Entropic Uncertainty Relations (EUR)

• Entropic uncertainty relations (EUR) are rigorous information-theoretic bounds that quantify the trade-off in predictability for incompatible quantum measurements.
• EURs extend to continuous variables, quantum memory, and multipartite systems, enhancing applications in cryptography and experimental quantum protocols.
• Experimental validations using stabilizer bases and multipartite setups provide actionable insights for designing secure quantum communication and efficient quantum batteries.

Entropic uncertainty relations (EURs) articulate the quantum mechanical constraints on the simultaneous predictability of measurement outcomes for incompatible observables, expressed in rigorous information-theoretic language. EURs have evolved into foundational tools for quantum information, cryptography, and the delineation of quantum correlations, unifying measurement-theoretic and entanglement-based perspectives of uncertainty. The formalism encompasses both finite- and infinite-dimensional systems, generalizes to multiple observables and measurement scenarios, and encompasses the effects of quantum memory as well as relativistic (non-inertial) reference frames.

1. Foundations: Canonical Forms and Generalizations

The archetypal EUR for observables $X$ and $Z$ on a finite-dimensional Hilbert space $\mathcal{H}$ is the Maassen–Uffink inequality,

$$H(X) + H(Z) \geq -\log c, \quad c = \max_{x, z} |\langle x|z\rangle|^2,$$

where $H(X)$ is the Shannon entropy for the measurement probabilities $p_X(x) = \langle x | \rho | x \rangle$, and $c$ quantifies the maximal overlap of the measurement bases (Coles et al., 2015, Niekamp et al., 2011). This bound reflects the fundamental limitation of obtaining sharp outcomes for both observables: the more incompatible $X$ and $Z$ (the smaller $c$), the greater the total uncertainty. The relation admits generalization to Rényi, min/max, and Tsallis entropies, with comparable structural form but modified coefficients and duality relations (Coles et al., 2015, Coles et al., 2011).

For continuous variables, the Hirschman–Beckner–Białynicki-Birula form is

$h(Q) + h(P) \geq \log (e \pi),$

with $h(Q)$ the differential entropy of the position distribution and $h(P)$ the momentum distribution, reflecting the information-theoretic strengthening of the Heisenberg–Robertson inequality (Coles et al., 2015, Corichi et al., 24 Mar 2025). For discretized or polymeric (non-regular) Hilbert spaces, the bound adapts to

$H_{\text{pos}} + H_{\text{mom}} \geq \ln(L/\mu_0)$

where $\mu_0$ is the lattice spacing and $L$ the total length, smoothly recovering the continuum result as $\mu_0 \to 0$ (Corichi et al., 24 Mar 2025).

2. Entropic Uncertainty Relations with Quantum Memory

A central advance is the quantum-memory-assisted EUR (QMA-EUR), which reduces the lower bound via entropy contributions from quantum side information (Coles et al., 2015, Xu et al., 6 Jan 2025, Du et al., 11 Dec 2025): $S(Q|B) + S(R|B) \geq \log_2 \frac{1}{c} + S(A|B),$ with $S(Q|B) = S(\rho_{QB}) - S(\rho_B)$ the conditional von Neumann entropy after measurement of $Q$ on $A$, where $B$ holds quantum memory. The overlap parameter $c$ and the conditional entropy $S(A|B)$ (which may be negative for entangled states) jointly determine the bound (Du et al., 11 Dec 2025, Hu et al., 2013).

This formulation extends to multipartite scenarios (Xu et al., 6 Jan 2025, Wang et al., 2022). For $m$ measurements on $A$ and $m$ quantum memories $(B_i)$,

$$\sum_{i=1}^m S(M_i|B_i) \geq -\frac{1}{m-1} \sum_{i<j} \log_2 c(M_i, M_j) + \frac{m-1}{2} S(A) - \sum_{i} \mathcal{I}(M_i: B_i)$$

where $\mathcal{I}(M: B)$ is the Holevo information quantifying the quantum correlations between $A$ and $B$, and the bound tightens when genuine multipartite quantum correlations are present (Wang et al., 2022, Xu et al., 6 Jan 2025). This is operationally significant for quantum cryptography, as inclusion of the Holevo terms improves the security thresholds in key distribution (Wang et al., 2022).

EURs also provide tight bounds for quantum discord and constraints on its shareability. For any quantum state $\rho_{AB}$: $$D_A(\rho_{AB}) \leq \min \left\{ S(\rho_A), I(\rho_{AB}), \frac{\delta_T + I(\rho_{AB})}{2} \right\},\quad \delta_T = S(Q|B) + S(R|B) - \log_2\frac{1}{c} - S(A|B)$$ showing explicit control of discord via entropic uncertainty (Hu et al., 2013).

3. EURs Beyond Two Observables, Weightings, and Extensions

Standard EURs extend to multiple measurements via majorization, weightings, and state-dependent parameters. For a collection $\{M_\theta\}$ of POVMs with measurement probabilities $w_\theta$ and Rényi entropy $H_\alpha$: $$\sum_\theta w_\theta H_\alpha(p_{\cdot|\theta}) \geq Q_\alpha[l, \bar{c}]$$ where $\bar{c}$ depends on the average index of coincidence and POVM measurement details. Weight optimization via semidefinite programming enables strictly stronger bounds than uniform weighting, directly benefiting quantum steering protocols and biased-basis QKD (Huang et al., 2023). Additionally, the tightness of EURs for multiple observables reflects nontrivial additivity properties – strict additivity holds only for $L=2$ observables, while for $L>2$ or for multipartite joint entropies, violation of naive additivity directly signals quantum correlations (2207.13469).

