Mutually Unbiased Bases: Theory & Applications

• Mutually Unbiased Bases are families of orthonormal bases that yield uniform outcome distributions when measurements are made in different bases, with a complete set consisting of d+1 bases in prime-power dimensions.
• They are constructed using algebraic methods such as finite fields, generalized Pauli operators, and graph-state formalisms, which provide explicit structures and symmetry insights.
• MUBs underpin key operational tasks in quantum information including state tomography, communication protocols, and entanglement detection, and are validated experimentally especially in challenging dimensions like 6.

Mutually unbiased bases (MUBs) are families of orthonormal bases in a finite-dimensional Hilbert space that realize maximal complementarity: if a system is prepared in a vector from one basis, then measurement in any different basis from the family yields a uniform outcome distribution. Formally, two orthonormal bases $B=\{\lvert e_i\rangle\}_{i=1}^d$ and $C=\{\lvert f_j\rangle\}_{j=1}^d$ are mutually unbiased when

$$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$

A set of MUBs is complete if it contains $d+1$ bases, which is the maximal possible number in dimension $d$ (Spengler et al., 2013, Te'eni et al., 21 Aug 2025). MUBs sit at the intersection of finite-field methods, generalized Pauli groups, complex Hadamard matrices, projective and Hjelmslev geometries, operator algebras, and several operational tasks in quantum information.

1. Definition, maximality, and the existence problem

The defining condition of mutual unbiasedness is equivalent to uniform transition probabilities between distinct bases, or, in projective language, to the rank-one projector relation $\operatorname{Tr}(P_iQ_j)=1/d$ for all $i,j$ (Te'eni et al., 21 Aug 2025). A classical bound shows that no set of MUBs in $\mathbb{C}^d$ can contain more than $d+1$ bases, and a set achieving that bound is called complete (Te'eni et al., 21 Aug 2025).

Complete sets are known to exist whenever $d$ is a prime power, $C=\{\lvert f_j\rangle\}_{j=1}^d$0. Standard constructions proceed either through finite fields, in the style of Wootters–Fields, or through commuting classes of the generalized Pauli group, as in Bandyopadhyay, Boykin, Roychowdhury, and Vatan (Spengler et al., 2013). In prime dimension $C=\{\lvert f_j\rangle\}_{j=1}^d$1, the complete set of $C=\{\lvert f_j\rangle\}_{j=1}^d$2 stabilizer bases is essentially unique up to unitary equivalence, whereas in prime-power dimension $C=\{\lvert f_j\rangle\}_{j=1}^d$3 many inequivalent complete stabilizer MUBs exist (Zhu, 2015).

Outside prime-power dimensions, the existence problem remains central and unresolved in general. The smallest non-prime-power case, $C=\{\lvert f_j\rangle\}_{j=1}^d$4, is the canonical example: only three MUBs are known, and strong evidence suggests that no more than three simultaneous MUBs exist (Spengler et al., 2013, D'Ambrosio et al., 2013). A stronger nonexistence statement for completeness was established by relating MUBs to mutually orthogonal Latin squares: if a complete set of $C=\{\lvert f_j\rangle\}_{j=1}^d$5 MUBs exists in $C=\{\lvert f_j\rangle\}_{j=1}^d$6, then a complete set of $C=\{\lvert f_j\rangle\}_{j=1}^d$7 MOLS of order $C=\{\lvert f_j\rangle\}_{j=1}^d$8 must exist; since no complete set of MOLS exists for $C=\{\lvert f_j\rangle\}_{j=1}^d$9, a complete set of seven MUBs cannot exist in dimension six (Joka, 5 Nov 2025). The same implication rules out complete MUB sets in dimension ten as well (Joka, 5 Nov 2025).

2. Algebraic constructions in prime-power dimension

A standard algebraic route begins from generalized Pauli operators. On $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$0, one defines

$$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$1

On $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$2 qupits, tensor products of $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$3 generate the generalized Pauli group, and commuting classes of size $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$4 yield MUBs through their common eigenbases (Spengler et al., 2013). This stabilizer perspective underlies most known explicit complete constructions and also the symmetry analysis of stabilizer MUBs (Zhu, 2015).

