Presentation is loading. Please wait.

Presentation is loading. Please wait.

Fixing the lower limit of uncertainty in the presence of quantum memory Archan S. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata Collaborators:

Similar presentations


Presentation on theme: "Fixing the lower limit of uncertainty in the presence of quantum memory Archan S. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata Collaborators:"— Presentation transcript:

1 Fixing the lower limit of uncertainty in the presence of quantum memory Archan S. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata Collaborators: Tanumoy Pramanik, Priyanka Chowdhury, Siladitya Mal

2 Plan: Various forms of uncertainty relations (Heisenberg, Robertson- Schrodinger, Entropic, Error-disturbance…) Quantum memory (Information theoretic task: quantum memory as a tool for reducing uncertainty) Fine-graining & Optimal lower limit (Connection with winning probability of a memory game) Examples (pure & mixed entangled states: Werner, Bell-diagonal, etc..) Applications: Key generation (lower limit of key extraction rate) Classical information (Physical resource for reducing uncertainty in terms of a new uncertainty relation)

3 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:

4 Entropic uncertainty relations:

5 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

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

7 Fine-grained uncertainty relation and nonlocality of tripartite systems: [T. Pramanik & ASM, Phys. Rev. A 85, (2012)] FUR determines nonlocality of tripartite systems as manifested by the Svetlichny inequality, discriminating between classical physics, quantum physics and superquantum (nosignalling) correlations. Fine-grained uncertainty relations and biased nonlocal games: [A. Dey, T. Pramanik & ASM, Phys. Rev. A 87, (2013)] FUR discriminates between the degree of nonlocal correlations in classical, quantum and superquantum theories for a range [not all] of biasing parameters.

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

9 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.

10 Experimental reduction of uncertainty

11 Tighter lower bound of uncertainty: [Pati et al., Phys. Rev. A 86, (2012)] Role of more general quantum correlations, viz., discord in memory Discord: Mutual information: Classical information:

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

13 Examples: [TP, PC, ASM, PRL 110, (2013)] Singlet state: (Uncertainty reduces to zero) Werner state: Fine-grained lower limit: Lower limit using EUR (Berta et al.):

14 Examples: …….[TP, PC, ASM, PRL 110, (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

15 Application: Security of key distribution protocols: Uncertainty principle bounds bounds secret key extraction per state Rate of key extraction per state: [Ekert, PRL (1991); Devetak & Winter, PROLA (2005); Renes & Boileau, PRL (2009); Berta et al., Nat. Phys. (2010)] Rate of key extraction using fine-graining: [TP, PC, ASM, PRL (2013)] FUR: Optimal lower bound on rate of key extraction:

16 Explanation of optimal lower limit in terms of physical resources: [T. Pramanik, S. Mal, ASM, arXiv: ] In any operational situation, fine-graining provides the bound to which uncertainty may be reduced maximally. Q: What are the physical resources that are responsible for this bound ? not just entanglement ---- Is it discord ? [c.f., Pati et al.] : However, FUR optimal lower bound is not always same, e.g., for A: Requires derivation of a new uncertainty relation

17 The memory game: Bob prepares a bipartite state and sends one particle to Alice. Alice performs a measurement on one of two observables R and S, and communicates her choice [not the outcome] to Bob. Bob’s task is to infer the outcome of Alice’s measurement by performing some operation on his particle (memory). Q: What information can Bob extract about Alice’s measurement outcome ? Classical information contains information about Alice’s outcome when she measures alsong a particular direction that maximizes In the absence of correlations, Bob’s uncertainty about Alice’s outcome is When Bob measure the observable R, the reduced uncertainty is where

18 Derivation of a new uncertainty relation (memory game): [TP, SM, ASM, arXiv: ] When Alice and Bob measure the same observable R, the reduced uncertainty given by the conditional entropy becomes Extractable classical information: Similarly, for S: Apply to EUR: New uncertainty relation:

19 Lower bounds using different uncertainty relations: Entropic uncertainty relation [Berta et al., Nat. Phys. (2010)] (Entanglement as memory) Modified EUR [Pati et al., PRA (2012)] (Role of Discord) Modified EUR through fine-graining [TP, PC, ASM, PRL (2013)] Modified EUR [TP, SM, ASM, arXiv: ] (Extractable classical information)

20 Quantum memory and Uncertainty L Comparison of various lower bounds

21 Summary Various forms of uncertainty relations: Heisenberg, Robertson-Schrodinger, Entropic, Error-disturbance, etc… Reduction of uncertainty using quantum memory [Berta et al, Nat. Phys. (2010); Pati et al., PRA (2012)] Fine-grained uncertainty relation: linking uncertainty with nonlocality; bipartite, tripartite systems, biased games [Oppenheim & Wehner, Science (2010); TP & ASM, PRA (2012); AD, TP, ASM, PRA (2013)] Fine-graining leads to optimal lower bound of uncertainty in the presence of quantum memory [TP, PC, ASM, PRL (2013)] Application in privacy of quantum key distribution Maximum possible reduction of uncertainty is given by extractable classical information [TP, SM, ASM, arXiv: ]


Download ppt "Fixing the lower limit of uncertainty in the presence of quantum memory Archan S. Majumdar S. N. Bose National Centre for Basic Sciences, Kolkata Collaborators:"

Similar presentations


Ads by Google