Tensor-Product Mutually Unbiased Bases

• Tensor-product MUBs are defined by imposing the unbiasedness condition on multipartite product states, ensuring each cross-overlap equals 1/d.
• They employ local factorization, graph-state formalisms, and finite-field constructions to generate bases in both prime-power and composite dimensions.
• Entangled constructions and nonextendibility results, especially in dimension six, highlight the limits of product-based approaches in composite systems.

Tensor-product mutually unbiased bases arise when the mutually unbiased basis condition is imposed on Hilbert spaces with explicit multipartite structure, such as $$\mathbb C^{d_1}\otimes\cdots\otimes \mathbb C^{d_n}$$, or when basis vectors are replaced by rank-one projectors admitting a tensor-product-type factorization in $\mathbb C^{d^2}$. Two orthonormal bases are mutually unbiased when every cross-overlap has modulus squared $1/d$. In tensor-product settings, this basic condition leads to several distinct but related objects: direct and indirect product bases, graph-state bases, maximally entangled bases, hybrid bases obtained from unextendible maximally entangled bases, and vectorized projector formulations in enlarged spaces. The resulting theory combines local-factor criteria, prime-power constructions, entanglement-sensitive classifications, and strong nonexistence or nonextendibility results in composite dimensions, especially dimension six (McNulty et al., 2015, Kibler, 2014, Shi et al., 2019).

1. Local factorization of the unbiasedness condition

For a multipartite system

$$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$

a product basis is a basis whose vectors factorize as

$$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$

The tensor-product MUB problem begins with the observation that mutual unbiasedness can either be inherited directly from local MUBs or be constrained by them. For direct product bases $\{|j_a,J_a\rangle\}$ and $\{|k_b,K_b\rangle\}$ in $\mathbb C^p\otimes\mathbb C^q$, the criterion is exact: the two bases are mutually unbiased if and only if

$$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$

This is the clean tensor-product criterion for direct product bases in composite dimension (McNulty et al., 2011).

The corresponding statement for arbitrary orthogonal product bases is stronger than a mere sufficient condition. If

$$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$

is a product state and

$\mathbb C^{d^2}$0

is an orthogonal product basis, then the state is mutually unbiased to $\mathbb C^{d^2}$1 if and only if, for each subsystem $\mathbb C^{d^2}$2,

$\mathbb C^{d^2}$3

In other words, the global product state is MU to the global product basis exactly when each local component is MU to all local vectors that appear in that subsystem of the basis. This factorized characterization is the central structural statement for multipartite product MUBs (McNulty et al., 2015).

The same line of work distinguishes direct product bases, where one literally takes the tensor product of local orthonormal bases, from indirect product bases, where every basis vector is still a product vector but different vectors may use different local bases in one subsystem. This distinction is essential in low-dimensional classifications, especially in $\mathbb C^{d^2}$4, because indirect product bases are not always locally equivalent to direct ones (McNulty et al., 2015, McNulty et al., 2012).

A major consequence is a sharp restriction on the number of mutually unbiased product bases. If

$\mathbb C^{d^2}$5

with $\mathbb C^{d^2}$6 or $\mathbb C^{d^2}$7, and $\mathbb C^{d^2}$8 for $\mathbb C^{d^2}$9, then there exist at most

$1/d$0

mutually unbiased product bases in $1/d$1. Thus a qubit factor limits the number of MU product bases to $1/d$2, and a qutrit factor limits it to $1/d$3 (McNulty et al., 2015).

2. Canonical product structures for qubits, qutrits, and multipartite systems

When the smallest subsystem is a qubit or qutrit, the local-factor theorem leads to complete classification results. In $1/d$4, there is, up to local equivalence, a unique triple of MU product bases, built from tensor products of eigenbases of the Pauli operators $1/d$5. In $1/d$6, there is, up to local equivalence, a unique quadruple of MU product bases, constructed from the four local MU bases in dimension $1/d$7, namely the eigenbases of

$1/d$8

These uniqueness statements identify the maximal product-only MU structures for multiple qubits and qutrits (McNulty et al., 2015).

