1. Fine-grained uncertainty and security of key generation Archan S. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata Collaborators: T. Pramanik ; M. Kaplan (Paris Tech.); P. Chowdhury (SNBNCBS) Phys. Rev. A 90, 050305 (R) (2014); Phys. Rev. A 92, 042317 (2015)

2. Plan: Course-grained (Variance based URs, e.g., Heisenberg; Robertson- Schrodinger; Maccone-Pati; Entropic ) Vs Fine-grained Uncertainty relations Applications: Nonlocality (Quantum Steering ) Key generation (lower limit of key extraction rate) [1s-DIQKD] Higher security for more uncertain CV systems

3. Uncertainty Relations Heisenberg uncertainty relation (HUR) : For any two non-commuting observables, the bounds on the uncertainty of the precision of measurement outcome is given by Robertson-Schrodinger uncertainty relation: For any two arbitrary observables, the bounds on the uncertainty of the precision of measurement outcome is given by Applications : ➢ Entanglement detection. (PRA 78, 052317 (2008).) ➢ Witness for mixedness. (PRA 87, 012105 (2013).) Drawbacks : (1) The lower bound is state dependent. (2) Captures correlations only up to 2 nd order (variances)

4. Sum of variances [Maccone & Pati, PRL 113, 260401 (2014)] Tightens the product, as well !

5. Entropic uncertainty relation (EUR) : Where denotes the Shannon entropy of the probability distribution of the measurement outcomes of the observable Applications : ➢ Used to detect steering. [ PRL 106, 130402 (2011); PRA 89 (2014).] ➢ Reduction of uncertainty using quantum memory [Nature Phys. 2010] Drawback : ➢ Unable to capture the non-local strength of quantum physics.

6. Coarse-grained uncertainty relation In both HUR and EUR we calculate the average uncertainty where average is taken over all measurement outcomes Fine-grained uncertainty relation ➢ In fine-grained uncertainty relation, the uncertainty of a particular measurement outcome or any any combination of outcomes is considered. ➢ Uncertainty for the measurement of i-th outcome is given by Advantage ➢ FUR is able to discriminate different no-signaling theories on the basis of the non-local strength permitted by the respective theory. J. Oppenheim and S. Wehner, Science 330, 1072 (2010)

7. Fine-grained uncertainty relation [Oppenheim and Weiner, Science 330, 1072 (2010)] (Entropic uncertainty relations provide a coarse way of measuring uncertainty: they do not distinguish the uncertainty inherent in obtaining any combination of outcomes for different measurements) Measure of uncertainty: If or, then the measurement is certain corresponds to uncertainty in the measurement FUR game: Alice & Bob receive binary questions and (projective spin measurements along two different directions at each side), with answers `a’ and `b’. Winning Probability: : set of measurement settings : measurement of observable A is some function determining the winning condition of the game

8. FUR in single qubit case To describe FUR in the single qubit case, let us consider the following game Input Output Winning condition Alice wins the game if she gets spin up (a=0) measurement outcome. Winning probability Measurement settings

9. FUR for two-qubit CHSH game Connecting uncertainty with nonlocality Classification of physical theory with respect to maximum winning probability

10. Application of fine-grained uncertainty relation Fine-grained uncertainty relation and nonlocality of tripartite systems: [T. Pramanik & ASM, Phys. Rev. A 85, 024103 (2012)] FUR determines nonlocality of tripartite systems as manifested by the Svetlichny inequality, discriminating between classical physics, quantum physics and superquantum (nosignalling) correlations. [Tripartite case: ambiguity in defining correlations; e.g., Mermin, Svetlichny types]

11. Nonlocality in biased games For both bipartite and tripartite cases the different no- signaling theories are discriminated when the players receive the questions without bias. Now the question is that if each player receives questions with some bias then what will be the winning probability for different no-signaling theories

12. Fine-grained uncertainty relations and biased nonlocal games: [A. Dey, T. Pramanik & ASM, Phys. Rev. A 87, 012120 (2013)] FUR discriminates between the degree of nonlocal correlations in classical, quantum and superquantum theories for a range [not all] of biasing parameters.

