Entropic Uncertainty Relations

• Entropic uncertainty relations are quantum formulations that use entropy measures (e.g., Shannon, Rényi) to quantify the intrinsic unpredictability of non-commuting observables.
• They extend traditional variance-based uncertainty by providing state-independent, operational bounds useful in assessing quantum correlations and improving cryptographic security.
• These relations underpin applications in quantum key distribution, entanglement verification, and generalized measurement frameworks in both finite and infinite-dimensional systems.

Entropic uncertainty relations (EURs) formalize the fundamental quantum limitation that outcomes of non-commuting observables cannot all be predicted with certainty. Unlike variance-based uncertainty principles, EURs utilize information-theoretic entropy measures—most notably Shannon, Rényi, min- and max-entropies—to quantify intrinsic unpredictability of quantum measurement statistics, both in finite and infinite dimensional systems, and across scenarios involving quantum memory, multiple observers, and generalized probabilistic theories.

1. Foundations and Mathematical Formulation

The standard Heisenberg–Robertson uncertainty relations bound the product of standard deviations, but these relations can become trivial or inapplicable for certain states or observables. Entropic uncertainty relations overcome these deficiencies by using the entropy of measurement outcome distributions as a basis-independent and operationally meaningful quantifier of uncertainty (Bialynicki-Birula et al., 2010, Coles et al., 2015).

For a quantum system in state $\rho$ and two observables $A$ and $B$ with orthonormal eigenbases $\{|a_i\rangle\}$ and $\{|b_j\rangle\}$, the outcome probability distributions for measurements are $p_i = \langle a_i|\rho|a_i\rangle$ and $q_j = \langle b_j|\rho|b_j\rangle$. The (Shannon) entropy of measurement $X$ is $H(X) = -\sum_j p_j \log_2 p_j$. The core Maassen–Uffink relation then reads (Bialynicki-Birula et al., 2010, Niekamp et al., 2011, Coles et al., 2015):

$H(A) + H(B) \geq -2\log_2 c$

where

$c = \max_{i,j} |\langle a_i | b_j \rangle|\,.$

This lower bound is state-independent and depends only on the maximal overlap between the two measurement bases; for mutually unbiased bases (MUBs), $c = 1/\sqrt{d}$ and the bound becomes $H(A) + H(B) \geq \log_2 d$.

Rényi ($H_\alpha$) and min/max-entropy extensions allow further flexibility and can yield strictly stronger or operationally relevant uncertainty quantifiers (Bialynicki-Birula et al., 2010, Coles et al., 2015). For instance, min-entropy $H_{\min}(X) = -\log \max_x p_x$ governs optimal guessing probabilities, essential in cryptographic contexts.

2. Quantum-Memory-Assisted and Multipartite EURs

When a quantum system $A$ is entangled with a quantum memory $B$, the intrinsic uncertainty in measurement outcomes can be reduced. The landmark relation of Berta et al. [Berta et al., Nat. Phys. 6, 659 (2010)] generalizes EURs as:

$S(X|B) + S(Z|B) \geq -\log_2 c + S(A|B)$

Here, $S(X|B)$ is the conditional von Neumann entropy of the post-measurement state, and $S(A|B)$ quantifies entanglement ($S(A|B)<0$ iff $A$ and $B$ are entangled). This form implies that entanglement (negative $S(A|B)$) can dilute the bound and allow sharper predictions (Wang et al., 2019, Ding et al., 2019).

Recent works have further generalized these relations to multipartite settings. For a system $A$ with multiple distributed quantum memories $B_1,\dots, B_n$ and a collection of $m$ observables partitioned among the memories, Zhang & Fei derive complementary inequalities:

$$\sum_{t=1}^n \sum_{M_i \in S_t} S(M_i|B_t) \geq -\frac{1}{m-1} \log_2 \prod_{i<j} c_{ij} + \frac{1}{m-1}\sum_{t=1}^n \frac{m_t(m_t-1)}{2} S(A|B_t) + \max\{0,\delta_{m,n}\}$$

with $\delta_{m,n}$ explicitly incorporating mutual information terms and accessible outcome information (Holevo quantities), and $c_{ij}$ the pairwise complementarities (Zhang et al., 2023).

These multipartite EURs quantify how quantum correlations shared across multiple memories reduce measurement uncertainty and provide a unified framework that strictly improves on previous bounds, directly informing multipartite quantum cryptography, entanglement witnessing, and steering.

3. Multi-Measurement, State-Dependent, and Device-Independent EURs

Extensions to more than two measurements, as in the Liu–Mu–Fan framework, lead to relations of the form (Liu et al., 2014):

$\sum_{m=1}^N H(M_m) \geq -\log b + (N-1)S(\rho)$

Here, $b$ is a generalized overlap parameter depending on the structure of the $N$ measurement bases and their mutual relationships. This approach also accommodates additional features such as weighted combinations and conditional entropy terms when quantum memory is present, recovering classic two-measurement EURs as special cases.

