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Quantum Unertainty Relations and Some Applications

Quantum Unertainty Relations and Some Applications. Archan S. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata. Plan:. Various forms of uncertainty relations: Heisenberg Robertson-Schrodinger Entropic Fine-grained …… Error-disturbance

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Quantum Unertainty Relations and Some Applications

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  1. Quantum Unertainty Relationsand Some Applications Archan S. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata

  2. Plan: • Various forms of uncertainty relations: Heisenberg Robertson-Schrodinger Entropic Fine-grained …… Error-disturbance …………… • Applications: Purity & mixedness EPR paradox and steering Nonlocality(bipartite, tripartite & biased games) Quantum memory (Information theoretic task: quantum memory as a tool for reducing uncertainty) Key generation (lower limit of key extraction rate)

  3. Uncertainty Principle:Uncertainty in observable A: Consider two self-adjoint operators Now,Minimizing l.h.s w.r.t one gets , thus Hence, or For canonically conjugate pairs of observables, e.g., (Heisenberg Uncertainty Relation)

  4. Heisenberg uncertainty relation:Scope for improvement:State dependence of r.h.s. ? higher order correlations not captured by variance ? Effects for mixed states ?Various tighter relations, e.g., Robertson-Schrodinger:

  5. Application of RS uncertainty relation: detecting purity and mixednessS. Mal, T. Pramanik, A. S. Majumdar, Phys. Rev. A 87, 012105 (2013)Problem: Set of all pure states not convex.Approach: Consider generalized Robertson-Schrodinger uncertainty relationChoose operators A and B such thatFor pure states: For mixed states:[Linear Entropy]

  6. For pure states: Qubits: Choose, . ,Generalized uncertainty as measure of mixedness (linear entropy) For mixed states: gives Results extendable to n-qubits and single and bipartite qutrits

  7. Examples:2-qubits: (Observables) Linear entropy: Qutrits: Isotropic states:Observables: Linear entropy:

  8. Detecting mixedness of qubits & qutrits

  9. Entropic uncertainty relations:

  10. Application of HUR & EURDemonstration of EPR Paradox&Steering

  11. EPR Paradox[Einstein, Podolsky, Rosen, PRA 47, 777 (1935)] Assumptions: (i) Spatial separability & locality: no action at a distance (ii) reality:“if without in any way disturbing the system, we can predict with certainty the value of a physical quantity, then there exists an element of physical reality corresponding to this quantity.” EPR considered two spatially separated particles with maximum correlations in their positions and momenta Measurement of position of 1 implies with certainty the position of 2 (definite predetermined value of position of 2 without disturbing it) Similarly, measurement of momentum of 1 implies momentum of 2 (again, definite predetermined value of momentum of 2 without disturbing it) Hence, particle 2 in a state of definite position and momentum. Since no state in QM has this property, EPR conclude that QM gives an incomplete description of the state of a particle.

  12. EPR Paradox & SteeringEinstein’s later focus on separability and locality versus completenessConsider 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.

  13. Steering:[Schrodinger, Proc. Camb. Phil. Soc. 31, 555 (1935)Alice can steer Bob’s state into either or depending upon her choice of measurement“It is rather discomforting that the theory should allow a system to be steered …… into one or the other type of state at the experimenter’s mercy in spite of having no access to it.”(Shrodinger: Steering not possible experimentally, hence QM not correct for delocalized (entangled) systems)

  14. EPR Paradox: a testable formulation [M. Reid, Phys. Rev. A 40, 913 (1989)][Application of Uncertainty Relation]Analogous to position and momentum, consider quadratures of two correlated and spatially separated light fields. Correlations : (with some error)Estimated amplitudes:

  15. EPR paradox (Reid formulation….) [Tara & Agarwal, PRA (1994)]Average errors of inferences: chosen for highest possible accuracyUncertainty principle: > 1 EPR paradox occurs if above inequality is violated due to correlations.(c.f., experimental violation with light modes, Ou et al. PRL (1992))

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

  17. 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 himExperimental demonstration: Using mixed entangled states [Saunders et al. Nature Phys. 6, 845 (2010)]

  18. 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 classicallyAlice’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.