13. Uncertainty in the presence of correlations [Berta et al., Nature Physics 6, 659 (2010)]

14. Reduction of uncertainty: a memory game [Berta et al., Nature Physics 6, 659 (2010)] Bob prepares a bipartite state and sends one particle to Alice Alice performs a measurement and communicates to Bob her choice of the observable P or Q, but not the outcome By performing a measurement on his particle (memory) Bob’s task is to reduce his uncertainty about Alice’s measurement outcome The amount of entanglement reduces Bob’s uncertainty Example: Shared singlet state: Alice measures spin along, e.g., x- or z- direction. Bob perfectly successful; no uncertainty.

15. Optimal lower bound of entropic uncertainty using FUR [T. Pramanik, P. Chowdhury, ASM, Phys. Rev. Lett. 110, 020402 (2013)] Derivation: Consider EUR for two observables P and Q: Fix (without loss of generality) and minimize entropy w.r.t Q FUR:

16. Examples: …….[TP, PC, ASM, PRL 110, 020402 (2013)] State with maximally mixed marginals: Fine-grained lower bound: EUR lower bound (Berta et al.): Optimal lower limit achieveble in any real experiment not attained in practice

17. Application of FUR in Quantum Steering ASM, T. Pramanik, P. Chowdhury Phys. Rev. A 90, 050305 (R) (2014); Phys. Rev. A 92, 042317 (2015)

18. EPR Paradox & Steering [Schrodinger, Proc. Camb. Phil. Soc. 31, 555 (1935) Consider nonfactorizable state of two systems: If Alice measures in she instantaneously projects Bob’s system into one of the states and similarly, for the other basis. Since the two systems no longer interact, no real change can take place in Bob’s system due to Alice’s measurement. However, the ensemble of is different from the ensemble of EPR: nonlocality is an artefact of the incompleteness of QM. Schrodinger: Steering: Alice’s ability to affect Bob’s state through her choice of measurement basis.

19. Steering: a modern perspective [Wiseman et al., PRL (2007)] Steering as an information theoretic task. Leads to a mathematical formulation Steering inequalities, in the manner of Bell inequalities

20. Steering as a task [Wiseman, Jones, Doherty, PRL 98, 140402 (2007); PRA (2007)] (Asymmetric task) Local Hidden State (LHS): Bob’s system has a definite state, even if it is unknown to him Experimental demonstration: Using mixed entangled states [Saunders et al. Nature Phys. 6, 845 (2010)]

21. Steering task: (inherently asymmetric) Alice prepares a bipartite quantum state and sends one part to Bob (Repeated as many times) Alice and Bob measure their respective parts and communicate classically Alice’s taks: To convince Bob that the state is entangled (If correlations between Bob’s measurement results and Alice’s declared results can be explained by LHS model for Bob, he is not convinced. – Alice could have drawn a pure state at random from some ensemble and sent it to Bob, and then chosen her result based on her knowledge of this LHS). Conversely, if the correlations cannot be so explained, then the state must be entangled. Alice will be successful in her task of steering if she can create genuinely different ensembles for Bob by steering Bob’s state.

22. Steering inequalities: [various types] Demonstration of EPR paradox (Reid inequalities) based on correlations up to second order Similarly, Heisenberg uncertainty relation based on variances Several CV states do not violate Reid inequality, (e.g., 2d harmonic oscillator) Correlations may be hidden in higher order moments of observables (e.g., LG beams) Extension to higher orders: Entropic uncertainty relation

23. (iii) N00N states: (utility for precise interferometric measurements) [P. Chowdhury, T. Pramanik, ASM, G. S. Agarwal, Phys. Rev. A 89,012104 (2014)] ESR: is violated even though Reid inequality is not. However, ESR NOT violated for N > 2 ! (N00N states are Bell non-local)

24. Three distinct correlations: : positive probability distribution, : quantum states (a) Entangled states: (b) Steerable states: (c) Bell-nonlocal state: {C} strict subset of {b} strict subset of {a} (b) (c) (a)

25. Steering inequalities: Statements of Uncertainty relations: EPR Paradox (Reid Inequality): (HUR) Entropic Steering (ESR) : (EUR) Fine-grained steering: (FUR) for qubits

26. FUR for discrete variables (2-qubit case) T. Pramanik, M. Kaplan, ASM, PRA 90, 050305 (R) (2014) Consider steering scenario between Alice and Bob (local hidden state mode) algebraic: Similarly: FUR: (2-qubit case)

27. Fine-grained steering: examples T. Pramanik, M. Kaplan, ASM, PRA 90, 050305 (R) (2014) Pure entangled state: CHSH violation: Werner state: Steerable for ( for known observables p > 1/√2 Shown using two settings on each side (tight result) (improvement over earlier results requiring infinite settings)– fine graining performs optimization over all settings !