Tomographic protocols for continuous-variable and hybrid systems exploit statistical properties of optical tomograms to construct experimentally accessible EURs. The inverse-participation-ratio–based tomographic indicator gives the highest correlation with entropic sum-uncertainty in both integrable and strongly nonlinear regimes (Paul et al., 2023).

4. Measurement Disturbance, Reversibility, and Relative-Entropy EURs

Recent analysis unifies uncertainty, quantum memory, and measurement disturbance by introducing a measurement-reversibility correction: $$H(Z|B)_\omega + H(X|B)_\sigma \geq -\log c - \log f + H(A|B)_\rho$$ where $f$ is the fidelity between the pre-measurement state and the best-recovered post-measurement state via an explicit recovery channel (Berta et al., 2015). The term $-\log f$ quantifies the irreversibility of measurement, strictly tightens conventional EUR bounds, and admits experimental validation in current superconducting-qubit platforms (Berta et al., 2015).

Relative entropy generalizations (REURs) take the form: $$D(P_A\|Q_A) + D(P_B\|Q_B) \leq -\ln(1/c) - S(\rho) + H(Q_A) + H(Q_B)$$ where $D(P\|Q)$ is the Kullback-Leibler divergence of the measurement outcome distribution from a maximum-entropy reference, and $S(\rho)$ the von Neumann entropy. These relations directly sharpen the standard EURs and admit extension to Rényi or Tsallis relative entropies, with utility for quantum fields and situations where classical entropy is ill-defined (Floerchinger et al., 2020).

5. Physical Applications: Cryptography, Quantum Batteries, and Relativistic Regimes

EURs are foundational in quantum key distribution (QKD): the conditional min- and max-entropy versions

$$H_\infty^\epsilon(X|E) + H_{\max}^\epsilon(Z|E) \geq -\log_2 c(X, Z)$$

form the security backbone for prepare-and-measure and entanglement-based protocols. For projective rank-one POVMs the overlap $c<1$ gives nontrivial security, yet for more general POVMs, $c$ can reach unity, collapsing the lower bound and thereby precluding security proofs; this necessitates either measurement restriction or sampling-based effective overlaps (Bae, 2022).

In quantum thermodynamics, EUR tightness correlates quantitatively with the efficiency of quantum batteries under various environmental conditions. For a bipartite quantum battery–charger model, the difference $\Delta^{xz}(t)$ between the EUR left and right sides serves as a universal indicator of extractable work and energy-conversion efficiency (Song et al., 2024).

In relativistic frameworks, acceleration-induced decoherence (Unruh effect) and spacetime curvature (de Sitter, black-hole analogues) influence the available quantum correlations and thus the effective bound in QMA-EURs. Notably, depending on initial quantum correlations, acceleration can both tighten or loosen the EUR lower bound; decoherence-driven loss of quantum discord or minimal missing information mediates this behavior (Du et al., 11 Dec 2025, Jia et al., 2015, Feng et al., 2013). Indefinite causal ordering of quantum operations, as implemented with the quantum switch or time-flip, can mitigate noise-induced degradation of EURs and thus extend operational quantum information protocols (Karpat, 18 Nov 2025).

6. Hierarchy, Tightness, and Experimental Realizations

Tightness of entropic bounds is achieved for specific pairs of observables: the Maassen-Uffink relation is saturated for mutually unbiased bases and for any pair of stabilizer bases in the stabilizer formalism (Niekamp et al., 2011). In dichotomic anticommuting observables, variance-based and entropic bounds coincide for Tsallis entropies at $q=2,3$, and the entropic uncertainty for multiple measurements achieves its minimum for maximally entangled or maximally mixed states, depending on the scenario (Niekamp et al., 2011, 2207.13469). The experimental landscape is mature: tight multipartite EURs have been observed in four-photon GHZ-type platforms, demonstrating directly measurable enhancement of security proofs and witnessing strong quantum correlations beyond simple additivity (Wang et al., 2022).

7. Outlook and Open Directions

Current research focuses on strengthening EURs via majorization, conditional relative entropies, and entropic error-disturbance frameworks. Extensions to field-theoretic and quantum-gravity contexts, as well as to resource theories of coherence and multipartite steering, are active areas (Floerchinger et al., 2020, 2206.13218). The operational role of tightness—as both a diagnostic of quantum correlation and an efficiency indicator in energy-conversion protocols—suggests broader applications spanning fundamental and applied quantum science (Song et al., 2024, Xu et al., 6 Jan 2025).

Open challenges include the characterization of optimal measurement weightings for biased EURs, formulation in highly singular or infinite-dimensional Hilbert spaces, and the design of device-independent and self-testing protocols based on EURs (Huang et al., 2023, Corichi et al., 24 Mar 2025). The unification of entropy-based, variance-based, and operational/game-theoretic uncertainty relations remains a central conceptual and technical thread permeating the field.