  19. Wiseman et al., Nature Physics (2010)

  20. Steering inequalities:[Motivations] Demonstration of EPR paradox (Reid inequalities) based on correlations up to second order Several CV states do not violate Reid inequality Correlations may be hidden in higher order moments of observables Similarly, Heisenberg uncertainty relation based on variances Extension to higher orders: Entropic uncertainty relation

  21. Entropic steering inequality[Walborn et al., PRL (2011)]Condition for non-steerability: (1)Now, the conditional probability (2) (follows from (1) – LHS for Bob, and rule for conditional probabilities: for Hence, (3)[(2), (3) are non-steering conditions equivalent to (1)]

  22. Entropic steering inequality [Walborn et al., PRL (2011)](some definitions):Relative entropy: H(p(X)||q(X)) = Conditional entropy:H(X|Y) = Now, using: H(X|Y) = (can be negative for entangled states) H(X|Y) = H(XY) - H(Y)Shannon Entropy (or, von-Neuman, for quantum case)

  23. Entropic inequalityConsider relative entropy between the probability distributions:Positivity of relative entropy: (variables are and given )

  24. Entropic steering inequalityUse non-steering condition (2): It follows that: Hence,

  25. Entropic steering inequality… Averaging over all Now, consider conjugate variable pairs: Similarly, Hence, (5)

  26. Entropic steering relationsEntropic uncertainty relation for conjugate variables R and S: (Bialynicki-Birula & Mycielski, Commun. Math. Phys. (1975))(6)LHS model for Bob: [(6) holds for each state marked by ]: Averaged over all hidden variables:Hence, using (5), ESR:

  27. Examples:(by choosing variables s.t. correlations between and )(i) two-mode squeezed vaccum state:ESR is violated for TMSV

  28. (ii) LG beams: (Entangled states of harmonic oscillator) [P. Chowdhury, T. Pramanik, ASM, G. S. Agarwal, Phys. Rev. A 89, 012104 (2014) ESR: is violated even though Reid inequality is not.

  29. 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 2nd order (variances)

  30. 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.

  31. 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)

  32. 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 measurementFUR 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

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

  34. FUR in bipartite case Unbiased case Winning condition Winning probability

  35. FUR for two-qubit CHSH gameConnecting uncertainty with nonlocalityClassification of physical theory withrespect to maximum winningprobability

  36. Application of fine-grained uncertainty relationFine-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]

  37. FUR in Tripartite case Winning conditions

  38. FUR in tripartite case Winning probability is the probability corresponding winning condition Svetlichny-box : • Maximum winning probability • Classical theory : shared randomness : • Quantum theory : quantum state : • Super quantum correlation :

  39. 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

  40. Fine-grained uncertainty relationsand 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.

  41. FUR in biased bipartite case Consider the case where • Maximum winning probability • Classical theory : • Quantum theory : • i. For , • ii. For , • Super quantum correlation : Note that the result for the case where is same as above

  42. FUR in Biased Tripartite case Winning condition To get the winning probability, we consider a trick called bi-partition model where Alice and Bob play a bipartite game with probability r and another unitarily equivalent game with probability (1-r). At the end they calculate the average winning probability where average is taken over probability r. PRL 106, 020405 (2011). A. Dey, T. Pramanik, and A. S. Majumdar, PRA 87, 012120 (2013)

  43. FUR in Biased Tripartite case Consider the case where • Maximum average winning probability of the game • Classical theory : • Quantum theory : • i. For , • ii. For , • Super quantum correlation : Note that the result for the case where is same as above A. Dey, T. Pramanik, and A. S. Majumdar, PRA 87, 012120 (2013)

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

  45. 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 outcomeBy performing a measurement on his particle (memory) Bob’s task is to reduce his uncertainty about Alice’s measurement outcomeThe 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.

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