---
title: Exact Uncertainty Relations
url: https://www.emergentmind.com/topics/exact-uncertainty-relation
type: topic
---

# Exact Uncertainty Relations

The exact uncertainty relation is not a single formula but a family of rigorously formulated uncertainty statements in which the qualifier “exact” has a specific technical meaning. In the literature summarized here, it may denote an equality, a tight state-independent majorization bound, or a closed form whose dependence on incompatibility data is completely specified. The best-known instances are the Hall–Reginatto exact uncertainty principle, which identifies a Fisher-information-based equality underlying the Schrödinger equation; Hall’s operator equality for pure states; tight universal majorization relations for measurement outcome distributions; and exact operational formulations for entropic, energy–time, error–disturbance, and single-observable settings [1602.05004][2512.20359][1304.6351][1604.04944].

## 1. Terminological scope and senses of exactness

In this literature, “exact” is used in several non-equivalent but related senses.

| Setting | Exact statement | Sense of exactness |
|---|---|---|
| Hall–Reginatto EUP | $\delta x_l\,\Delta N_l=\hbar/2$ | Equality |
| Hall operator relation | $\delta_B A\,\Delta B_{nc}=1/2$ | Equality |
| Universal UUR | $p(\rho)\otimes q(\rho)\prec \omega$ | Tight state-independent bound |
| Memory-assisted entropic UR | dependence on all $d$ largest overlaps $c_{(1)},\dots,c_{(d)}$ | Closed-form exact dependence |

The Hall–Reginatto formulation starts from a classical ensemble and postulates momentum fluctuations whose scale is fixed exactly by the Fisher information of the position density. The resulting equality is then stronger than the usual Heisenberg inequality in the sense that the latter follows only after using $\Delta p_l\ge \Delta N_l$ [1602.05004].

In Hall’s operator formulation for pure states, exactness refers to a literal equality between a reciprocal error-like quantity $\delta_B A$ and the nonclassical fluctuation $\Delta B_{nc}$. In the later Krylov-space reformulation, that equality becomes the statement that the operator amplitude vector moves on a unit sphere with constant speed $b_1$ [2512.20359].

In universal majorization and entropic formulations, exactness does not mean equality for every state. Rather, it means that the right-hand side is the optimal state-independent vector bound, or that the lower bound depends on all relevant overlap data rather than only on the largest overlap or the first two overlaps [1304.6351][1604.04944].

A further operational use appears in arrival-time theory, where “exact” designates sharp energy–time theorems for an explicitly defined arrival-time POVM generated by absorption, namely
$$
\Delta T\,\Delta E \ge \frac{\sqrt p\,\hbar}{2},\qquad
\langle T\rangle\,\Delta E \ge 1.37\,\sqrt p\,\hbar,
$$
under the assumption that the absorbing term vanishes on the initial wave function [1109.5087].

## 2. Hall–Reginatto exact uncertainty principle

The original exact uncertainty principle is formulated for a classical ensemble in configuration space $x\in\mathbb{R}^n$ described by a probability density $P(x,t)$ and a Hamilton principal function $S(x,t)$. These satisfy the continuity equation
$$
\frac{\partial P}{\partial t}
+\nabla\!\cdot\!\bigl(P\,\tfrac{1}{m}\nabla S\bigr)=0
$$
and the Hamilton–Jacobi equation
$$
\frac{\partial S}{\partial t}
+\frac{1}{2m}\,\nabla S\!\cdot\!\nabla S
+V(x)=0.
$$
The key postulate is that the physical momentum field is not simply $\nabla S$ but
$$
p_l(x,t)=\partial_{x_l}S+N_l(x,t),
$$
where $N_l$ is a fluctuation with unbiasedness conditions
$$
\langle N_l\rangle=0,\qquad
\langle \partial_{x_l}S\,N_l\rangle=0.
$$
The classical action then acquires an extra fluctuation term,
$$
\mathcal A_Q=\mathcal A_C+\frac{1}{2m}\sum_{l=1}^n\int dt\,(\Delta N_l)^2.
$$
Hall and Reginatto impose that there is a direct position-uncertainty measure $\delta x_l$ determined solely by $P$, and that momentum fluctuations scale inversely under rescaling. This yields
$$
(\Delta N)^2=\frac{\hbar^2}{4}\sum_{l=1}^n F_l[P],\qquad
F_l[P]=\int d^n x\,\frac{1}{P}\bigl(\partial_{x_l}P\bigr)^2,
$$
with the natural choice
$$
\delta x_l=\frac{1}{\sqrt{F_l[P]}}.
$$
The defining exact equality is therefore
$$
\delta x_l\,\Delta N_l=\frac{\hbar}{2}.
$$
Since the actual momentum uncertainty obeys $\Delta p_l\ge \Delta N_l$, the usual Heisenberg inequality follows as a corollary rather than as the primitive statement [1602.05004].