In dimension six, the tensor-product decomposition

$1/d$9

is the standard setting. All orthonormal product bases in this space are equivalent to four families, denoted

$$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$0

and every MU product pair is equivalent to one of four corresponding families

$$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$1

From these, exactly two inequivalent MU product triples survive: $$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$2 and

$$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$3

No quadruple of mutually unbiased product bases exists in dimension six (McNulty et al., 2011).

The dimension-six classifications also make explicit the local Heisenberg–Weyl structure. In $$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$4, one uses the three MU bases

$$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$5

while in $$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$6 one uses the four MU bases

$$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$7

with the qutrit Fourier matrix

$$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$8

serving as the standard representative of the $$\mathbb{C}^d=\mathbb{C}^{d_1}\otimes \cdots \otimes \mathbb{C}^{d_n},\qquad d=d_1d_2\cdots d_n,$$9-basis (McNulty et al., 2011, McNulty et al., 2011).

This classification shows that tensor-product MUBs in composite dimensions are not merely “local MUBs tensored together.” Indirect product bases exist, and in dimension six they are indispensable for exhausting the product-state sector, even though they do not lead to larger MU families.

3. Prime-power constructions: graph states, finite fields, and maximal families

In prime-power dimensions $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$0, complete sets of $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$1 MUBs are known to exist, and tensor-product structure can be encoded very compactly. One construction is the graph-state formalism for $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$2 $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$3-level systems. In this approach, one basis is the computational basis $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$4, and the other $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$5 bases are graph-state bases. Each graph-state basis is generated by a symmetric adjacency matrix $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$6 over $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$7, and for two bases with adjacency matrices $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$8 and $$|a_i\rangle = |a_i^{(1)}\rangle\otimes \cdots \otimes |a_i^{(n)}\rangle.$$9,

$\{|j_a,J_a\rangle\}$0

implies mutual unbiasedness. The key constructive result is that one can choose the $\{|j_a,J_a\rangle\}$1 adjacency matrices as all polynomial combinations of powers of a single symmetric matrix $\{|j_a,J_a\rangle\}$2,

$\{|j_a,J_a\rangle\}$3

provided $\{|j_a,J_a\rangle\}$4 has irreducible characteristic polynomial over $\{|j_a,J_a\rangle\}$5. Then the powers of $\{|j_a,J_a\rangle\}$6 represent the finite field $\{|j_a,J_a\rangle\}$7, and a single $\{|j_a,J_a\rangle\}$8-dimensional vector associated with this graph can be used to generate a complete set of MUBs (Spengler et al., 2013).

The graph-state basis itself is produced from

$\{|j_a,J_a\rangle\}$9

by applying phase gates determined by the graph, and then using local $\{|k_b,K_b\rangle\}$0 shifts,

$\{|k_b,K_b\rangle\}$1

In this formalism, tensor-product MUBs appear exactly as the product-separable graph-state bases: whenever the adjacency matrix has no edges between certain vertices, the corresponding basis factorizes. The graph method therefore embeds product bases into a larger complete set that also contains entangled bases, while making the entanglement pattern directly readable from the adjacency matrix (Spengler et al., 2013).

A complementary structural result characterizes maximal families of MUBs in arbitrary dimension by partitioned unitary error bases, up to a choice of a family of Hadamards. A partitioned UEB contains the identity and is partitioned into commuting classes

$\{|k_b,K_b\rangle\}$2

where each class has exactly $\{|k_b,K_b\rangle\}$3 operators and, together with the identity, forms a maximal commuting set of $\{|k_b,K_b\rangle\}$4 commuting operators. The common eigenbases of the commuting classes form a maximal family of MUBs. The same work gives a new finite-field construction of partitioned UEBs and hence of maximal MUBs in prime-power dimensions, expressed in tensor-diagrammatic language (Musto, 2017).

