Mutually Unbiased Projector-Valued Measurements

• MUPVMs are families of projective measurements defined by uniform Hilbert–Schmidt overlap conditions, generalizing the concept of mutually unbiased bases.
• The operator reformulation replaces vector modulus conditions with trace identities, clarifying the geometry and rank-one factorization crucial for quantum tomography and process reconstruction.
• MUPVMs play key roles in operational quantum protocols such as Bell certification and QRACs, and extend uncertainty relations and separability criteria in quantum information.

Searching arXiv for recent and foundational papers on mutually unbiased projector-valued measurements, MUBs, and MUMs. Mutually Unbiased Projector-Valued Measurements (MUPVMs) are families of projective measurements whose outcome projectors satisfy a uniform cross-overlap condition under the Hilbert–Schmidt inner product. In the rank-$1$ case, MUPVMs coincide exactly with the familiar notion of mutually unbiased bases (MUBs): each orthonormal basis defines a rank-$1$ projector-valued measurement, and distinct bases are mutually unbiased precisely when all cross-traces of the associated projectors equal $1/d$ in dimension $d$ (Kibler, 2014). More generally, the projector formulation isolates mutual unbiasedness as an operator-geometric property, replacing vector-overlap conditions by trace identities between projectors; this viewpoint is central both to operator-space reformulations of the MUB problem (Kibler, 2014) and to later operational definitions of mutually unbiased projective measurements beyond the rank-$1$ setting (Tavakoli et al., 2019, Farkas et al., 2022).

1. Rank-$1$ origin and projector formulation

A family of orthonormal bases

$$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$

in $\mathbb C^d$ is mutually unbiased when, for distinct bases,

$$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$

Within the same basis one has orthonormality. Writing

$$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$

each basis defines a rank-$1$0 projective measurement satisfying

$1$1

The MUB condition is then equivalently

$1$2

so MUBs are exactly rank-$1$3 MUPVMs (Kibler, 2014).

This projector reformulation is conceptually important because it shifts the basic object from vectors to projectors. In that language, orthogonality within one measurement and unbiasedness across different measurements are both encoded by trace relations in operator space. The same move underlies quantum state tomography based on MUB projectors, where each setting is a rank-$1$4 PVM and the full collection of $1$5 such settings yields $1$6 projectors in dimension $1$7 (Lima et al., 2010). The same structure is used again in quantum process reconstruction, where the MUB projectors act simultaneously as input states, measurement effects, and an overcomplete operator expansion (Fernández-Pérez et al., 2011).

A common misconception is that mutually unbiasedness is intrinsically a basis-vector notion. The projector formulation shows that, at least in the rank-$1$8 case, the essential data are the Hilbert–Schmidt overlaps of projectors rather than amplitudes between vectors (Kibler, 2014). This observation is what makes MUPVMs the natural operator-level generalization of MUBs.

2. Operator-space geometry and exact rank-$1$9 constraints

A central operator-space reformulation expands each rank-$1/d$0 projector in the matrix-unit basis $1/d$1 of $1/d$2: $1/d$3 The coefficients $1/d$4 are regarded as a vector

$1/d$5

Finding $1/d$6 MUBs in $1/d$7 is then equivalent to finding $1/d$8 such vectors satisfying

$1/d$9

together with the factorization

$d$0

Equivalently, one may use matrices

$d$1

satisfying

$d$2

These are simply the projectors in matrix form (Kibler, 2014).

The coefficient arrays obey the exact conditions

$d$3

reflecting Hermiticity and trace $d$4, and the inner-product identity

$d$5

For $d$6, the vectors are therefore equiangular in $d$7 with common inner product $d$8 (Kibler, 2014).

The rank-$d$9 factorization is decisive: $1$0 Without it, one would merely have a family of operators with prescribed Hilbert–Schmidt overlaps. With it, one recovers basis vectors and hence the ordinary MUB problem. This distinction remains fundamental in any discussion of MUPVMs: rank-$1$1 MUPVMs admit one-vector parametrization, whereas higher-rank projectors generally do not (Kibler, 2014).