The same formalism reproduces standard nonrelativistic quantum mechanics. The Euler–Lagrange equations derived from $\mathcal A_Q$ give a modified Hamilton–Jacobi equation which, under the Madelung substitution
$$
\psi=\sqrt P\,e^{iS/\hbar},
$$
becomes exactly the linear Schrödinger equation
$$
i\hbar\,\frac{\partial\psi}{\partial t}
=
\Bigl(-\frac{\hbar^2}{2m}\nabla^2+V\Bigr)\psi.
$$
This gives the exact uncertainty principle a foundational role: the equality involving Fisher information is the structural input, while the Schrödinger dynamics emerges from it [1602.05004].

## 3. Generalized exact uncertainty principle and quasi-nonlinear dynamics

A later extension replaces the linear fluctuation scaling law by
$$
\Delta N_l\mapsto w^{-1}\!\bigl(\kappa\,w(\Delta N_l)\bigr),
$$
where $w(z)$ is smooth, strictly increasing, obeys $w(z)<z$ for $z>0$, and satisfies $w'(0)=1$. The same Fisher-information argument then gives
$$
w(\Delta N_l)=\sqrt{C\,F_l[P]},\qquad C=\frac{\hbar^2}{4},
$$
so that the modified exact uncertainty relation becomes
$$
\delta x_l\,\Delta N_l^{(w)}=\frac{\hbar}{2},\qquad
\Delta N_l^{(w)}=w(\Delta N_l).
$$
Because $\Delta p_l\ge \Delta N_l$ and $w$ is increasing, one obtains
$$
\Delta x_l\,w(\Delta p_l)\ge \frac{\hbar}{2}.
$$
This provides a systematic embedding of generalized uncertainty relations into the exact-uncertainty framework [1602.05004].

For the quadratic generalized uncertainty principle,
$$
\Delta x\,\Delta p\ge \frac{\hbar}{2}\bigl(1+\beta(\Delta p)^2\bigr),
$$
the choice
$$
w(z)=\frac{z}{1+\beta z^2}
$$
makes the modified exact relation algebraically equivalent to the quadratic GUP. The fluctuation term in the action is then altered accordingly, and the Madelung transformation yields a quasi-nonlinear Schrödinger equation,
$$
i\hbar\,\frac{\partial\psi}{\partial t}
=
\biggl[
-\frac{\hbar^2}{2m}\nabla^2+V
-\frac{\hbar^2}{2m}\sum_{l=1}^n
W\!\bigl(C\,F_l[|\psi|^2]\bigr)\,
\frac{1}{|\psi|}\frac{\partial^2|\psi|}{\partial x_l^2}
\biggr]\psi.
$$
Here
$$
W_l[P]=W\!\bigl(C\,F_l[P]\bigr),\qquad
W(z)=\frac{d}{dz}\Bigl[w^{-1}(\sqrt z)\Bigr]^2-1.
$$
The equation is described as separable for non-interacting subsystems, homogeneity-preserving, and Galilei-invariant; plane-wave states with $F_l=0$ remain exact solutions. On that basis, the paper argues that a modification of a free-particle dispersion relation due to quantum gravity might not occur in reality [1602.05004].

For the one-dimensional harmonic oscillator ground state, the modified theory yields Gaussian solutions with a state-dependent width, and the position variance has a lower bound. In the GUP case,
$$
\min(\Delta x)^2=\hbar^2\beta=\beta_0\,l_p^2,
$$
or equivalently $\min\Delta x=l_p\sqrt{\beta_0}$. This realizes the familiar Planck-length-scale minimal position uncertainty within the exact-uncertainty construction [1602.05004].

