the Heisenberg uncertainty principle, as they are not phrased in terms of the state to be measured, but in terms of c. In scenarios such as quantum key distribution, we aim for measurement bases such that full knowledge of a state with respect to one basis implies minimal knowledge of the state with respect to the other bases.
This implies a high entropy of measurement outcomes, and thus we call these strong entropic uncertainty relations.
In fact, for a d-dimensional space, we have:[25] for any pair of mutually unbiased bases
If the dimension of the space is a prime power, we can construct d + 1 MUBs, and then it has been found that[27] which is stronger than the relation we would get from pairing up the sets and then using the Maassen and Uffink equation.
Thus we have a characterization of d + 1 mutually unbiased bases as those for which the uncertainty relations are strongest.
[29] This means merely being mutually unbiased does not lead to high uncertainty, except when considering measurements in only two bases.
[27][30] While there has been investigation into mutually unbiased bases in infinite dimension Hilbert space, their existence remains an open question.
Weigert and Wilkinson[31] were first to notice that also a linear combination of these operators have eigenbases, which have some features typical for the mutually unbiased bases.