A compact reformulation is given by the graph-state formalism. For $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$5 $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$6-level systems, with $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$7 prime, a generalized graph is specified by a symmetric adjacency matrix $$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$8, with self-loops allowed. The associated graph state is

$$|\langle e_i,f_j\rangle|^2=\frac1d \quad\text{for all }i,j.$$9

where $d+1$0 is a one-body phase gate and $d+1$1 for $d+1$2 is a controlled-phase gate (Spengler et al., 2013). The corresponding graph-state basis is generated by local $d+1$3-shifts. The decisive criterion is that if $d+1$4 and $d+1$5 are symmetric adjacency matrices, then the associated bases are mutually unbiased whenever

$d+1$6

This converts mutual unbiasedness into invertibility of matrix differences over $d+1$7 (Spengler et al., 2013).

The same paper shows how a single symmetric matrix $d+1$8 with irreducible characteristic polynomial over $d+1$9 generates a matrix representation of the finite field $d$0: $d$1 Because $d$2 is a field, any two distinct matrices in $d$3 differ by an invertible matrix, so the corresponding graph-state bases are pairwise mutually unbiased. Together with the computational basis, they form a complete set of $d$4 MUBs (Spengler et al., 2013). In many cases $d$5 may be chosen tridiagonal with unit super- and subdiagonals and diagonal entries specified by a single vector $d$6, so one graph, or even one vector, suffices to encode the full complete set (Spengler et al., 2013).

This algebraic economy is closely related to an older unifying viewpoint in terms of modules and finite geometries. Several standard complete constructions—planar-function, Alltop, symplectic-spread, and Galois-ring constructions—admit a description in which the non-standard MUB vectors form a module under componentwise multiplication, while the corresponding exponent patterns realize subspaces of projective geometries $d$7 or projective Hjelmslev geometries $d$8 (Hall et al., 2012). This suggests that the known prime-power constructions share a common higher-dimensional geometric substrate rather than merely a counting analogy.

3. Geometric, operator-theoretic, and classification frameworks

Beyond explicit construction, MUBs admit several complementary structural descriptions. One is geometric: the space of unordered projective orthonormal bases of $d$9 can be modeled as

$\operatorname{Tr}(P_iQ_j)=1/d$0

where $\operatorname{Tr}(P_iQ_j)=1/d$1 is the monomial unitary group (Te'eni et al., 21 Aug 2025). A $\operatorname{Tr}(P_iQ_j)=1/d$2-equivariant embedding into the Grassmannian of $\operatorname{Tr}(P_iQ_j)=1/d$3-planes in $\operatorname{Tr}(P_iQ_j)=1/d$4 induces a metric

$\operatorname{Tr}(P_iQ_j)=1/d$5

with $\operatorname{Tr}(P_iQ_j)=1/d$6 if and only if the two bases are mutually unbiased (Te'eni et al., 21 Aug 2025). In this framework, the familiar relation between a basis unbiased to the standard basis and a complex Hadamard matrix becomes a statement about the orbit decomposition of the subspace $\operatorname{Tr}(P_iQ_j)=1/d$7 under the left action of $\operatorname{Tr}(P_iQ_j)=1/d$8 (Te'eni et al., 21 Aug 2025).

The same geometric language organizes extension problems. Given a list of $\operatorname{Tr}(P_iQ_j)=1/d$9 MUBs, the candidate extensions form the intersection

$i,j$0

and inequivalent extensions are precisely the distinct orbits of this space under the simultaneous stabilizer of the given list (Te'eni et al., 21 Aug 2025). In dimension $i,j$1, this orbit analysis yields new symmetries that reduce the parameter space of MUB triples by a factor of $i,j$2 (Te'eni et al., 21 Aug 2025).

A second structural viewpoint replaces bases by commuting classes of matrices. There is a one-to-one correspondence between MUBs in $i,j$3 and maximal commuting classes of orthogonal normal matrices in $i,j$4: from $i,j$5 MUBs one obtains $i,j$6 commuting classes, each consisting of $i,j$7 commuting orthogonal normal matrices, and conversely a maximal commuting basis of $i,j$8 ensures a complete set of MUBs (Banerjee et al., 3 Jul 2025). The extendability problem then becomes equivalent to finding real points on a quadratic affine variety encoding orthogonality and unbiasedness constraints, with the unbiasedness-to-standard-basis sector forming a complete intersection prime ideal (Banerjee et al., 3 Jul 2025).