## 4. Hall’s operator equality and the Krylov-space geometric form

For two Hermitian operators $A$ and $B$ on a pure state $\rho=|\psi\rangle\langle\psi|$, Hall’s exact uncertainty relation is
$$
\delta_B A\,\Delta B_{nc}=\frac12.
$$
The reciprocal quantity $\delta_B A$ is defined by
$$
(\delta_B A)^{-2}
=
\sum_k
\frac{\bigl\langle a_k\bigm|\,i[B,\rho]\,\bigm|a_k\bigr\rangle^2}
{\langle a_k|\rho|a_k\rangle},
$$
while
$$
B_{nc}=B-B_{cl},
$$
with
$$
B_{cl}
=
\sum_k
\frac{\langle a_k|\tfrac12\{B,\rho\}|a_k\rangle}
{\langle a_k|\rho|a_k\rangle}\,
|a_k\rangle\langle a_k|.
$$
The decomposition satisfies
$$
\langle B_{cl}\rangle=\langle B\rangle,\qquad
\langle B_{nc}\rangle=0,\qquad
\langle B^2\rangle=\langle B_{cl}^2\rangle+\langle B_{nc}^2\rangle.
$$
The exactness here is literal: the relation is an equality, not a lower bound [2512.20359].

A 2025 reformulation gives this relation a geometric interpretation in Krylov space. Let the Liouvillian superoperator be
$$
\mathcal L:A\mapsto [H,A],
$$
and represent operators as vectors $|A)$ with Hilbert–Schmidt inner product $(A|B)=\mathrm{Tr}(A^\dagger B)$. Starting from a normalized seed $|K_0)$, the Lanczos recursion
$$
\mathcal L|K_n)=b_{n+1}|K_{n+1})+b_n|K_{n-1}),\qquad b_0=0,
$$
generates an orthonormal Krylov basis with Lanczos coefficients $b_n\ge 0$ [2512.20359].

Heisenberg evolution,
$$
|{\cal O}(t))=e^{-i\mathcal L t}|{\cal O}(0)),
$$
is expanded as
$$
|{\cal O}(t))=\sum_{n=0}^{D-1} i^n\,\phi_n(t)\,|K_n),
$$
where the amplitudes are real and satisfy
$$
\sum_{n=0}^{D-1}\phi_n(t)^2=1.
$$
Thus the amplitude vector
$$
\mathbf a(t)=\bigl(\phi_0(t),\phi_1(t),\ldots,\phi_{D-1}(t)\bigr)^T
$$
lives on a real unit sphere. Its components obey
$$
\dot\phi_n=b_n\phi_{n-1}-b_{n+1}\phi_{n+1},
$$
and the squared speed is
$$
\|\dot{\mathbf a}(t)\|^2=\sum_n \dot\phi_n^2(t)=b_1^2.
$$
Hence
$$
\Bigl\|\frac{d\mathbf a}{dt}\Bigr\|=b_1
$$
at all times. The arc length therefore satisfies
$$
\ell_K(t)=\int_0^t \|\dot{\mathbf a}(t')\|\,dt'=b_1 t,
$$
and the geometric distance obeys the exact linear speed limit
$$
d[A(0),A(t)]\equiv \arccos\bigl(\phi_0(t)\bigr)\le b_1 t.
$$
Only the first Lanczos coefficient enters the speed; higher coefficients determine curvature and torsion of the trajectory, not the instantaneous norm of the velocity [2512.20359].

The same work identifies the Mandelstam–Tamm relation as a special case with $\Delta E=b_1$, and rewrites the exact quantum speed limit of Pati et al. as
$$
\delta_{\mathcal L}A=\frac{1}{2b_1},\qquad
\Delta\mathcal L_{nc}=b_1,
$$
so that $\delta_{\mathcal L}A\,\Delta\mathcal L_{nc}=1/2$ becomes the constant-speed identity in Krylov space [2512.20359].

