Reverse Uncertainty Relations

• Reverse Uncertainty Relations are quantum inequalities that set state-dependent upper bounds on the sum of variances of incompatible observables.
• They arise from simple vector geometry and operator inequalities, offering finite bounds even when individual variances approach zero.
• Recent developments extend these bounds to multiple observables and link them to practical applications like purity detection, quantum control, and metrology.

Searching arXiv for papers on reverse uncertainty relations to ground the article. Reverse uncertainty relation denotes a class of quantum uncertainty inequalities that provide upper bounds on joint fluctuation measures, most commonly the sum of variances of incompatible observables. In contrast to the Heisenberg–Robertson and Robertson–Schrödinger paradigm, which constrains how small fluctuations can be, reverse uncertainty relations constrain how large they can be for a fixed state and observable pair. In the recent formulation of Girolami and Fadel, the reverse bound arises from a simple norm inequality in an inner-product space and yields finite, algebraically elementary upper bounds on $\Delta A^2+\Delta B^2$ in terms of $\Delta(A-B)$ and $\operatorname{Re}\operatorname{Cov}(A,B)$ (Urbanowski, 4 Mar 2025). Earlier work had already developed state-dependent reverse bounds using the Dunkl–Williams inequality (Mondal et al., 2016), uncertainty-matrix constructions involving fidelity and relative entropy (Mukhopadhyay et al., 2018), and experimentally tested reverse bounds in photonic qutrits (Xiao et al., 2019). Subsequent developments addressed divergence issues, extended the framework to multiple observables, and connected reverse uncertainty bounds to purity detection, uncertainty intervals, and quantum-control-assisted settings (Zheng et al., 2023).

1. Concept and formal definition

For two observables $A$ and $B$ in a quantum state $\rho$ or $|\psi\rangle$, a reverse uncertainty relation is an inequality of the form

$\Delta A^2+\Delta B^2 \le U(A,B;\rho),$

where $U$ is a state-dependent upper bound. The central conceptual point, stated explicitly in several works, is that incompatible observables are subject not only to lower restrictions on simultaneous sharpness but also to upper restrictions on simultaneous spread (Mondal et al., 2016, Xiao et al., 2019).

In the formulation of Girolami and Fadel, for a normalized state $|\psi\rangle$ and Hermitian operators $\Delta(A-B)$0, one introduces

$\Delta(A-B)$1

so that $\Delta(A-B)$2, $\Delta(A-B)$3, and $\Delta(A-B)$4 with

$\Delta(A-B)$5

From this construction they derive two reverse uncertainty bounds: $\Delta(A-B)$6 and

$\Delta(A-B)$7

These were presented as new, simple upper bounds on the sum of variances (Urbanowski, 4 Mar 2025).

A distinct line of work defines reverse bounds through an “uncertainty matrix” $\Delta(A-B)$8 with $\Delta(A-B)$9 and $\operatorname{Re}\operatorname{Cov}(A,B)$0. In that framework, for arbitrary mixed states one obtains

$\operatorname{Re}\operatorname{Cov}(A,B)$1

where $\operatorname{Re}\operatorname{Cov}(A,B)$2, $\operatorname{Re}\operatorname{Cov}(A,B)$3, $\operatorname{Re}\operatorname{Cov}(A,B)$4 is the Uhlmann fidelity, and $\operatorname{Re}\operatorname{Cov}(A,B)$5 is the quantum relative entropy (Mukhopadhyay et al., 2018). This places reverse uncertainty relations in a broader information-theoretic setting.

A plausible implication is that “reverse uncertainty relation” is not a single inequality but a family of upper-bound constructions with different structural inputs: vector geometry, operator inequalities, auxiliary-operator methods, and state-space distinguishability.

2. Vector-inequality origin of the 2025 formulation

The 2025 construction begins with an inner-product-space statement independent of quantum mechanics. For any two vectors $\operatorname{Re}\operatorname{Cov}(A,B)$6 in a real or complex inner-product space $\operatorname{Re}\operatorname{Cov}(A,B)$7,

$\operatorname{Re}\operatorname{Cov}(A,B)$8

and also

$\operatorname{Re}\operatorname{Cov}(A,B)$9

The second inequality is explicitly stronger whenever $A$0 is not real and positive (Urbanowski, 4 Mar 2025).

Two proofs are given. The first uses the polarization identity

$A$1

followed by the Cauchy–Schwarz estimate

$A$2

The second uses the triangle inequality, squaring, and the parallelogram law (Urbanowski, 4 Mar 2025).