Taken together, these constructions show that prime-power tensor-product MUBs are controlled by global algebraic data: a single symmetric matrix $\{|k_b,K_b\rangle\}$5, a finite-field representation, or a partitioned UEB. The local tensor factors remain visible, but the complete MUB family is generated by a higher-level algebraic structure rather than by independent local choices basis by basis.

4. Tensor-product-type reformulation in $\{|k_b,K_b\rangle\}$6

A different use of tensor-product structure replaces basis vectors by rank-one projectors and studies MUBs in the enlarged space $\{|k_b,K_b\rangle\}$7. In this reformulation, the usual problem of finding a complete set of $\{|k_b,K_b\rangle\}$8 mutually unbiased bases in $\{|k_b,K_b\rangle\}$9 is equivalent to finding $\mathbb C^p\otimes\mathbb C^q$0 vectors

$\mathbb C^p\otimes\mathbb C^q$1

with components $\mathbb C^p\otimes\mathbb C^q$2, $\mathbb C^p\otimes\mathbb C^q$3, satisfying

$\mathbb C^p\otimes\mathbb C^q$4

together with the factorization condition

$\mathbb C^p\otimes\mathbb C^q$5

This is the paper’s central tensor-product-like structure: each vector in $\mathbb C^p\otimes\mathbb C^q$6 is built from a single vector $\mathbb C^p\otimes\mathbb C^q$7 by an outer product of components (Kibler, 2014).

The equivalence is obtained by associating to each MUB vector $\mathbb C^p\otimes\mathbb C^q$8 the rank-one projector

$\mathbb C^p\otimes\mathbb C^q$9

expanding $$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$0 in the basis $$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$1 of $$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$2 matrices, and reading the expansion coefficients as $$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$3. The matrix-valued form is

$$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$4

with Hilbert–Schmidt overlap relation

$$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$5

Conversely, if one finds $$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$6 vectors in $$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$7 obeying the inner-product condition and the factorization

$$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$8

then the original MUB vectors are recovered simply by taking the column vector built from the $$|\langle j_a \mid k_b\rangle|^2=\frac1p,\qquad |\langle J_a \mid K_b\rangle|^2=\frac1q.$$9 (Kibler, 2014).

This formulation makes explicit that the vectors $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$0 are not arbitrary elements of $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$1; they lie in the subset corresponding to pure-state projectors. The same note briefly treats symmetric informationally complete POVMs in exactly the same vein, reinforcing the general principle that basis and measurement problems in $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$2 can be recast as structured vector problems in $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$3 with rank-one factorization encoding purity (Kibler, 2014).

A plausible implication is that the tensor-product-type reformulation separates two distinct difficulties in the MUB problem: the geometric overlap constraint in the enlarged space and the nonlinear rank-one factorization back to $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$4. The note explicitly suggests this viewpoint as potentially useful in difficult cases such as $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$5 (Kibler, 2014).

5. Bipartite entangled constructions: MUMEBs, MUSEB$$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$6s, and difference matrices

In bipartite spaces $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$7, the tensor-product MUB problem includes bases made of entangled states. A pure state is a special entangled state with Schmidt number $$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$8 if it has Schmidt decomposition

$$|u_1,u_2,\dots,u_n\rangle = |u_1\rangle\otimes\cdots\otimes|u_n\rangle$$9

with orthonormal sets in the two factors. A basis of $\mathbb C^{d^2}$00 pairwise orthogonal such states is an SEB$\mathbb C^{d^2}$01, and two SEB$\mathbb C^{d^2}$02s are mutually unbiased if every cross-overlap has modulus $\mathbb C^{d^2}$03. A family of such bases is a MUSEB$\mathbb C^{d^2}$04; when $\mathbb C^{d^2}$05, these are MUMEBs, mutually unbiased maximally entangled bases (Shi et al., 2019).