## 5. Exact majorization and entropic formulations

The universal uncertainty-relation program replaces entropy-specific statements by a vector majorization inequality. For two orthonormal bases $\{|a_m\rangle\}$ and $\{|b_n\rangle\}$ in dimension $d$, and any state $\rho$, define
$$
p_m(\rho)=\langle a_m|\rho|a_m\rangle,\qquad
q_n(\rho)=\langle b_n|\rho|b_n\rangle,
$$
and form the $d^2$-component vector $p(\rho)\otimes q(\rho)$. Then
$$
p(\rho)\otimes q(\rho)\prec \omega,
$$
where $\omega$ is constructed from
$$
\Omega_k
=
\max_{\substack{I_k\subset\{1,\dots,d\}\times\{1,\dots,d\}\\|I_k|=k}}
\;\max_\rho\;
\sum_{(m,n)\in I_k} p_m(\rho)q_n(\rho).
$$
The vector is
$$
\omega=
\bigl(\Omega_1,\Omega_2-\Omega_1,\Omega_3-\Omega_2,\dots,\Omega_d-\Omega_{d-1},0,\dots,0\bigr).
$$
This is the exact universal uncertainty relation: for each $k$ there exists a subset $I_k$ and a pure $\rho$ achieving the corresponding maximum, so no strictly larger state-independent vector can replace $\omega$ [1304.6351].

The first entries admit closed form. Writing
$$
c=\max_{m,n}|\langle a_m|b_n\rangle|,
$$
one has
$$
\Omega_1=\frac14(1+c)^2.
$$
Similarly,
$$
\Omega_2=\frac14(1+c')^2,
$$
with $c'$ obtained from the appropriate two-projector optimization. Any Schur-concave uncertainty quantifier $\Phi$ then yields
$$
\Phi(p\otimes q)\ge \Phi(\omega),
$$
and if $\Phi$ is additive under tensor products,
$$
\Phi(p)+\Phi(q)\ge \Phi(\omega).
$$
This immediately produces exact Shannon, Rényi, and Tsallis uncertainty relations [1304.6351].

A different notion of exactness appears in the improved entropic uncertainty relation with quantum memory. For a $d$-dimensional system $A$, arbitrary memory $B$, and two projective measurements $X=\{X_i\}$ and $Z=\{Z_j\}$ on $A$, define the overlap matrix
$$
c_{ij}=|\langle x_i|z_j\rangle|^2,
$$
and order the first $d$ largest overlaps as
$$
c_{(1)}\ge c_{(2)}\ge \cdots \ge c_{(d)}.
$$
Then
$$
H(X|B)+H(Z|B)
\ge
H(A|B)
-\log c_{(1)}
+\sum_{k=1}^{d-1}\frac{2-\Omega_{2k}}{2}\,
\log\frac{c_{(k)}}{c_{(k+1)}}.
$$
The proof combines the relative-entropy method of Berta et al., successive mass rearrangement, and the direct-sum majorization bound $P\oplus Q\prec \{1\}\oplus W$ associated with Rudnicki–Puchała–Życzkowski. The resulting lower bound depends on all $d$ largest overlaps, not only on the maximum overlap or the first two overlaps [1604.04944].

The memory-free corollary is
$$
H(X)+H(Z)
\ge
-\log c_{(1)}
+\sum_{k=1}^{d-1}\frac{2-\Omega_{2k}}{2}\,
\log\frac{c_{(k)}}{c_{(k+1)}}.
$$
The paper states that these bounds are strictly tighter than previous entropic bounds in the presence of quantum memory and gives applications to quantum cryptography with entanglement witnesses and quantum key distributions. It also notes that the new closed form is complementary to direct-sum majorization bounds and is stronger than Berta et al. and Coles–Piani except in the maximally unbiased case [1604.04944].