28. Uncertainty, nonlocality and security of QKD First QKD protocol [BB84]: security based on HUR : Non-commuting spin operators Entanglement based QKD [E91]: security based on demonstration of entanglement, viz., violation of Bell’s inequality Recent QKD protocols: Importance of Device Independence Security of DIQKD (neither Alice, nor Bob trusted) --- nonlocality Security of 1s-DIQKD (only Alice is trusted ) -- steering [asymmetric situation]

29. Secret key rate under individual attack AliceBob Eve Part of I(B:A) that is useful to share secret key between Alice and Bob Part of I(B:A) is available to the eavesdropper, Eve. I. Csiszar and J. Korner, IEEE Trans. Inf. Theory 24, 339 (1978).

30. One-sided device independent QKD A A B Bob’s assumption System B is quantum. System A is not trusted as quantum system. Key rate The above key rate is independent of the property of the system A, i.e., one-sided device independent QKD.

31. Monogamy relations and security in 1sDIQKD T. Pramanik, M. Kaplan, ASM, PRA 90, 050305 (R) (2014) 3-party scenario (Alice, Bob, Eve) Define similarly Monogamy relation: Violation of steering inequality by Alice and Bob: Key rate: For maximum violation, key rate 0.5 (significant improvement on results using other approaches)

32. Steering inequalities for Continuous variable systems: EPR Paradox (Reid Inequality): (HUR) Entropic Steering (ESR) : (EUR) Fine-grained steering: (FUR) for qubits Discrete variables: Continuous variables: ?? (CV FUR) ?

33. Fine-grained uncertainty relation for CV states T. Pramanik, P. Chowdhury, ASM, PRA 92, 042317 (2015) Measurement analogy between spin ½ projectors and parity operators (dichotomic measurement outcomes) Wigner function: expectation value of product of displaced parity operators (used in deriving Bell-CHSH inequalities for CV systems) [c.f., Banaszek & Wodkievic, PRL (1987); ASM, G S. Agarwal et al., PRA (2013)] Minimum uncertainty states in phase space are Coherent states. Bound in FUR obtained by using coherent states.

34. Bell’s inequality for CV systems, e.g., LG beams [ ASM, G S. Agarwal et al., PRA (2013)] Wigner function for LG beams: Correlations BI through Wigner fn for CV sytems, Banaszek & Wodkievich, PRA (1998) Wigner fn. : expectation value of product of displaced parity operators

35. Fine-grained uncertainty relation for continuous variables [PC, TP, ASM, PRA 2015] lower and upper bounds calculated using minimum uncertainty coherent states Fine-grained steering relation: FUR for discrete variables: [TP, M. Kaplan, ASM, PRA 90, 050305 R (2014)] Comparison of range of certainty: Discrete variables: Continuous variables: Lower !

36. Higher security of key generation for more uncertain CV systems Steerability of N00N states revealed for N > 2 using FUR Upper bound of FUR violated by Lower bound of key rate: e.g., for N00N states, r = 1 ( key rate ideal ) in 1sDIQKD (comparison: key rate ½ using discrete variable maximally entangled states) PC, TP, MK, ASM, Phys. Rev. A 90, 050305 (R) 2014; Phys. Rev. A 92, 042317 (2015)

37. Summary (Fine-grained URs & security of 1s-DIQKD) Phys. Rev. A 90, 050305 (R) (2014); Phys. Rev. A 92, 042317 (2015) Various forms of uncertainty relations: Heisenberg, Robertson-Schrodinger, Entropic, Fine-grained, etc… Physical content of uncertainty relations (state (in- )dependent bounds, correlations, relation with nonlocality, etc. Uncertainty relations used for deriving steering inequalities Fine-graining leads to optimal (or tight) bounds, c.f., Reduction of uncertainty using quantum memory [Berta et al, Nat. Phys. (2010); [TP, PC, ASM, PRL (2013)] Fine-grained steering relation for discrete variables: leads to maximal steerability for Werner states with two settings on each side. [TP, MK, ASM, PRA (R) (2014)] Application in the lower bound of key rate of 1sDIQKD protocol. FUR for continuous variable systems: New manifestation of higher range of uncertainty for CV systems. Higher security in-principle. [TP, PC, ASM, PRA (2015)]