---
title: Mutually Unbiased Projector-Valued Measurements
url: https://www.emergentmind.com/topics/mutually-unbiased-projector-valued-measurements-mupvms
type: topic
---

# Mutually Unbiased Projector-Valued Measurements

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 \(d\) [1401.4643]. 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 [1401.4643] and to later operational definitions of mutually unbiased projective measurements beyond the rank-\(1\) setting [1912.03225, 2204.11886].

## 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-\(1\) projective measurement satisfying
\[
\Pi_{a\alpha}\Pi_{a\beta}=\delta_{\alpha\beta}\Pi_{a\alpha},
\qquad
\sum_{\alpha=0}^{d-1}\Pi_{a\alpha}=I.
\]
The MUB condition is then equivalently
\[
\operatorname{Tr}(\Pi_{a\alpha}\Pi_{b\beta})
=
|\langle a\alpha\mid b\beta\rangle|^2
=
\frac1d
\qquad (a\neq b),
\]
so MUBs are exactly rank-\(1\) MUPVMs [1401.4643].

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-\(1\) PVM and the full collection of \(D+1\) such settings yields \(D(D+1)\) projectors in dimension \(D\) [1003.2125]. 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 [1104.2888].

A common misconception is that mutually unbiasedness is intrinsically a basis-vector notion. The projector formulation shows that, at least in the rank-\(1\) case, the essential data are the Hilbert–Schmidt overlaps of projectors rather than amplitudes between vectors [1401.4643]. This observation is what makes MUPVMs the natural operator-level generalization of MUBs.

## 2. Operator-space geometry and exact rank-\(1\) constraints

A central operator-space reformulation expands each rank-\(1\) projector in the matrix-unit basis \(E_{pq}\) of \(\mathrm{GL}(d,\mathbb C)\):
\[
\Pi_{a\alpha}=\sum_{p=0}^{d-1}\sum_{q=0}^{d-1} w_{pq}(a\alpha)\,E_{pq}.
\]
The coefficients \(w_{pq}(a\alpha)\) are regarded as a vector
\[
\mathbf w(a\alpha)\in\mathbb C^{d^2}.
\]
Finding \(d+1\) MUBs in \(\mathbb C^d\) is then equivalent to finding \(d(d+1)\) such vectors satisfying
\[
\sum_{p,q}\overline{w_{pq}(a\alpha)}\,w_{pq}(b\beta)
=
\delta_{\alpha,\beta}\delta_{a,b}
+\frac1d(1-\delta_{a,b}),
\tag{1}
\]
together with the factorization
\[
w_{pq}(a\alpha)=\omega_p(a\alpha)\,\overline{\omega_q(a\alpha)}.
\tag{2}
\]
Equivalently, one may use matrices
\[
(M_{a\alpha})_{pq}=\omega_p(a\alpha)\,\overline{\omega_q(a\alpha)}
\]
satisfying
\[
\operatorname{Tr}(M_{a\alpha}M_{b\beta})
=
\delta_{\alpha,\beta}\delta_{a,b}
+\frac1d(1-\delta_{a,b}).
\tag{3}
\]
These are simply the projectors in matrix form [1401.4643].

The coefficient arrays obey the exact conditions
\[
\overline{w_{pq}(a\alpha)}=w_{qp}(a\alpha),
\qquad
\sum_p w_{pp}(a\alpha)=1,
\]
reflecting Hermiticity and trace \(1\), and the inner-product identity
\[
\mathbf w(a\alpha)\cdot \mathbf w(b\beta)
=
\delta_{\alpha,\beta}\delta_{a,b}
+\frac1d(1-\delta_{a,b}).
\]
For \(a\neq b\), the vectors are therefore equiangular in \(\mathbb C^{d^2}\) with common inner product \(1/d\) [1401.4643].

The rank-\(1\) factorization is decisive:
\[
w_{pq}(a\alpha)=\omega_p(a\alpha)\,\overline{\omega_q(a\alpha)}.
\]
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-\(1\) MUPVMs admit one-vector parametrization, whereas higher-rank projectors generally do not [1401.4643].