## 6. Operational, relativistic, and measurement-theoretic exact relations

An exact operational energy–time relation is obtained when arrival is modeled by absorption. Let $H$ be self-adjoint, let the detector be a positive operator $D\ge 0$, define
$$
K=H-iD,
$$
and let
$$
B_t=e^{-iKt/\hbar},\qquad \|B_t\|\le 1.
$$
The loss of normalization determines the arrival-time POVM, and if the initial state $\psi$ satisfies $D\psi=0$, then with absorption probability
$$
p=1-\lim_{t\to\infty}\|B_t\psi\|^2
$$
the exact theorems are
$$
\Delta T\,\Delta E\ge \frac{\sqrt p\,\hbar}{2},
\qquad
\langle T\rangle\,\Delta E\ge C\sqrt p\,\hbar,
\qquad
C=2(-Z_1/3)^{3/2}\approx 1.376.
$$
The first bound can be approached arbitrarily closely, while the second is sharp and is governed by the Airy problem on the half-line. A two-level system nearly saturates both bounds, and a trapped-ion realization with a single ${}^{40}\mathrm{Ca}^+$ ion is proposed [1109.5087].

Measurement uncertainty can also be formulated exactly through quantum estimation theory. For a POVM $M=\{M_i\}$, observable $X=x_0 I+x\cdot\Lambda$, and unknown state $\rho$, the measurement error is defined by
$$
\varepsilon(X;M)
=
\min_{\text{consistent estimators }\hat X}
\Bigl[\lim_{n\to\infty}n\,\mathrm{Var}[\hat X]-\Delta X^2\Bigr]
=
x^T\bigl[J(M;\rho)^{-1}-J_Q(\rho)^{-1}\bigr]x
\ge 0.
$$
For any two observables $X_1,X_2$ and any POVM $M$,
$$
\varepsilon(X_1;M)\,\varepsilon(X_2;M)
\ge
\frac14\bigl|\mathrm{Tr}\bigl(\rho[X_1,X_2]\bigr)\bigr|^2.
$$
For the class of noisy random PVMs, a stronger attainable bound holds:
$$
\varepsilon(X_1;M)\,\varepsilon(X_2;M)
\ge
(\Delta_Q X_1)^2(\Delta_Q X_2)^2-\bigl[C_Q(X_1,X_2)\bigr]^2,
$$
and equality is achieved by randomly measuring suitable projective observables $Y_1$ and $Y_2$ with probabilities $q_1$ and $q_2$ [1010.3571].

A universal error–disturbance framework defines, for any measurement $M$ and process $\Theta$,
$$
\varepsilon(A;M,\rho)
=
\sqrt{\|A\|_\rho^2-\|M_\rho A\|_{M(\rho)}^2},
\qquad
\eta(B;\Theta,\rho)
=
\sqrt{\|B\|_\rho^2-\|\Theta_\rho B\|_{\Theta(\rho)}^2}.
$$
The exact error–disturbance inequality is
$$
\varepsilon(A;M,\rho)\,\eta(B;\Theta,\rho)\ge \sqrt{\mathcal R^2+\mathcal I^2},
$$
with
$$
\mathcal I=\frac{1}{2i}\langle[A,B]\rangle_\rho
$$
and an explicit real term $\mathcal R$ built from $M_\rho$ and $\Theta_\rho$. The parallel error-only relation has the same structure, and the trivial measurement case reproduces the Kennard–Robertson relation. The paper emphasizes that the bound may exceed the traditional naive $\hbar/2$ product while remaining consistent with Heisenberg’s original observation-effect intuition [2004.06099].

A compact two-factor reformulation defines
$$
\bar\varepsilon(A)=\varepsilon(A)+\sigma(A),\qquad
\bar\eta(B)=\eta(B)+\sigma(B),
$$
and proves
$$
(\varepsilon(A)+\sigma(A))(\eta(B)+\sigma(B))
\ge |\langle[A,B]\rangle|.
$$
This combines Ozawa’s universally valid error–disturbance relation with Robertson’s intrinsic fluctuation bound and is presented as a universally valid Heisenberg uncertainty relation [1205.1360].

In the relativistic photon case, where no genuine position operator exists, the center-of-energy operator
$$
\hat R_i=\frac12(\hat H^{-1}\hat N_i+\hat N_i\hat H^{-1})
$$
satisfies
$$
[\hat R_i,\hat P_j]=i\hbar\,\delta_{ij},\qquad
[\hat R_i,\hat R_j]\neq 0.
$$
This noncommutativity raises the finite-energy three-dimensional bound to
$$
\Delta R_i\,\Delta P_i\ge \frac{3}{2}\hbar,
$$
while the infinite-momentum-frame limit returns to the nonrelativistic $\hbar/2$ per axis [1205.0516].