A first landmark result is the construction of two MUMEBs in

$\mathbb C^{d^2}$06

the first example of MUMEBs in $\mathbb C^{d^2}$07 when $\mathbb C^{d^2}$08. The paper also proves a recursive theorem: if there are $\mathbb C^{d^2}$09 MUSEB$\mathbb C^{d^2}$10s in $\mathbb C^{d^2}$11 and $\mathbb C^{d^2}$12 MUSEB$\mathbb C^{d^2}$13s in $\mathbb C^{d^2}$14, then there exist

$\mathbb C^{d^2}$15

MUSEB$\mathbb C^{d^2}$16s in

$\mathbb C^{d^2}$17

Equivalently,

$\mathbb C^{d^2}$18

The same paper derives that three MUMEBs exist for all $\mathbb C^{d^2}$19 with $\mathbb C^{d^2}$20, and that two MUMEBs exist for infinitely many pairs with $\mathbb C^{d^2}$21 (Shi et al., 2019).

A second large family comes from difference matrices in combinatorial design theory. For any prime power $\mathbb C^{d^2}$22, one can construct $\mathbb C^{d^2}$23 mutually unbiased bases in

$\mathbb C^{d^2}$24

consisting of one product basis and $\mathbb C^{d^2}$25 maximally entangled bases. The same method yields, for a prime $\mathbb C^{d^2}$26, $\mathbb C^{d^2}$27 mutually unbiased bases in

$\mathbb C^{d^2}$28

with $\mathbb C^{d^2}$29 maximally entangled bases and one product basis. In addition, the construction improves lower bounds in several non-prime-power cases, including

$\mathbb C^{d^2}$30

and further examples at $\mathbb C^{d^2}$31 (Zang et al., 2022).

A third bipartite mechanism starts from unextendible maximally entangled bases. In $\mathbb C^{d^2}$32, one begins with a UMEB of $\mathbb C^{d^2}$33 maximally entangled states,

$\mathbb C^{d^2}$34

and completes it by adding two product states to obtain a full orthonormal basis. The paper derives a necessary and sufficient condition for constructing a pair of MUBs in $\mathbb C^{d^2}$35, generalizes the construction to $\mathbb C^{d^2}$36 for $\mathbb C^{d^2}$37, and gives an explicit $\mathbb C^{d^2}$38 example (Zhao et al., 2020).

These constructions show that tensor-product MUBs in bipartite systems are not restricted to separable bases. Product bases, maximally entangled bases, and hybrid bases can all participate, and the number of entangled bases can dominate the product sector even in constructions that retain one preferred product basis.

6. Dimension six as the principal obstruction regime

Dimension six is the central composite-dimension test case because

$\mathbb C^{d^2}$39

is the smallest non-prime-power dimension, and the existence of a complete set of seven MUBs remains open. The tensor-product viewpoint is especially informative here because the strongest known structural obstructions are phrased in terms of product bases and product vectors.

The first obstruction is exhaustive: all mutually unbiased product bases in dimension six can be constructed, yielding several continuous families of pairs and two triples, but no quadruple. This already implies that a complete set of seven MUBs, if it exists, cannot contain a triple of mutually unbiased product bases (McNulty et al., 2011).

The second obstruction is analytic rather than classificatory. No triple of MU product bases in dimension six can be extended by a single MU vector. Equivalently, once one has any triple of MU product bases in $\mathbb C^{d^2}$40, there does not exist even one additional state mutually unbiased to all states in that triple. The same paper strengthens this to the statement that the constellation

$\mathbb C^{d^2}$41

cannot be part of a complete set of seven MU bases (McNulty et al., 2012).

The third obstruction sharpens the hypothetical structure of a complete set. If a complete set of seven mutually unbiased bases in dimension six exists, it contains at most one product basis. Since one basis can always be mapped to the standard computational basis, this is the strongest possible upper bound in the product-basis setting. The theorem does not rule out the existence of a complete set, but it rules out any complete set composed largely of product bases and shows that at least six of the seven bases must contain entangled states (McNulty et al., 2011).

