Entropic Uncertainty Relations in Quantum Systems

Entropic uncertainty relations recast the familiar Heisenberg principle into an information-theoretic language, quantifying the unpredictability of measurement outcomes by means of entropy measures. Rather than relying solely on variances, these relations employ Shannon, Rényi or other entropic functions to establish lower bounds on the sum of uncertainties associated with incompatible observables. The formalism naturally incorporates scenarios in which a quantum system is correlated with an ancillary memory, leading to tighter bounds when entanglement is present. Extensions further account for multiple measurements, continuous variables, and open‐system dynamics. Such generalisations underpin modern advances in quantum cryptography, precision metrology and fundamental tests of quantum mechanics by clarifying how correlations, system–environment interactions and measurement disturbance determine the ultimate limits of knowledge.

Research from Nature Portfolio

Recent advances have delivered tighter entropic bounds for sequences of incompatible measurements in the presence of a quantum memory, revealing how multipartite correlations can mitigate uncertainty even for three or more observables. Another strand has explored the time-dependent behaviour of entropic uncertainty under structured bosonic reservoirs, contrasting Markovian decay with non-Markovian revivals and demonstrating strategies—such as weak‐measurement reversal—to suppress total uncertainty in open systems. Parallel efforts have reformulated trade-off relations into state-independent inequalities, providing geometrical insights into variance and entropy bounds and overcoming the “triviality’’ problem of vanishing lower limits for certain pure states. Together, these contributions establish a unified and experimentally accessible framework for entropic uncertainty in both closed and dissipative quantum setups.

Entropic Uncertainty Relations in Quantum Systems publication trend

The graph below shows the total number of articles in entropic uncertainty relations in quantum systems across all publications each year (not limited to Nature Index journals).

Technical terms

Entropic uncertainty relation: An inequality that bounds the sum of entropies of measurement outcome distributions for non-commuting observables, replacing variance-based limits with information-theoretic measures.

Quantum memory: An ancillary system entangled with the measured system that can reduce uncertainty bounds by providing side information about measurement outcomes.

Majorization: A partial ordering of probability distributions used to compare the spread of uncertainties and establish universal bounds beyond pairwise entropic sums.

Markovian/non-Markovian processes: Classifications of open-system dynamics in which the system’s evolution is either memoryless (Markovian) or exhibits information backflow from the environment (non-Markovian).

Relative entropy: A measure of distinguishability between two quantum states, defined as the expectation of the logarithmic ratio of their state probabilities, often used to derive entropic bounds.

References

1. Strong majorization uncertainty relations and experimental verifications. npj Quantum Information (2023).
2. Entropic distinguishability of quantum fields in phase space. Quantum (2024).
3. Entropic Uncertainty Relation and Information Exclusion Relation for multiple measurements in the presence of quantum memory. Scientific Reports (2015).
4. Entropic uncertainty relations for Markovian and non-Markovian processes under a structured bosonic reservoir. Scientific Reports (2017).
5. Entropic uncertainty relation in neutrino oscillations. European Physical Journal C (2020).
6. Entropic uncertainty and measurement reversibility. New Journal of Physics (2016).
7. Tight State-Independent Uncertainty Relations for Qubits. Mathematics (2016).