Mapping these vector identities to quantum mechanics via $A$3 and $A$4 yields

$A$5

which directly produces the reverse uncertainty bounds. The derivation is notable because it avoids the more elaborate normalization structure that earlier reverse relations inherited from Dunkl–Williams-type arguments. The paper also states that neither of the resulting bounds diverges if $A$6 or $A$7, in contrast to some earlier bounds (Urbanowski, 4 Mar 2025).

This suggests a shift in the literature from reverse bounds built by nontrivial geometric normalization toward bounds extracted from elementary Hilbert-space identities.

3. Earlier reverse bounds and the uncertainty-interval viewpoint

A foundational state-dependent reverse relation was proposed by Mondal, Bagchi, and Pati. Using the Dunkl–Williams inequality on the vectors $A$8 and $A$9, with $B$0 and $B$1, they derived an upper bound on the sum of variances of the form

$B$2

with

$B$3

The same work also derived an upper bound on the product of variances using the reverse Cauchy–Schwarz inequality (Mondal et al., 2016).

That paper emphasized a “sandwich” structure: Robertson–Schrödinger furnishes a lower bound on $B$4, while the reverse relation furnishes an upper bound; similarly, reverse sum-form bounds can be paired with lower bounds such as Maccone–Pati to produce a two-sided fluctuation interval (Mondal et al., 2016). The later “near-optimal variance-based uncertainty relations” program made this interval idea explicit by constructing upper bounds $B$5 and lower bounds $B$6 such that

$B$7

for any chosen basis expansion of $B$8 and $B$9 (Xiao et al., 2016).

The upper sum-form bound in that near-optimal framework is

$\rho$0

with

$\rho$1

derived from the triangle inequality plus the parallelogram law (Xiao et al., 2016).

A plausible implication is that reverse uncertainty relations evolved along two distinct tracks: one centered on direct observable-space formulas such as $\rho$2 and covariance, and another centered on basis-dependent decompositions designed to achieve near-optimality.

4. Removing divergence and unifying lower and upper bounds

A persistent issue with the 2017 Dunkl–Williams-type reverse relation is that the denominator

$\rho$3

can vanish, causing the upper bound to diverge even when the observables and state are well defined. This “infinity” problem was made explicit in later work on stronger reverse relations for multiple incompatible observables (Zheng et al., 2023).

To cure this, Zheng, Ji, and Zhang introduced a weighted Hilbert–Schmidt bilinear form

$\rho$4

together with positivity and Cauchy–Schwarz properties, and specialized to the maximally mixed state $\rho$5. For two observables they derived

$\rho$6

equivalently

$\rho$7

The right-hand side is built from finite quantities in finite dimension and therefore does not diverge (Zheng et al., 2023).

The same paper also proposed a unified framework in which lower- and upper-bound uncertainty relations arise from the same auxiliary-operator construction. With suitably chosen auxiliary operators $\rho$8 and associated quantities $\rho$9, generalized commutators, and generalized anticommutators, a master inequality produces lower bounds; by choosing $|\psi\rangle$0 one recovers the new reverse upper bound, whereas choosing $|\psi\rangle$1 recovers Robertson- or Schrödinger-type lower bounds (Zheng et al., 2023).

For multiple observables $|\psi\rangle$2, they introduced

$|\psi\rangle$3

and obtained a general reverse bound with auxiliary contributions $|\psi\rangle$4 whose inclusion monotonically decreases the upper bound. The paper states that the bound can thereby be made arbitrarily tight by choosing more auxiliary operators (Zheng et al., 2023).

This suggests that the modern reverse-uncertainty program is not limited to finding a single improved inequality, but increasingly treats upper bounds as elements of a tunable hierarchy.

5. Alternative strengthening strategies and multi-observable extensions

A separate strengthening program uses refined norm inequalities rather than auxiliary operators. Abd-Rabbou and Qiao derived four families of two-observable forward–reverse inequalities from Maligranda’s angular-distance inequality, and extended them to $|\psi\rangle$5 observables using the sharp $|\psi\rangle$6-vector triangle refinement of M. Kato et al. (Abd-Rabbou et al., 4 Aug 2025).

For two observables, the reverse part has the generic form

$|\psi\rangle$7

The four families split into covariance-sensitive and commutator-sensitive versions, with or without an orthogonal state $|\psi\rangle$8 (Abd-Rabbou et al., 4 Aug 2025). For example, the covariance-sensitive family without orthogonal state is

$|\psi\rangle$9

with

$\Delta A^2+\Delta B^2 \le U(A,B;\rho),$0

The commutator-sensitive families replace $\Delta A^2+\Delta B^2 \le U(A,B;\rho),$1 by $\Delta A^2+\Delta B^2 \le U(A,B;\rho),$2 and use $\Delta A^2+\Delta B^2 \le U(A,B;\rho),$3 in the angular-distance term (Abd-Rabbou et al., 4 Aug 2025).

