REVIEWS OF MODERN PHYSICS, VOLUME 89, JANUARY–MARCH 2017 Entropic uncertainty relations and their applications Patrick J. Coles* Institute for Quantum Computing and Department of Physics and Astronomy, University of Waterloo, N2L3G1 Waterloo, Ontario, Canada Mario Berta† Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, California 91125, USA Marco Tomamichel‡ School of Physics, The University of Sydney, Sydney, NSW 2006, Australia Stephanie Wehner§ QuTech, Delft University of Technology, 2628 CJ Delft, Netherlands (published 6 February 2017) Heisenberg’s uncertainty principle forms a fundamental element of quantum mechanics. Uncertainty relations in terms of entropies were initially proposed to deal with conceptual shortcomings in the original formulation of the uncertainty principle and, hence, play an important role in quantum foundations. More recently, entropic uncertainty relations have emerged as the central ingredient in the security analysis of almost all quantum cryptographic protocols, such as quantum key distribution and two-party quantum cryptography. This review surveys entropic uncertainty relations that capture Heisenberg’s idea that the results of incompatible measurements are impossible to predict, covering both finite- and infinite-dimensional measurements. These ideas are then extended to incorporate quantum correlations between the observed object and its environment, allowing for a variety of recent, more general formulations of the uncertainty principle. Finally, various applications are discussed, ranging from entanglement witnessing to wave-particle duality to quantum cryptography. DOI: 10.1103/RevModPhys.89.015002 CONTENTS 1. Shannon entropy 9 2. Rényi entropies 10 I. Introduction 2 3. Maassen-Uffink proof 10 A. Scope of this review 4 4. Tightness and extensions 10 II. Relation to Standard Deviation Approach 5 5. Tighter bounds for qubits 10 A. Position and momentum uncertainty relations 5 6. Tighter bounds in arbitrary dimension 10 B. Finite spectrum uncertainty relations 5 7. Tighter bounds for mixed states 11 C. Advantages of entropic formulation 6 D. Arbitrary measurements 11 1. Counterintuitive behavior of standard deviation 6 E. State-dependent measures of incompatibility 12 2. Intuitive entropic properties 6 F. Relation to guessing games 13 3. Framework for correlated quantum systems 7 G. Multiple measurements 14 4. Operational meaning and information applications 7 1. Bounds implied by two measurements 14 III. Uncertainty Without a Memory System 7 2. Complete sets of MUBs 14 A. Entropy measures 7 3. General sets of MUBs 15 1. Surprisal and Shannon entropy 7 4. Measurements in random bases 15 2. Rényi entropies 8 5. Product measurements on multiple qubits 16 3. Examples and properties 8 6. General sets of measurements 16 B. Preliminaries 9 7. Anticommuting measurements 17 1. Physical setup 9 8. Mutually unbiased measurements 17 2. Mutually unbiased bases 9 H. Fine-grained uncertainty relations 18 C. Measuring in two orthonormal bases 9 I. Majorization approach to entropic uncertainty 18 1. Majorization approach 18 2. From majorization to entropy 19 *[email protected] 3. Measurements in random bases 19 † [email protected] 4. Extensions 19 ‡ [email protected] IV. Uncertainty Given a Memory System 19 §[email protected] A. Classical versus quantum memory 20 0034-6861=2017=89(1)=015002(58) 015002-1 © 2017 American Physical Society Coles et al.: Entropic uncertainty relations and their … B. Background: Conditional entropies 20 D. Entanglement witnessing 43 1. Classical-quantum states 20 1. Shannon entropic witness 44 2. Classical-quantum entropies 20 2. Other entropic witnesses 44 3. Quantum entropies 21 3. Continuous variable witnesses 45 4. Properties of conditional entropy 22 E. Steering inequalities 45 C. Classical memory uncertainty relations 22 F. Wave-particle duality 45 D. Bipartite quantum memory uncertainty relations 23 G. Quantum metrology 46 1. Guessing game with quantum memory 23 H. Other applications in quantum information theory 46 2. Measuring in two orthonormal bases 23 1. Coherence 47 3. Arbitrary measurements 24 2. Discord 47 4. Multiple measurements 25 3. Locking of classical correlations 47 5. Complex projective two-designs 25 4. Quantum Shannon theory 48 6. Measurements in random bases 26 VII. Miscellaneous Topics 48 7. Product measurements on multiple qubits 27 A. Tsallis and other entropy functions 48 8. General sets of measurements 27 B. Certainty relations 49 E. Tripartite quantum memory uncertainty relations 27 C. Measurement uncertainty 49 1. Tripartite uncertainty relation 27 1. State-independent measurement-disturbance 2. Proof of quantum memory uncertainty relations 28 relations 50 3. Quantum memory tightens the bound 28 