Quantum Unertainty Relations and Some Applications

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)

3
Uncertainty PrincipleUncertainty 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 relationScope for
improvementState dependence of r.h.s. ? higher
order correlations not captured by variance ?
Effects for mixed states ?Various tighter
relations, e.g., Robertson-Schrodinger
5
(No Transcript)
6
(No Transcript)
7
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
8
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
9
Examples2-qubits (Observables) Linear
entropy Qutrits Isotropic
statesObservables Linear entropy
10
Detecting mixedness of qubits qutrits

11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
Entropic uncertainty relations
15
Application of HUR EURDemonstration of EPR
ParadoxSteering
16
EPR ParadoxEinstein, 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.

17
EPR Paradox SteeringEinsteins later focus on
separability and locality versus
completenessConsider nonfactorizable state of
two systems If Alice measures in
she instantaneously projects Bobs 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 Bobs
system due to Alices measurement.However, the
ensemble of is different from the
ensemble of EPR nonlocality is an artefact
of the incompleteness of QM.Schrodinger
Steering Alices ability to affect Bobs state
through her choice of measurement basis.
18
SteeringSchrodinger, Proc. Camb. Phil. Soc.
31, 555 (1935)Alice can steer Bobs state into
either or depending upon her
choice of measurementIt is rather
discomforting that the theory should allow a
system to be steered into one or the other
type of state at the experimenters mercy in
spite of having no access to it.(Shrodinger
Steering not possible experimentally, hence QM
not correct for delocalized (entangled) systems)
19
EPR Paradox a testable formulation M. Reid,
Phys. Rev. A 40, 913 (1989) Application of
Uncertainty RelationAnalogous to position and
momentum, consider quadratures of two correlated
and spatially separated light fields.
Correlations

(with some error) Estimated
amplitudes
20
EPR paradox (Reid formulation.) Tara Agarwal,
PRA (1994)Average errors of inferences
chosen for highest
possible accuracyUncertainty principle
gt 1 EPR
paradox occurs if above inequality is violated
due to correlations.(c.f., experimental
violation with light modes, Ou et al. PRL (1992))
21
Steering a modern perspective Wiseman et al.,
PRL (2007)Steering as an information theoretic
task.Leads to a mathematical formulationSteeri
ng inequalities, in the manner of Bell
inequalities
22
Steering as a taskWiseman, Jones, Doherty, PRL
98, 140402 (2007) PRA (2007)(Asymmetric
task)Local Hidden State (LHS) Bobs 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)
23
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
Alices taks To convince Bob that the state is
entangled(If correlations between Bobs
measurement results and Alices 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 Bobs state.
24
Wiseman et al., Nature Physics (2010)
25
Steering inequalitiesMotivations

• 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

26
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)
27
Entropic steering inequality Walborn et al., PRL
(2011) (some definitions)Relative entropy
H(p(X)q(X)) Conditional entropy H(XY)
Now, using H(XY)

(can be negative for


entangled states) H(XY)
H(XY) - H(Y)
Shannon Entropy (or, von-Neuman, for quantum
case)
28
Entropic inequalityConsider relative entropy
between the probability distributionsPositivi
ty of relative entropy (variables are
and given )
29
Entropic steering inequalityUse non-steering
condition (2) It follows that
Hence,
30
Entropic steering inequality Averaging over
all Now, consider conjugate variable pairs
Similarly, Hence,

(5)
31
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 variablesHence,
using (5), ESR
32
Examples(by choosing variables s.t.
correlations between and
)(i) two-mode squeezed vaccum
stateESR is violated for TMSV
33

(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.
34
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)
35
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.

36
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)

37
Fine-grained uncertainty relationOppenheim 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
38
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 (a0) measurement outcome.
Winning probability
Output
39
FUR in bipartite case
Unbiased case
Winning condition
Winning probability
40
FUR for two-qubit CHSH game Connecting
uncertainty with nonlocalityClassific
ation of physical theory withrespect to maximum
winningprobability
41
Application of fine-grained uncertainty
relationFine-grained uncertainty relation and
nonlocality of tripartite systemsT. 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
42
FUR in Tripartite case
Winning conditions
43
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

44
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
45
Fine-grained uncertainty relationsand biased
nonlocal gamesA. 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.
46
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
47
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)
48
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)
49
Uncertainty in the presence of correlations
Berta et al., Nature Physics 6, 659
(2010)
50
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) Bobs task is to reduce
his uncertainty about Alices measurement
outcome The amount of entanglement
reduces Bobs uncertainty
Example Shared singlet state Alice
measures spin along, e.g., x- or z- direction.

Bob
perfectly successful no uncertainty.
51
Experimental reduction of uncertainty

52
Tighter lower bound of uncertaintyPati et al.,
Phys. Rev. A 86, 042105 (2012)Role of more
general quantum correlations, viz., discord in
memory Discord Mutual
informationClassical information
53
Optimal lower bound of entropic uncertainty using
FURT. 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
54
Examples TP, PC, ASM, PRL 110, 020402
(2013) Singlet state

(Uncertainty reduces to zero)
Werner state Fine-grained lower
limit Lower limit using EUR (Berta et
al.)
55
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
56
Application Security of key distribution
protocolsUncertainty principle bounds secret
key extraction per stateRate of key extraction
per stateEkert, PRL (1991) Devetak Winter,
PROLA (2005) Renes Boileau, PRL (2009) Berta
et al., Nat. Phys. (2010)
Rate of key extraction using fine-grainingTP,
PC, ASM, PRL (2013)FUR Optimal lower bound
on rate of key extraction
57
Explanation of optimal lower limit in terms of
physical resourcesT. Pramanik, S. Mal, ASM,
arXiv 1304.4506In 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
58
The memory gameBob 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. Bobs task is to infer
the outcome of Alices measurement by performing
some operation on his particle (memory). Q
What information can Bob extract about Alices
measurement outcome ? Classical
information contains
information about Alices outcome when she
measures alsong a particular direction that
maximizes In the absence of correlations,
Bobs uncertainty about Alices outcome is
When Bob measure the observable R, the reduced
uncertainty is where


59
Derivation of a new uncertainty relation (memory
game)TP, SM, ASM, arXiv 1304.4506When
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

60
Lower bounds using different uncertainty
relationsEntropic 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 EURTP, SM, ASM,
arXiv1304.4506(Extractable classical
information)
61
Quantum memory and
Uncertainty
LComparison of various lower bounds
62
Summary

• 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.
• Applications determination of purity/mixedness
  TP, SM , ASM, PRA (2013)
• Demonstration of EPR paradox and steering PC,
  ASM, GSA, PRA (2014)
• Reduction of uncertainty using quantum memory
  Berta et al, Nat. Phys. (2010) Pati et al., PRA
  (2012)
• 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