A frequently noted benefit of the operator formulation is that it removes the modulus from the basic constraints. The original MUB condition uses

$1$2

whereas the projector formulation uses

$1$3

In the paper’s words, when passing from equiangular lines in $1$4 to equiangular vectors in $1$5, “the modulus disappears and the $1$6 factor is replaced by $1$7” (Kibler, 2014). For MUPVMs, this is not merely algebraic convenience; it is the intrinsic way to state unbiasedness between projective measurements.

3. General projective definitions beyond bases

A broader operational definition of mutually unbiased projective measurements was given later for two $1$8-outcome projective measurements $1$9 and $1$0. They are called mutually unbiased if

$1$1

and

$1$2

for all $1$3. This operational condition is equivalent to the sandwich identities

$1$4

for all $1$5 (Tavakoli et al., 2019). In the rank-$1$6 nondegenerate case, these identities reduce to

$1$7

hence to the usual MUB relation.

This formulation is important because it explicitly remains within projective measurements rather than general POVMs. It therefore aligns closely with the notion of MUPVMs as mutually unbiased PVMs, not merely as a limit of a POVM formalism (Tavakoli et al., 2019). In dimensions $1$8 and $1$9, the operationally defined measurements reduce to direct sums of MUBs, but in dimensions $$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$0 and $$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$1 the paper exhibits mutually unbiased projective measurements satisfying the sandwich relations that are neither MUBs nor direct sums of MUBs (Tavakoli et al., 2019).

A closely related structural development defines mutually unbiased measurements by the same compression identities

$$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$2

for $$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$3-outcome measurements acting on an arbitrary Hilbert space $$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$4. In that framework, all such MUMs are actually projective, rank-$$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$5 MUBs are the special case on $$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$6, direct sums of MUBs form a natural higher-rank subclass, and there also exist projective examples that are not direct sums of MUBs (Farkas et al., 2022). The canonical form places the measurements on $$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$7 as

$$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$8

$$B_a=\{\lvert a\alpha\rangle:\alpha=0,1,\dots,d-1\},\qquad a=0,1,\dots,d$$9

with associated unitary data $\mathbb C^d$0 satisfying

$\mathbb C^d$1

Such a pair is a direct sum of MUBs if and only if all $\mathbb C^d$2 commute (Farkas et al., 2022).

This establishes an important point of interpretation. There are two distinct generalizations in the literature: one from MUBs to POVM-based mutually unbiased measurements, and one from rank-$\mathbb C^d$3 projective measurements to broader mutually unbiased projective measurements. MUPVMs belong to the second line. The projective developments show that higher-rank or block-projective mutual unbiasedness is not exhausted by direct sums of ordinary MUBs (Tavakoli et al., 2019, Farkas et al., 2022).

4. Relation to MUMs, purity, and nonprojective generalizations

Mutually unbiased measurements (MUMs) generalize MUBs by replacing projectors with POVM effects $\mathbb C^d$4 obeying

$\mathbb C^d$5

$\mathbb C^d$6

and

$\mathbb C^d$7

with

$\mathbb C^d$8

(Chen et al., 2014, Kalev et al., 2014). The efficiency parameter $\mathbb C^d$9 controls the same-measurement sharpness. The case

$$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$0

occurs if and only if all $$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$1 are rank one, yielding a complete set of $$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$2 MUBs (Chen et al., 2014). Thus MUPVMs in the rank-$$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$3 sense are precisely the sharp endpoint of the MUM family.

This POVM generalization matters because complete sets of $$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$4 MUBs are known in prime-power dimensions, but their existence is open in general, whereas complete sets of $$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$5 MUMs exist in every finite dimension (Chen et al., 2014, Kalev et al., 2014). For that reason, MUMs often function as an always-available surrogate for complete mutually unbiased projective families in uncertainty relations, tomography, entanglement detection, and witness construction (Chen et al., 2014, Liu et al., 2015, Siudzińska et al., 2021, Rastegin, 2014).

One strand of work further constructs a class of MUMs with arbitrary purity $$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$6, where $$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$7 is literally the purity

$$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$8

At $$|\langle a\alpha\mid b\beta\rangle|=\frac{1}{\sqrt d}, \qquad a\neq b.$$9, each effect equals $$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$0; at $$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$1, the effects become rank-$$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$2 projectors and the measurements reduce to von Neumann measurements associated with MUBs (Salehi et al., 2021). These arbitrary-purity MUMs are constructed so that effects within one measurement commute and share common spectra, making them structurally closer to projective measurements than generic POVMs (Salehi et al., 2021).

