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

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)

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:

Entropic uncertainty relations:

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

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

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.

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

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.

Experimental reduction of uncertainty

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:

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:

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.):

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

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:

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

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

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:

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)

Quantum memory and Uncertainty L Comparison of various lower bounds

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: ]