A third viewpoint concerns symmetry. Complete stabilizer MUBs are often group covariant, but sharp covariance is exceptionally rare. Among stabilizer MUBs, only the cases $i,j$9 and $\mathbb{C}^d$0 are sharply covariant, and this remains true even if antiunitary symmetries are allowed (Zhu, 2015). This sharply distinguishes MUBs from SICs, for which all known examples are sharply covariant (Zhu, 2015).

4. Operational roles in quantum information

MUBs are central to tomography, uncertainty, communication, and certification. In state tomography, a complete set of $\mathbb{C}^d$1 MUBs is informationally complete and minimal among orthonormal-basis measurement schemes. In the star-product formulation, if $\mathbb{C}^d$2 are the probabilities of the MUB projectors $\mathbb{C}^d$3, then

$\mathbb{C}^d$4

so knowledge of all MUB probabilities suffices for exact reconstruction (Filippov et al., 2010). For $\mathbb{C}^d$5-qubit systems, a complete set consists of $\mathbb{C}^d$6 bases, compared with $\mathbb{C}^d$7 Pauli observables, and an explicit Clifford-circuit synthesis with the three-stage form

$\mathbb{C}^d$8

constructs every MUB circuit in $\mathbb{C}^d$9 time (Yu et al., 2023).

The same $d+1$0-qubit MUB circuits can be used as the ensemble in classical shadow tomography. Uniform MUB sampling yields an informationally complete channel

$d+1$1

but for general observables the variance can be exponential in $d+1$2 (Wang et al., 2023). For approximately MUBs-average observables or states, and for MUBs-sparse instances under biased sampling, the variance becomes polynomial in $d+1$3 (Wang et al., 2023).

In prepare-and-measure self-testing, the $d+1$4 QRAC has optimal average success probability

$d+1$5

and this optimum is achieved if and only if Bob’s two measurements are rank-1 projective and mutually unbiased (Farkas et al., 2018). This gives a device-independent-style certification, within a dimension-bounded setting, of a pair of MUBs from operational performance alone (Farkas et al., 2018).

MUBs also support entanglement detection. For bipartite $d+1$6 systems, if $d+1$7 denotes the sum of mutual predictabilities across $d+1$8 local MUB settings, then all separable states satisfy

$d+1$9

and in particular $d$0 for a complete set (Hiesmayr et al., 2013). In the magic simplex of Bell-diagonal states, this criterion detects PPT-bound entanglement when a complete set of MUBs is used; for $d$1 the paper reports explicit PPT states with $d$2 (Hiesmayr et al., 2013).

At the same time, operational behavior can depend on which MUB set and which outcome labeling are used. Under coherent measurement choice, certain relabelings of Wootters–Fields MUBs can make the measurement outcome perfectly guessable, with $d$3 and $d$4, even though the measurements remain mutually unbiased in the usual overlap sense (Doda et al., 2020). This does not alter the mathematical definition of MUBs, but it shows that some security statements depend on implementation details beyond pairwise overlaps.

5. Approximate and generalized variants

Because exact complete sets are unavailable in many dimensions, several approximate notions have been developed. A $d$5-AMUB is a family of orthonormal bases for which all cross-basis overlaps satisfy $d$6, while an Almost Perfect MUB (APMUB) imposes the sharper biangular spectrum

$d$7

with sparse basis vectors whose nonzero components all have equal magnitude (Kumar et al., 2024, Kumar et al., 14 Dec 2025). For composite dimensions of the form $d$8, with $d$9 a prime power and $C=\{\lvert f_j\rangle\}_{j=1}^d$00, constructions based on resolvable block designs yield $C=\{\lvert f_j\rangle\}_{j=1}^d$01 APMUBs, often far more than the known exact MUBs in those dimensions (Kumar et al., 2024).

The quality of approximate families can be quantified by several measures. These include the $C=\{\lvert f_j\rangle\}_{j=1}^d$02-coherence

$C=\{\lvert f_j\rangle\}_{j=1}^d$03

the max and RMS deviations $C=\{\lvert f_j\rangle\}_{j=1}^d$04 and $C=\{\lvert f_j\rangle\}_{j=1}^d$05 from $C=\{\lvert f_j\rangle\}_{j=1}^d$06, Bengtsson’s distance $C=\{\lvert f_j\rangle\}_{j=1}^d$07, and the projective 2-design defect through the normalized frame potential $C=\{\lvert f_j\rangle\}_{j=1}^d$08 (Kumar et al., 14 Dec 2025). For exact MUB pairs, $C=\{\lvert f_j\rangle\}_{j=1}^d$09, $C=\{\lvert f_j\rangle\}_{j=1}^d$10, and $C=\{\lvert f_j\rangle\}_{j=1}^d$11; for APMUBs with $C=\{\lvert f_j\rangle\}_{j=1}^d$12, these quantities can be computed solely from $C=\{\lvert f_j\rangle\}_{j=1}^d$13 and $C=\{\lvert f_j\rangle\}_{j=1}^d$14, without using the internal construction details (Kumar et al., 14 Dec 2025).