2. State-dependent measurement-disturbance relations 50 4. Tripartite guessing game 29 VIII. Perspectives 51 5. Extension to Rényi entropies 29 Acknowledgments 51 6. Arbitrary measurements 29 Appendix A: Mutually Unbiased Bases 51 F. Mutual information approach 30 1. Connection to Hadamard matrices 52 1. Information exclusion principle 30 2. Existence 52 2. Classical memory 30 3. Simple constructions 52 3. Stronger bounds 30 Appendix B: Proof of Maassen-Uffink’s Relation 52 4. Quantum memory 31 Appendix C: Rényi Entropies for Joint Quantum Systems 53 5. A conjecture 31 1. Definitions 53 G. Quantum channel formulation 31 2. Entropic properties 53 1. Bipartite formulation 31 a. Positivity and monotonicity 53 2. Static-dynamic isomorphism 32 b. Data-processing inequalities 54 3. Tripartite formulation 32 c. Duality and additivity 54 V. Position-momentum Uncertainty Relations 32 3. Axiomatic proof of uncertainty relation with quantum A. Entropy for infinite-dimensional systems 33 memory 54 1. Shannon entropy for discrete distributions 33 References 55 2. Shannon entropy for continuous distributions 33 B. Differential relations 34 C. Finite-spacing relations 34 I. INTRODUCTION D. Uncertainty given a memory system 34 1. Tripartite quantum memory uncertainty relations 35 Quantum mechanics has revolutionized our understanding 2. Bipartite quantum memory uncertainty relations 36 of the world. Relative to classical mechanics, the most 3. Mutual information approach 36 dramatic change in our understanding is that the quantum E. Extension to min- and max-entropies 36 world (our world) is inherently unpredictable. 1. Finite-spacing relations 36 By far the most famous statement of unpredictability is 2. Differential relations 37 Heisenberg’s uncertainty principle (Heisenberg, 1927), which F. Other infinite-dimensional measurements 37 we treat here as a statement about preparation uncertainty. VI. Applications 37 Roughly speaking, it states that it is impossible to prepare a A. Quantum randomness 37 quantum particle for which both position and momentum are 1. The operational significance of conditional sharply defined. Operationally, consider a source that con- min-entropy 38 sistently prepares copies of a quantum particle in the same 2. Certifying quantum randomness 38 way, as shown in Fig. 1. For each copy, suppose we randomly B. Quantum key distribution 39 measure either its position or its momentum (but we never 1. A simple protocol 39 attempt to measure both quantities for the same particle1). 2. Security criterion for QKD 39 We record the outcomes and sort them into two sequences 3. Proof of security via an entropic associated with the two different measurements. The uncer- uncertainty relation 39 tainty principle states that it is impossible to predict both 4. Finite size effects and min-entropy 40 the outcome of the position and the momentum measure- 5. Continuous variable QKD 41 C. Two-party cryptography 41 ments: at least one of the two sequences of outcomes will be 1. Weak string erasure 41 unpredictable. More precisely, the better such a preparation 2. Bounded-storage model 42 3. Noisy-storage model 43 1Section I.A notes other uncertainty principles that involve 4. Uncertainty in other protocols 43 consecutive or joint measurements. Rev. Mod. Phys., Vol. 89, No. 1, January–March 2017 015002-2 Coles et al.: Entropic uncertainty relations and their … quantum mechanics should be to find simple, conceptually insightful statements like these. If this problem was only of fundamental importance, it would be a well-motivated one. Yet in recent years there is new motivation to study the uncertainty principle. The rise of quantum information theory has led to new applications of quantum uncertainty, for example, in quantum cryptography. In particular quantum key distribution is already commercially marketed and its security crucially relies on Heisenberg’s FIG. 1. Physical scenario relevant to preparation uncertainty uncertainty principle. (We discuss various applications in relations. Each incoming particle is measured using either Sec. VI.) There is a clear need for uncertainty relations that measurement P or measurement Q, where the choice of the are directly applicable to these technologies. measurement is random. An uncertainty relation says we cannot In Eqs. (1) and (2), uncertainty has been quantified using predict the outcomes of both P and Q. If we can predict the the standard deviation of the measurement results. This is, outcome of P well, then we are necessarily uncertain about the however, not the only way to express the uncertainty principle. outcome of measurement Q, and vice versa. It is instructive to consider what preparation uncertainty means in the most general setting. Suppose