A frequently noted benefit of the operator formulation is that it removes the modulus from the basic constraints. The original MUB condition uses
\[
|\langle a\alpha\mid b\beta\rangle|=\frac1{\sqrt d},
\]
whereas the projector formulation uses
\[
\operatorname{Tr}(\Pi_{a\alpha}\Pi_{b\beta})=\frac1d.
\]
In the paper’s words, when passing from equiangular lines in \(\mathbb C^d\) to equiangular vectors in \(\mathbb C^{d^2}\), “the modulus disappears and the \(1/\sqrt d\) factor is replaced by \(1/d\)” [1401.4643]. 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 \(n\)-outcome projective measurements \(\{P_a\}_{a=1}^n\) and \(\{Q_b\}_{b=1}^n\). They are called mutually unbiased if
\[
\langle\psi|P_a|\psi\rangle=1 \Rightarrow \langle\psi|Q_b|\psi\rangle=\frac1n,
\]
and
\[
\langle\psi|Q_b|\psi\rangle=1 \Rightarrow \langle\psi|P_a|\psi\rangle=\frac1n,
\]
for all \(a,b\). This operational condition is equivalent to the sandwich identities
\[
P_a=nP_aQ_bP_a,
\qquad
Q_b=nQ_bP_aQ_b
\]
for all \(a,b\) [1912.03225]. In the rank-\(1\) nondegenerate case, these identities reduce to
\[
\operatorname{Tr}(P_aQ_b)=\frac1n,
\]
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 [1912.03225]. In dimensions \(2\) and \(3\), the operationally defined measurements reduce to direct sums of MUBs, but in dimensions \(4\) and \(5\) the paper exhibits mutually unbiased projective measurements satisfying the sandwich relations that are neither MUBs nor direct sums of MUBs [1912.03225].

A closely related structural development defines mutually unbiased measurements by the same compression identities
\[
P_a=dP_aQ_bP_a,
\qquad
Q_b=dQ_bP_aQ_b,
\]
for \(d\)-outcome measurements acting on an arbitrary Hilbert space \(H\). In that framework, all such MUMs are actually projective, rank-\(1\) MUBs are the special case on \(\mathbb C^d\), direct sums of MUBs form a natural higher-rank subclass, and there also exist projective examples that are not direct sums of MUBs [2204.11886]. The canonical form places the measurements on \(\mathbb C^n\otimes\mathbb C^d\) as
\[
P_a = I_n\otimes |a\rangle\langle a|,
\]
\[
Q_b = \frac1d \sum_{j,k\in[d]} V^b_{jk}\otimes |j\rangle\langle k|,
\]
with associated unitary data \(U_j^b\) satisfying
\[
U_1^b=I,\qquad
\sum_b U_j^b(U_k^b)^\dagger=\delta_{jk}dI.
\]
Such a pair is a direct sum of MUBs if and only if all \(U_j^b\) commute [2204.11886].

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-\(1\) 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 [1912.03225, 2204.11886].

## 4. Relation to MUMs, purity, and nonprojective generalizations

Mutually unbiased measurements (MUMs) generalize MUBs by replacing projectors with POVM effects \(P_n^{(b)}\) obeying
\[
\operatorname{Tr}(P_n^{(b)})=1,
\]
\[
\operatorname{Tr}(P_n^{(b)}P_{n'}^{(b')})=\frac1d
\qquad (b\neq b'),
\]
and
\[
\operatorname{Tr}(P_n^{(b)}P_{n'}^{(b)})
=
\delta_{n,n'}\,\kappa
+
(1-\delta_{n,n'})\frac{1-\kappa}{d-1},
\]
with
\[
\frac1d<\kappa\le 1
\]
[1407.6816, 1401.2706]. The efficiency parameter \(\kappa\) controls the same-measurement sharpness. The case
\[
\kappa=1
\]
occurs if and only if all \(P_n^{(b)}\) are rank one, yielding a complete set of \(d+1\) MUBs [1407.6816]. Thus MUPVMs in the rank-\(1\) sense are precisely the sharp endpoint of the MUM family.

This POVM generalization matters because complete sets of \(d+1\) MUBs are known in prime-power dimensions, but their existence is open in general, whereas complete sets of \(d+1\) MUMs exist in every finite dimension [1407.6816, 1401.2706]. 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 [1407.6816, 1501.01717, 2109.14069, 1407.7333].

One strand of work further constructs a class of MUMs with arbitrary purity \(\kappa\in[1/d,1]\), where \(\kappa\) is literally the purity
\[
\operatorname{Tr}\!\left[(P_n^{(b)})^2\right]=\kappa.
\]
At \(\kappa=1/d\), each effect equals \(\mathbb I/d\); at \(\kappa=1\), the effects become rank-\(1\) projectors and the measurements reduce to von Neumann measurements associated with MUBs [2105.08831]. 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 [2105.08831].

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 \(\operatorname{Tr}(P_n^{(b)})=1\) is tailored to rank-\(1\) or nonprojective effects, not to higher-rank projectors [1401.2706]. A natural higher-rank projective analogue suggested by the overlap philosophy is
\[
\operatorname{Tr}(Q_n^{(b)}Q_{n'}^{(b')})=
\begin{cases}
r\,\delta_{nn'}, & b=b',\\[1mm]
\dfrac{r^2}{d}, & b\neq b',
\end{cases}
\]
for equal-rank projectors \(\operatorname{rank}(Q_n^{(b)})=r\), but this appears as an inferred extension rather than as a standard formalism in those MUM papers [1401.2706].

