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, RobertsonSchrodinger, 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
pi  1
or
pi
pi  0 , then the measurement is certain
0  pi  1 corresponds to uncertainty in the measurement
FUR game: Alice & Bob receive binary questions t A and t B (projective spin
measurements along two different directions at each side), with answers `a’ and `b’.
Winning Probability:
TA : set of measurement settings {t A}
V (a, b | t A , t B )
a : measurement of observable A
tA
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, 024103 (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, 012120 (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
S ( P | B)  S (Q | B)  log 2
1
 S ( A | B)
c
S ( A | B)  S (  AB )  S (  B )
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, 042105 (2012)]
Role of more general quantum correlations, viz., discord in memory
1
 S ( A | B)
c
 max{0, DA (  AB )  C AM (  AB )}
S ( P | B)  S (Q | B)  log2
Discord:
Mutual information:
Classical information:
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
FUR:
(without loss of generality) and minimize entropy w.r.t Q
Examples:
[TP, PC, ASM, PRL 110, 020402 (2013)]
Singlet state:
(Uncertainty reduces to zero)
Werner state:
Fine-grained lower limit:
1  p 
2H 

 2 
Lower limit using EUR (Berta et al.):
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
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: 1304.4506]
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.] :
S ( A | B)
max{0, DA (  AB )  CAM (  AB )}
However, FUR optimal lower bound is not always same, e.g., for
A: Requires derivation of a new uncertainty relation
MMM
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 CB (  AB ) contains information about Alice’s outcome
when she measures alsong a particular direction that maximizes CB (  AB )
M
In the absence of correlations, Bob’s uncertainty about Alice’s outcome is
When Bob measure the observable R, the reduced uncertainty is
where
S ( A )
Derivation of a new uncertainty relation (memory game):
[TP, SM, ASM, arXiv: 1304.4506]
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:1304.4506]
(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: 1304.4506]