For $\Delta A^2+\Delta B^2 \le U(A,B;\rho),$4 observables, the reverse extension reads

$\Delta A^2+\Delta B^2 \le U(A,B;\rho),$5

where

$\Delta A^2+\Delta B^2 \le U(A,B;\rho),$6

and $\Delta A^2+\Delta B^2 \le U(A,B;\rho),$7 is built from pairwise covariance-like terms with coefficients $\Delta A^2+\Delta B^2 \le U(A,B;\rho),$8 (Abd-Rabbou et al., 4 Aug 2025).

That paper reports numerical comparisons against the Mondal–Bagchi–Pati and Zheng–Ji–Zhang reverse bounds, claiming strictly tighter performance in the three examples studied, and exact saturation for special phase choices in the spin-1 SU(2) example (Abd-Rabbou et al., 4 Aug 2025). Because these claims are example-specific, they should not be generalized beyond the parameter regimes explicitly analyzed there.

6. Physical interpretation, examples, and experimental status

Several works interpret reverse uncertainty relations as establishing a finite interval for joint fluctuations. In the experimental study on photonic qutrits, forward and reverse bounds together were said to confine $\Delta A^2+\Delta B^2 \le U(A,B;\rho),$9 to a finite interval, described as a “quantum tract” (Xiao et al., 2019). The same study tested the state-dependent reverse sum relation for spin-1 observables $U$0 and $U$1 across the family

$U$2

In every tested case, the measured $U$3 lay below the reverse-bound curve, and for $U$4 the measured pairs

$U$5

$U$6

$U$7

agreed with the ideal values $U$8 within error bars (Xiao et al., 2019).

The 2025 vector-inequality paper also gives a qubit example with $U$9, $|\psi\rangle$0, and

$|\psi\rangle$1

There,

$|\psi\rangle$2

$|\psi\rangle$3

Reverse-UR II reduces to

$|\psi\rangle$4

which the paper states holds for all $|\psi\rangle$5, with equality at $|\psi\rangle$6 or $|\psi\rangle$7 (Urbanowski, 4 Mar 2025).

The information-theoretic uncertainty-matrix approach associates the normalized matrix state $|\psi\rangle$8 with purity and observable incompatibility, and uses the reverse inequality to derive an experimentally accessible lower bound on fidelity

$|\psi\rangle$9

provided $\Delta(A-B)$00 is engineered to coincide with a target $\Delta(A-B)$01 (Mukhopadhyay et al., 2018). This links reverse uncertainty relations to benchmarking without full tomography.

7. Applications, extensions, and open questions

Applications proposed across the literature include metrology, quantum control, thermodynamics, quantum speed limits, fidelity estimation, and purity detection. The uncertainty-matrix formalism was used to derive a reverse quantum speed limit for Markovian evolution,

$\Delta(A-B)$02

with $\Delta(A-B)$03 defined by a time average involving $\Delta(A-B)$04 and $\Delta(A-B)$05 (Mukhopadhyay et al., 2018). The same work also identified “worst-case” bounds on measurement fluctuations in metrology and device-efficient lower bounds on fidelity as applications (Mukhopadhyay et al., 2018).

Zheng, Ji, and Zhang showed that their nondivergent two-observable reverse bound can be rearranged into a lower bound on purity,

$\Delta(A-B)$06

thereby providing a protocol for estimating purity without full state tomography (Zheng et al., 2023).

A 2025 extension introduced a quantum-control-assisted reverse uncertainty relation in which conditional variances obey

$\Delta(A-B)$07

In that work, entanglement with a control system lowers the effective upper bound relative to the unconditional bound $\Delta(A-B)$08, and in a two-spin Heisenberg model with Dzyaloshinskii–Moriya interaction the authors reported a single-valued relation between the effective bound and mixedness $\Delta(A-B)$09 (Li et al., 21 Jul 2025). Because this is a specialized dynamical setting, a cautious interpretation is that reverse uncertainty bounds may depend systematically on mixedness in some controlled models, rather than that such dependence is universal in all formulations.

Open questions are explicitly noted in the recent literature. Girolami and Fadel ask which among their bounds, the covariance-refined version, and the Dunkl–Williams variant is tightest in a given physical scenario, and suggest that experimental tests on qubits or optical modes could adjudicate their practical strength (Urbanowski, 4 Mar 2025). More broadly, the coexistence of vector-inequality, uncertainty-matrix, auxiliary-operator, and angular-distance constructions indicates that reverse uncertainty relations remain an active area of structural comparison rather than a closed theory with a unique canonical form.