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

Rank-\(1\) MUPVMs appear naturally in tomography. In MUB-based state tomography of photonic qudits, each basis \(\mathcal B_\alpha\) determines a PVM with outcomes
\[
\Pi_m^{(\alpha)}=|\psi_m^{(\alpha)}\rangle\langle\psi_m^{(\alpha)}|,
\qquad
\sum_{m=1}^D \Pi_m^{(\alpha)}=I,
\]
and different settings satisfy
\[
\big|\langle \psi_m^{(\alpha)} \mid \psi_{m'}^{(\alpha')} \rangle\big|^2 = \frac{1}{D},
\qquad
\alpha\neq \alpha'.
\]
The probabilities
\[
p_m^{(\alpha)}=\mathrm{Tr}(\rho\Pi_m^{(\alpha)})
\]
determine the state via
\[
\rho=\sum_{\alpha=1}^{D+1}\sum_{m=1}^{D} p_m^{(\alpha)}\Pi_m^{(\alpha)}-I
\]
[1003.2125]. In dimensions \(7\) and \(8\), complete sets of \(8\) and \(9\) MUBs were implemented experimentally using spatial light modulators and Fourier-plane detection, achieving fidelities \(F_{7q}=0.96\pm0.03\), \(F_{8q}^{(1)}=0.93\pm0.03\), and \(F_{8q}^{(2)}=0.91\pm0.03\) after “Forced-purity” optimization [1003.2125].

The same rank-\(1\) MUPVM structure supports process tomography. A complete set of MUB projectors
\[
\mathcal P_m^{(\gamma)}=|\psi_m^{(\gamma)}\rangle\langle\psi_m^{(\gamma)}|
\]
satisfying
\[
\mathrm{Tr}\!\left(\mathcal P_m^{(\gamma)}\mathcal P_n^{(\beta)}\right)
=
\delta_{\beta\gamma}\delta_{mn}+\frac1D(1-\delta_{\beta\gamma})
\]
is used simultaneously as a family of PVMs, as input states, and as an overcomplete operator basis for the process matrix \(\chi\) [1104.2888]. Reconstruction reduces to a linear system
\[
p_{\eta s}^{(\gamma,l)}
=
\sum_{\alpha,\beta,m,n}
\chi_{mn}^{(\alpha,\beta)}
\,
\mathrm{Tr}\!\left(
\mathcal P_m^{(\alpha)}
\mathcal P_l^{(\gamma)}
\mathcal P_n^{(\beta)}
\mathcal P_s^{(\eta)}
\right),
\]
followed by pseudoinverse recovery [1104.2888].

MUPVMs also govern uncertainty and incompatibility at the projector level. For two rank-\(1\) mutually unbiased projective measurements
\[
E_i=|e_i\rangle\langle e_i|,
\qquad
F_j=|f_j\rangle\langle f_j|,
\]
one has
\[
E_iF_jE_i=\frac1d E_i,
\qquad
F_jE_iF_j=\frac1d F_j.
\]
Consequently, the nonselective Lüders channels satisfy
\[
\mathcal M_F\circ\mathcal M_E(\rho)=\mathcal M_E\circ\mathcal M_F(\rho)=\frac{\mathbb I}{d},
\]
so two sequential mutually unbiased projective measurements erase all information in fixed causal order [1909.13125]. 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 [1909.13125].

Within semi-device-independent prepare-and-measure settings, a pair of mutually unbiased rank-\(1\) PVMs is exactly the optimal solution of the \(2^d\to1\) QRAC. The optimal average success probability
\[
\bar p_Q=\frac12\left(1+\frac1{\sqrt d}\right)
\]
is achievable if and only if Bob’s measurements are rank-\(1\) projective and mutually unbiased; no nonprojective POVMs can do better [1803.00363]. This gives a self-testing characterization of a pair of rank-\(1\) MUPVMs.

Uncertainty and entanglement-detection results obtained for MUMs specialize directly to MUPVMs at \(\kappa=1\). For example, for a complete set of \(d+1\) MUMs the total index of coincidence is
\[
C(\kappa,\rho)=
\frac{(d\kappa-1)\bigl[d\,\operatorname{Tr}(\rho^2)-1\bigr]+d^2-1}{d(d-1)},
\]
which reduces at \(\kappa=1\) to
\[
C(1,\rho)=\operatorname{Tr}(\rho^2)+1
\]
[1407.6816]. Similarly, the Shannon uncertainty bound
\[
\frac{1}{d+1}\sum_{b=1}^{d+1} H(\mathcal P^{(b)}|\rho)
\ge
\log_2\frac{d+1}{\kappa+1}
\]
reduces in the projective case to
\[
\frac{1}{d+1}\sum_{b=1}^{d+1} H(\mathcal P^{(b)}|\rho)
\ge
\log_2\frac{d+1}{2}
\]
[1407.6816]. MUM-based separability criteria likewise reduce to the MUB/MUPVM case when \(\kappa=1\) [1501.01717, 1407.7333].