Tighter state-dependent bounds may be realized by leveraging majorization techniques, convex or piecewise-linear approximations to entropy, or quadratic function bounds, which often outperform classic EURs for relevant regimes (Chen et al., 2018). For sets of binary observables, effective anti-commutators evaluated in the pre-measurement state furnish device-independent EURs, advantageous in cryptographic scenarios requiring certification without trusting device details (Kaniewski et al., 2014).

4. Operational and Cryptographic Applications

Entropic uncertainty relations have direct operational significance. In quantum key distribution (QKD), EURs quantitatively limit the eavesdropper's knowledge about the raw key and thereby establish strong security guarantees (Coles et al., 2015, Ding et al., 2019). The min-entropy versions provide tight lower bounds for privacy amplification, even in finite-key or adversarially biased measurement settings (Bouman et al., 2011, Krawec, 2023).

Multipartite and memory-assisted EURs underpin security proofs for advanced QKD protocols, password-based identification in bounded/noisy storage models, and randomness expansion. Sampling-based EURs and all-but-one min-entropy bounds have been developed to ensure positive key rates even in cases where traditional overlap-based EURs fail (e.g., for certain POVMs with $c=1$) (Bae, 2022, Bouman et al., 2011).

Furthermore, EURs are foundational in entanglement witnessing, EPR steering detection, quantum metrology (connecting phase or number variance to entropic uncertainty), and characterizations of quantum coherence.

5. Extensions Beyond Standard Quantum Theory

Entropic uncertainty relations are not exclusive to quantum mechanics but extend to generalized probabilistic theories (GPTs). For any GPT satisfying convexity, transitivity of pure states, and self-duality, analogous preparation and measurement EURs exist—parameterized by the geometry of the state and effect spaces, with the entropic structure persisting through non-Hilbertian scenarios (Takakura et al., 2020). This underlines the universality of entropic uncertainty as a constraint arising from fundamental non-classicality.

Device-independent EURs certified via effective anti-commutators or Bell inequalities further illustrate robustness against system/model assumptions, and sampling-based methods enable entropic security in cases where overlap-based relations become trivial or inapplicable (Kaniewski et al., 2014, Bae, 2022).

6. Experimental Verification and Advanced Topics

Entropic uncertainty relations have been explored and confirmed in various physical systems, notably in all-optical quantum platforms preparing Bell-like and Bell-diagonal states and measuring entropies across sets of MUBs. Experimental data match theoretical predictions, and the role of additional parameters (Holevo quantity, mutual information) in tightening the EURs is evident (Ding et al., 2019).

Contemporary developments include:

• Thermodynamic EURs: Newer results relate entropy production and information-theoretic entropy in stochastic thermodynamics, revealing informational costs of symmetry-breaking and trajectory observables (Hasegawa et al., 10 Feb 2025).
• Scrambling and Many-Body Physics: EURs generalized to protocols involving weak/strong measurements, with bounds set by out-of-time-order correlator (OTOC) quasiprobabilities, connect directly to quantum chaos and information scrambling (Halpern et al., 2018).
• Pointer-based simultaneous measurements: Apparatus noise and squeezing can be included explicitly in state-dependent EURs for continuous variables (Heese et al., 2013).

Open directions span sharpening bounds for finite resolutions, multivariate systems, generalized measurement frameworks (POVMs), and further quantification of uncertainty in hybrid time–energy, phase–number, or non-Hermitian observables.

7. Summary Table: Key Entropic Uncertainty Relations

Setting	Relation	Primary Reference
Two observables (Shannon)	$H(A) + H(B) \geq -2\log_2 c$	(Bialynicki-Birula et al., 2010)
Two observables (Quantum memory)	$S(X|B) + S(Z|B) \geq -\log_2 c + S(A|B)$	(Coles et al., 2015)
Multiple observables	$\sum_{m=1}^N H(M_m) \geq -\log b + (N-1)S(\rho)$	(Liu et al., 2014)
Multipartite quantum memories	See Section 2 above for full bound, involving overlaps, conditional entropies, and correlators	(Zhang et al., 2023)
Min-entropy (all-but-one)	$$H_\infty(X\,|\,J=j,\,J'\neq j,\,\Psi)\geq (\delta/2 - 2\epsilon)n - 1$$ (see details)	(Bouman et al., 2011)

Entropic uncertainty relations thus constitute a central quantitative tool in quantum foundations and information science, conveying deep operational consequences for the predictability of incompatible quantum observables, resource-theoretic trade-offs, and security in quantum technologies.