For MUPVMs this has a dual significance. On the one hand, it clarifies that the projector case is an extremal point in a continuous family of unbiased measurements. On the other hand, it highlights a limitation: most MUM results do not amount to a theory of higher-rank MUPVMs, because the defining normalization $$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$3 is tailored to rank-$$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$4 or nonprojective effects, not to higher-rank projectors (Kalev et al., 2014). A natural higher-rank projective analogue suggested by the overlap philosophy is

$$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$5

for equal-rank projectors $$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$6, but this appears as an inferred extension rather than as a standard formalism in those MUM papers (Kalev et al., 2014).

5. Operational roles: tomography, process reconstruction, uncertainty, and entanglement

Rank-$$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$7 MUPVMs appear naturally in tomography. In MUB-based state tomography of photonic qudits, each basis $$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$8 determines a PVM with outcomes

$$\Pi_{a\alpha}=\lvert a\alpha\rangle\langle a\alpha\rvert,$$9

and different settings satisfy

$1$00

The probabilities

$1$01

determine the state via

$1$02

(Lima et al., 2010). In dimensions $1$03 and $1$04, complete sets of $1$05 and $1$06 MUBs were implemented experimentally using spatial light modulators and Fourier-plane detection, achieving fidelities $1$07, $1$08, and $1$09 after “Forced-purity” optimization (Lima et al., 2010).

The same rank-$1$10 MUPVM structure supports process tomography. A complete set of MUB projectors

$1$11

satisfying

$1$12

is used simultaneously as a family of PVMs, as input states, and as an overcomplete operator basis for the process matrix $1$13 (Fernández-Pérez et al., 2011). Reconstruction reduces to a linear system

$1$14

followed by pseudoinverse recovery (Fernández-Pérez et al., 2011).

MUPVMs also govern uncertainty and incompatibility at the projector level. For two rank-$1$15 mutually unbiased projective measurements

$1$16

one has

$1$17

Consequently, the nonselective Lüders channels satisfy

$1$18

so two sequential mutually unbiased projective measurements erase all information in fixed causal order (Gupta et al., 2019). If the order is coherently superposed by a quantum switch, the reduced system is still completely mixed, but the joint control-system output retains input-dependent coherence (Gupta et al., 2019).

Within semi-device-independent prepare-and-measure settings, a pair of mutually unbiased rank-$1$19 PVMs is exactly the optimal solution of the $1$20 QRAC. The optimal average success probability

$1$21

is achievable if and only if Bob’s measurements are rank-$1$22 projective and mutually unbiased; no nonprojective POVMs can do better (Farkas et al., 2018). This gives a self-testing characterization of a pair of rank-$1$23 MUPVMs.

Uncertainty and entanglement-detection results obtained for MUMs specialize directly to MUPVMs at $1$24. For example, for a complete set of $1$25 MUMs the total index of coincidence is

$1$26

which reduces at $1$27 to

$1$28

(Chen et al., 2014). Similarly, the Shannon uncertainty bound

$1$29

reduces in the projective case to

$1$30

(Chen et al., 2014). MUM-based separability criteria likewise reduce to the MUB/MUPVM case when $1$31 (Liu et al., 2015, Rastegin, 2014).

6. Measurement geometry, Bell certification, and QRAC-based higher-rank MUPVMs

Bell-theoretic work has elevated mutually unbiased projective measurements from a structural notion to an operationally certified one. A family of Bell inequalities was constructed so that the maximal quantum violation is achieved by a maximally entangled state of local dimension $1$32 together with any pair of $1$33-dimensional MUBs (Tavakoli et al., 2019). Maximal violation certifies that Bob’s projectors satisfy

$1$34

which is exactly the operational projector-algebraic form of mutual unbiasedness (Tavakoli et al., 2019). In dimensions $1$35 and $1$36, this forces a direct-sum-of-MUB structure; in higher dimensions, the certified mutually unbiased projective measurements can be more general (Tavakoli et al., 2019).