## 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 \(d\) together with any pair of \(d\)-dimensional MUBs [1912.03225]. Maximal violation certifies that Bob’s projectors satisfy
\[
P_a=dP_aQ_bP_a,
\qquad
Q_b=dQ_bP_aQ_b,
\]
which is exactly the operational projector-algebraic form of mutual unbiasedness [1912.03225]. In dimensions \(2\) and \(3\), this forces a direct-sum-of-MUB structure; in higher dimensions, the certified mutually unbiased projective measurements can be more general [1912.03225].

A more recent development introduces MUPVMs explicitly in the context of QRAC decoding geometry. There, a \(D\)-dimensional family of \(d\)-outcome PVMs \(\{P_a^{(n)}\}_{a=1}^d\) with equal projector rank \(D/d\) is called mutually unbiased when
\[
\operatorname{tr}\bigl(P_a^{(n)}P_b^{(m)}\bigr)
=
\frac{D}{d^2}
\qquad (n\neq m),
\]
while for the same measurement
\[
\operatorname{tr}\bigl(P_a^{(n)}P_b^{(n)}\bigr)=\frac{D}{d}\,\delta_{ab}
\]
[2606.12700]. When \(D=d\), each projector has rank \(1\), so ordinary MUBs are recovered exactly. This formalizes MUPVMs as the direct higher-rank generalization of MUBs.

In the binary-output case \(d=2\), writing
\[
M_\pm^{(n)}=\frac12(I\pm \sigma^{(n)}),
\]
the MUPVM condition becomes
\[
(\sigma^{(n)})^2=I,
\qquad
\operatorname{tr}\sigma^{(n)}=0,
\qquad
\operatorname{tr}\bigl(\sigma^{(n)}\sigma^{(m)}\bigr)=2^M\delta_{nm}
\]
for \(M\)-qubit observables [2606.12700]. 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 [2606.12700].

The same work shows that any MUPVM, assisted by one ancillary qubit, yields a QRAC with
\[
p_{\rm wc}\ge \frac12\left(1+\frac1{\sqrt N}\right),
\]
and constructs a new MUPVM-based \((M+2,M)\)-QRAC family attaining
\[
p=\frac12\left(1+\sqrt{\frac{M}{M+2}}\right)
\]
[2606.12700]. 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 \(n\), sharp MUB-based PVMs are deformed into noisy POVMs, and joint measurability emerges below thresholds such as \(1/\sqrt3\) for qubits and \(1/\sqrt8\) for qutrits, with the general pattern \(\eta\le (n^2-1)^{-1/2}\) for the constructed families [2004.05547]. 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-\(1\) MUPVMs is the usual MUB problem. The maximum number of MUBs in \(\mathbb C^d\) is \(d+1\); this maximum is attained when \(d\) is a prime power; for non-prime-power \(d\), existence of \(d+1\) MUBs is open in general, and dimension \(6\) remains the most prominent unresolved case, where only three MUBs are known while seven would be required for completeness [1401.4643]. The operator-space reformulation suggests that disproving the existence of the corresponding \(d(d+1)\) operator-space vectors would settle the nonexistence of complete MUBs in such dimensions [1401.4643].

For pairwise mutually unbiased projective measurements, the situation is broader. Any orthonormal basis has a mutually unbiased partner, so pairwise rank-\(1\) MUPVMs exist in every finite dimension. Higher-rank projective generalizations also exist, and they need not all be direct sums of MUBs [1912.03225, 2204.11886, 2606.12700]. 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 [1003.2125], process reconstruction [1104.2888], Bell certification [1912.03225], and QRAC geometry [2606.12700], 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 [1407.6816, 1401.2706], but they are not identical to MUPVMs except at the sharp endpoint \(\kappa=1\). Conversely, higher-rank MUPVMs lie only partially within standard MUM formalisms because the latter typically normalize each effect to trace \(1\) rather than to fixed rank [1401.2706]. The two notions therefore overlap at rank \(1\) but diverge beyond it.

In summary, MUPVMs occupy a boundary region between classical MUB theory and broader operator-valued measurement geometry. In rank \(1\), 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 [1912.03225, 2606.12700]. 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.

Source: https://www.emergentmind.com/topics/mutually-unbiased-projector-valued-measurements-mupvms