A different generalization relaxes bases to measurements. Mutually unbiased measurements (MUMs) are projective measurements satisfying

$C=\{\lvert f_j\rangle\}_{j=1}^d$15

for all outcomes $C=\{\lvert f_j\rangle\}_{j=1}^d$16 (Farkas et al., 2022). MUMs are strictly more general than MUBs because the number of outcomes need not equal the Hilbert-space dimension, and there exist MUMs that are not direct sums of MUBs (Farkas et al., 2022). This broader complementarity notion supports superdense coding protocols and admits constructions from quaternionic Hadamard matrices (Farkas et al., 2022).

6. Special dimensions, implementations, and open directions

Dimension $C=\{\lvert f_j\rangle\}_{j=1}^d$17 remains the most studied exceptional case. Experimentally, three MUBs have been implemented and tested for single-photon six-dimensional states, encoded either in a hybrid polarization–orbital-angular-momentum Hilbert space or in pure orbital angular momentum. The measured $C=\{\lvert f_j\rangle\}_{j=1}^d$18 overlap matrices showed close agreement with the ideal MUB block structure, with reported similarities $C=\{\lvert f_j\rangle\}_{j=1}^d$19 and $C=\{\lvert f_j\rangle\}_{j=1}^d$20 in the two encodings (D'Ambrosio et al., 2013). These experiments confirm the practical availability of MUB triples in $C=\{\lvert f_j\rangle\}_{j=1}^d$21, but do not alter the theoretical evidence that no fourth basis exists (D'Ambrosio et al., 2013).

The nonexistence question in $C=\{\lvert f_j\rangle\}_{j=1}^d$22 has also been reformulated numerically through Bell inequalities. For every pair $C=\{\lvert f_j\rangle\}_{j=1}^d$23, one can construct a Bell inequality whose maximal quantum violation in dimension $C=\{\lvert f_j\rangle\}_{j=1}^d$24 is attainable if and only if $C=\{\lvert f_j\rangle\}_{j=1}^d$25 MUBs exist in that dimension (Colomer et al., 2022). Three numerical approaches—see-saw optimization, nonlinear semidefinite programming, and Monte Carlo—correctly recover the known prime-power cases and all suggest that four MUBs do not exist in dimension six; the numerical optimizers coincide with the “four most distant bases” found earlier by optimizing a distance measure (Colomer et al., 2022). The same Monte Carlo evidence suggests that at most three MUBs exist in dimension ten (Colomer et al., 2022).

There are also infinite-dimensional analogues, but their behavior depends strongly on the underlying field. For states over the real line $C=\{\lvert f_j\rangle\}_{j=1}^d$26, only three MUBs are known, whereas over the $C=\{\lvert f_j\rangle\}_{j=1}^d$27-adic numbers $C=\{\lvert f_j\rangle\}_{j=1}^d$28, for every prime $C=\{\lvert f_j\rangle\}_{j=1}^d$29 there are at least $C=\{\lvert f_j\rangle\}_{j=1}^d$30 mutually unbiased generalized bases in the rigged Hilbert-space sense (Dam et al., 2011). The paper attributes this contrast to the different harmonic analysis of $C=\{\lvert f_j\rangle\}_{j=1}^d$31 and $C=\{\lvert f_j\rangle\}_{j=1}^d$32, especially the ultrametric structure and finite residue field in the $C=\{\lvert f_j\rangle\}_{j=1}^d$33-adic case (Dam et al., 2011).

Several open directions remain active. Non-prime-power dimensions continue to motivate both nonexistence criteria and alternative constructions (Joka, 5 Nov 2025, Colomer et al., 2022). The classification of inequivalent complete sets, particularly under local Clifford or graph operations, remains open even in prime-power dimensions (Spengler et al., 2013). This suggests that MUBs are best understood not as a single construction problem, but as a network of equivalent extremal problems in finite geometry, operator theory, algebraic geometry, and quantum information.