A more recent development introduces MUPVMs explicitly in the context of QRAC decoding geometry. There, a $1$37-dimensional family of $1$38-outcome PVMs $1$39 with equal projector rank $1$40 is called mutually unbiased when

$1$41

while for the same measurement

$1$42

(Akibue et al., 10 Jun 2026). When $1$43, each projector has rank $1$44, so ordinary MUBs are recovered exactly. This formalizes MUPVMs as the direct higher-rank generalization of MUBs.

In the binary-output case $1$45, writing

$1$46

the MUPVM condition becomes

$1$47

for $1$48-qubit observables (Akibue et al., 10 Jun 2026). Thus binary MUPVMs are precisely traceless Hermitian involutions orthogonal in Hilbert–Schmidt geometry. Any optimal two-qubit QRAC attaining the conjectured bound must use decoding measurements forming MUPVMs, though additional cubic constraints are also required for full optimality (Akibue et al., 10 Jun 2026).

The same work shows that any MUPVM, assisted by one ancillary qubit, yields a QRAC with

$1$49

and constructs a new MUPVM-based $1$50-QRAC family attaining

$1$51

(Akibue et al., 10 Jun 2026). This places higher-rank MUPVMs squarely within finite-dimensional quantum information tasks rather than leaving them as a purely formal generalization.

The same theme of geometry appears in work on unsharp versions of MUB projective measurements. For qudits of prime or prime-power dimension $1$52, sharp MUB-based PVMs are deformed into noisy POVMs, and joint measurability emerges below thresholds such as $1$53 for qubits and $1$54 for qutrits, with the general pattern $1$55 for the constructed families (Rao et al., 2020). Although this concerns unsharp deformations rather than MUPVMs proper, it shows how the geometry of sharp mutually unbiased projective measurements controls compatibility and classicality regions.

7. Existence questions, scope, and conceptual status

The existence problem for complete rank-$1$56 MUPVMs is the usual MUB problem. The maximum number of MUBs in $1$57 is $1$58; this maximum is attained when $1$59 is a prime power; for non-prime-power $1$60, existence of $1$61 MUBs is open in general, and dimension $1$62 remains the most prominent unresolved case, where only three MUBs are known while seven would be required for completeness (Kibler, 2014). The operator-space reformulation suggests that disproving the existence of the corresponding $1$63 operator-space vectors would settle the nonexistence of complete MUBs in such dimensions (Kibler, 2014).

For pairwise mutually unbiased projective measurements, the situation is broader. Any orthonormal basis has a mutually unbiased partner, so pairwise rank-$1$64 MUPVMs exist in every finite dimension. Higher-rank projective generalizations also exist, and they need not all be direct sums of MUBs (Tavakoli et al., 2019, Farkas et al., 2022, Akibue et al., 10 Jun 2026). What remains comparatively underdeveloped is a unified general theory of arbitrary-rank MUPVMs parallel in maturity to the extensive MUB literature.

A second conceptual issue is terminology. Much of the literature speaks of MUBs or MUMs even when the natural mathematical object is a family of projective measurements. In tomography (Lima et al., 2010), process reconstruction (Fernández-Pérez et al., 2011), Bell certification (Tavakoli et al., 2019), and QRAC geometry (Akibue et al., 10 Jun 2026), the projector-valued structure is explicit and often central. The term MUPVM serves to foreground that operator-algebraic content.

A third issue is the distinction between projective and POVM generalizations. MUMs provide a robust and dimension-independent extension of the unbiasedness idea (Chen et al., 2014, Kalev et al., 2014), but they are not identical to MUPVMs except at the sharp endpoint $1$65. Conversely, higher-rank MUPVMs lie only partially within standard MUM formalisms because the latter typically normalize each effect to trace $1$66 rather than to fixed rank (Kalev et al., 2014). The two notions therefore overlap at rank $1$67 but diverge beyond it.

In summary, MUPVMs occupy a boundary region between classical MUB theory and broader operator-valued measurement geometry. In rank $1$68, they are exactly MUBs. At the projector level, they encode mutual unbiasedness by constant Hilbert–Schmidt overlaps rather than by vector moduli. In higher rank, they describe subspace-level unbiasedness and already appear in operational settings such as Bell tests and QRACs (Tavakoli et al., 2019, Akibue et al., 10 Jun 2026). The literature therefore supports two complementary readings of MUPVMs: as the projector reformulation of MUBs, and as the beginning of a broader theory of mutually unbiased projective subspace measurements.