A related analysis assumes only four MUBs in $\mathbb C^{d^2}$42 and requires that one of them be a product-vector basis. In most cases the number of product vectors in each of the remaining three MUBs is at most two. There is an exceptional case in which the three remaining MUBs respectively contain at most three, two, and two product vectors, giving the extremal distribution

$\mathbb C^{d^2}$43

The exceptional case is explicitly constructed, but it remains highly constrained (Chen et al., 2017).

Even entangled-basis constructions do not evade the obstruction automatically. The two MUMEBs constructed in $\mathbb C^{d^2}$44 cannot be extended to four MUBs in $\mathbb C^{d^2}$45. This does not prove that four MUBs do not exist in dimension six, but it shows that this particular entangled pair cannot be embedded into such a quartet (Shi et al., 2019).

The combined picture is unambiguous: in dimension six, tensor-product methods reveal rigid local and entanglement-theoretic constraints, but they do not produce a route to a complete MUB family. Product constructions are severely limited, and even promising entangled constructions encounter nonextendibility.

7. Related tensor-product uses: coherence, Hadamard matrices, and structured nonextendibility

The tensor-product MUB framework also appears in problems that are adjacent to basis existence. One example is the autotensor of mutually unbiased bases (AMUB), defined from a set of MUBs $\mathbb C^{d^2}$46 in $\mathbb C^{d^2}$47 by

$\mathbb C^{d^2}$48

For qubit MUBs $\mathbb C^{d^2}$49, the corresponding two-qubit AMUBs $\mathbb C^{d^2}$50 organize the $\mathbb C^{d^2}$51-norm coherence of Bell-diagonal states. The paper finds that Werner states and isotropic states have equal coherence in all three AMUBs,

$\mathbb C^{d^2}$52

and

$\mathbb C^{d^2}$53

For Bell-diagonal states, the paper also studies the sum of AMUB coherences and describes its level surfaces as tetrahexahedron-like (Wang et al., 2019).

A second application turns mutually unbiased product bases into complex Hadamard matrices. In composite dimensions

$\mathbb C^{d^2}$54

if one has two orthonormal product bases in standard form that are mutually unbiased, then the unitary change-of-basis matrix between them is a complex Hadamard matrix. The central theorem states that if $\mathbb C^{d^2}$55 and $\mathbb C^{d^2}$56 are unitary $\mathbb C^{d^2}$57 matrices such that every

$\mathbb C^{d^2}$58

is a complex Hadamard matrix, then the block matrix $\mathbb C^{d^2}$59 built from these blocks and a Hadamard matrix $\mathbb C^{d^2}$60 of order $\mathbb C^{d^2}$61 is a complex Hadamard matrix of order $\mathbb C^{d^2}$62. This construction produces at least $\mathbb C^{d^2}$63 new isolated Butson-type matrices with orders ranging from $\mathbb C^{d^2}$64 to $\mathbb C^{d^2}$65, and the paper reports $\mathbb C^{d^2}$66 isolated complex Hadamard matrices in total, of which $\mathbb C^{d^2}$67 are new (McNulty et al., 2012).

A third adjacent direction concerns strongly unextendible MUBs in $\mathbb C^{d^2}$68. For each integer $\mathbb C^{d^2}$69, there exists a set of

$\mathbb C^{d^2}$70

MUBs in $\mathbb C^{d^2}$71 that is strongly $\mathbb C^{d^2}$72-unextendible. These bases arise from bent functions and the Kerdock bent set rather than from an ad hoc tensor-product construction. The paper explicitly emphasizes that they are not simply obtained by taking tensor products of lower-dimensional MUBs. This suggests that tensor-product viewpoints capture only part of the structured MUB landscape, even in dimensions where complete sets exist (Jedwab et al., 2016).

The broader significance is that tensor-product mutually unbiased bases are simultaneously a construction mechanism, a diagnostic for entanglement structure, a source of combinatorial and Hadamard-theoretic objects, and a setting in which nonextendibility phenomena become especially sharp. In prime-power dimensions they support complete algebraic constructions; in composite dimensions, and most notably in dimension six, they instead expose the limits of separable and partially separable organization.