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
……………
• 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)
Uncertainty Principle:
Uncertainty in observable A:
( A) 2  A2  A
2
 A2
A  A A
Consider two self-adjoint operators
A ,B
Now,
(A - iB )( A  iB )  0  A 2  2 B 2   C  0
iC  A ,B   A, B

Minimizing l.h.s w.r.t  one gets
Hence,
A2 B2 
1
C
4
2
or
C
2 B2
e.g.,

2
A2 
( A) 2 ( B ) 2  
For canonically conjugate pairs of observables,
( x )( p ) 
, thus
A, B  i
(Heisenberg Uncertainty Relation)
1
4
C
4B
2
2
0
A, B  2
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:
Application of RS uncertainty relation: detecting
purity and mixedness
S. 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
relation
Choose operators A and B such that
For pure states:
For mixed states:
Q0
Q0
Q  Sl (  ) [Linear Entropy]
Qubits:
Choose,
.
,
Generalized uncertainty as measure of mixedness (linear entropy)
For pure states:
For mixed states:
Q0
gives
Q0
Q  Sl (  )
Results extendable to n-qubits and single and bipartite qutrits
Examples:
2-qubits: (Observables)
Linear entropy:
Qutrits:
Isotropic states:
Observables:
Linear entropy:
Q  Sl (  )
Detecting mixedness of qubits & qutrits
Entropic uncertainty relations:
Application of HUR & EUR
Demonstration of EPR Paradox
&
Steering
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.
EPR Paradox & Steering
Einstein’s later focus on separability and locality versus completeness
Consider 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.
Steering:
[Schrodinger, Proc. Camb. Phil. Soc. 31, 555 (1935)
Alice can steer Bob’s state into either
upon her choice of measurement
or
depending
“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)
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:
EPR paradox (Reid formulation….) [Tara & Agarwal, PRA
(1994)]
Average errors of inferences:
chosen for highest possible accuracy
Uncertainty 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))
Steering: a modern perspective [Wiseman et al., PRL
(2007)]
Steering as an information theoretic task.
Leads to a mathematical formulation
Steering inequalities, in the manner of Bell
inequalities
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 him
Experimental demonstration: Using mixed entangled
states [Saunders et al. Nature Phys. 6, 845 (2010)]
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
Alice’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.
Wiseman et al., Nature Physics (2010)
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
Entropic steering inequality [Walborn et al., PRL (2011)]
Condition for non-steerability:
P(rA , rB )   P( ) P(rA |  ) PQ (rB |  )

(1)
Now, the conditional probability P(rB ,  | rA )  P( | rA )PQ (rB | )
(follows from (1) – LHS for Bob, and rule for conditional
probabilities:
P(a, b | c)  P(b | c) P(a | b) for {b}  {c}
Hence,
P(rB | rA )   P(rB ,  | rA )

(3)
[(2), (3) are non-steering conditions equivalent to (1)]
(2)
Entropic steering inequality [Walborn et al., PRL (2011)]
(some definitions):
Relative entropy: H(p(X)||q(X)) =
 px ln
x
Conditional entropy: H(X|Y) = 
px
qx
0
 p( x, y) ln p( x | y)
x, y
Now, using:
H(X|Y) = 
p ( x, y )
p( x | y) 
p( y )
 p( x, y) ln p( x, y)   p( y) ln p( y)
x, y
y
(can be negative for
entangled states)
H(X|Y) = H(XY) - H(Y)
Shannon Entropy (or, von-Neuman, for quantum case)
Entropic inequality
Consider relative entropy between the probability distributions:
P(rB ,  | rA )
P( | rA ) P(rB | rA )
Positivity of relative entropy:
P(rB ,  | rA )
drB P(rB ,  | rA ) ln
0


P( | rA ) P(rB | rA )

(variables are

and
rB
given
rA )
Entropic steering inequality
P(rB ,  | rA )  P( | rA )PQ (rB |  )
Use non-steering condition (2):
PQ (rB |  )
dr P(r ,  | r ) ln


P( r

B
B
A
B
It follows that:
| rA )
0
   drB P ( | rA ) PQ (rB |  ) lnPQ (rB |  )

   drB P (rB ,  | rA ) ln P (rB | rA )  0

Hence,
 P( | r )H
A
Q
(RB | )  H (RB | RA  rA )
Entropic steering inequality
… Averaging over all
rA
H ( RB | RA )   P( ) H Q ( RB |  )

Now, consider conjugate variable pairs:
Similarly,
S A , SB
H (S B | S A )   P( ) HQ (S B |  )

Hence,
H ( RB | RA )  H (S B | S A )   P( )[HQ ( RB |  ) HQ (S B |  )]

(5)
Entropic steering relations
Entropic uncertainty relation for conjugate variables R and S:
(Bialynicki-Birula & Mycielski, Commun. Math. Phys. (1975))
H ( RB )  H (S B )  ln  e
(6)
LHS model for Bob:
[(6) holds for each state marked by

]:
HQ (RB | )  HQ (SB | )  ln  e
Averaged over all hidden variables:
 P() H
Hence, using (5), ESR:
Q
( RB |  )  H Q (S B |  )  ln  e
H ( RB | RA )  H (S B | S A )  ln  e
Examples:
(by choosing variables s.t. correlations between
RA
and
RB
)
(i) two-mode squeezed vaccum state: R  X , S  P R  Y , S  P
B
B
X,
A
A
Y
H ( X | Y )  H ( PX | PY )  ln  e
ESR is violated for TMSV
XY  0
(ii) LG beams: (Entangled states of harmonic oscillator)
[P. Chowdhury, T. Pramanik,
ASM, G. S. Agarwal, Phys. Rev. A 89,
012104 (2014)
RB  X , SB  PX , RA  PY , S A  Y
 X PY   0
ESR:
H ( X | PY )  H ( PX | Y )  ln  e
is violated even though Reid inequality is not.
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.
➢ Witness for mixedness.
(PRA 78, 052317 (2008).)
(PRA 87, 012105 (2013).)
Drawbacks : (1) The lower bound is state dependent.
(2) Captures correlations only up to 2nd order (variances)
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.
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)
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 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.
Output
Winning probability
FUR in bipartite case
Unbiased case
Winning condition
Winning probability
FUR for two-qubit CHSH game
Connecting uncertainty with nonlocality
Classification of physical theory with
respect to maximum winning
probability
Application of fine-grained uncertainty relation
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.
[Tripartite case: ambiguity in defining correlations; e.g.,
Mermin, Svetlichny types]
FUR in Tripartite case
Winning conditions
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 :
Nonlocality in biased games
For both bipartite and tripartite cases the different nosignaling 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
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.
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
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)
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)
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 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, 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