## 7. Higher-order, single-observable, and trade-off extensions

A generalized exact uncertainty relation can be written directly in terms of cumulant-generating functionals. For observables $X,Y$ and state $|\psi\rangle$, define
$$
K[sX]=\log\langle\psi|e^{sX}|\psi\rangle,
\qquad
Z_{s,t}=\log(e^{sX}e^{tY}),
$$
with $Z_{s,t}$ expanded by the BCH formula. The exact generalized uncertainty relation is
$$
K[(s+s^*)X]+K[(t+t^*)Y]
\ge
K(Z_{s,t})+K^*(Z_{s,t}).
$$
Its second-order expansion reproduces Schrödinger’s variance relation and hence the Heisenberg form by discarding the covariance term. Third and higher orders generate skewness and higher-cumulant uncertainty relations through cross-cumulants and nested commutators. The paper presents examples in which the standard Schrödinger bound becomes trivial while the generalized relation remains nontrivial, and it gives applications to entanglement detection and skewness-based nonlocality [2003.08705].

A recent extension shows that uncertainty relations need not be pairwise. For a density operator $\rho$ with smallest and largest eigenvalues $\lambda_{\min}$ and $\lambda_{\max}$, and any observable $A$,
$$
\mathrm{Var}_\rho(A)
\ge
\frac{\lambda_{\max}+\lambda_{\min}}
{2(\lambda_{\max}-\lambda_{\min})^2}
\,\|[A,\rho]\|^2.
$$
This coefficient is stated to be optimal. After introducing the pinching
$$
\mathcal P_\rho(A)=\sum_\mu P_\mu A P_\mu
$$
and the classical variance
$$
U_{\rm cl}(A,\rho)=\mathrm{Var}_\rho(\mathcal P_\rho(A)),
$$
one obtains the sharpened relation
$$
\mathrm{Var}_\rho(A)
\ge
U_{\rm cl}(A,\rho)
+
\frac{\lambda_{\max}+\lambda_{\min}}
{2(\lambda_{\max}-\lambda_{\min})^2}
\,\|[A,\rho]\|^2.
$$
For qubits this becomes the exact identity
$$
\mathrm{Var}_\rho(A)=U_{\rm cl}(A,\rho)+U_{\rm nc}(A,\rho),
$$
with
$$
U_{\rm cl}(A,\rho)=(1-r^2)\cos^2\theta,\qquad
U_{\rm nc}(A,\rho)=\sin^2\theta.
$$
The same single-observable bounds yield improved product-form uncertainty relations for pairs of observables [2605.26331].

A related state-independent variance program uses Bloch-vector geometry. In the qubit case, for two observables $A=\vec\sigma\!\cdot\!\vec n_a$ and $B=\vec\sigma\!\cdot\!\vec n_b$ in a pure state, one obtains
$$
\Delta A\,\Delta B
\ge
\bigl|
\sqrt{1-\Delta A^2}\,\sqrt{1-\Delta B^2}
-\cos\theta_{ab}
\bigr|,
$$
and for three observables an exact surface,
$$
\Delta A^2+\Delta B^2+\sin^2\theta_{ab}\,\Delta C^2
+2\cos\theta_{ab}\sqrt{1-\Delta A^2}\sqrt{1-\Delta B^2}=2.
$$
This formulation is variance-based, state-independent once the observables are fixed, and presented as immune to the triviality problem associated with vanishing expectation values in Robertson-type bounds [1502.06292].

Taken together, these developments show that “exact uncertainty relation” designates a structured research program rather than a single theorem. Depending on context, exactness may mean equality, optimality, sharp operational attainability, or complete dependence on the relevant incompatibility data. A plausible implication is that the subject has shifted from a single variance product inequality to a hierarchy of exact formulations spanning Fisher information, operator geometry, majorization, conditional entropy, arrival-time POVMs, measurement error, and state–observable noncommutativity.

Source: https://www.emergentmind.com/topics/exact-uncertainty-relation