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The Uncertainty Principle
in the Presence of
Quantum Memory
Matthias Christandl, ETH Zurich
based on joint work with
M. Berta, R. Colbeck, J. Renes and R. Renner
arXiv:0909.0950
Nature Physics 6, 659 - 662 (2010)
The Uncertainty Principle
• „In quantum mechanics, the Heisenberg
uncertainty principle states that certain
pairs of physical properties, like position
and momentum, cannot both be known to
arbitrary precision.“
(Wikipedia)
The Uncertainty Principle
• „In quantum mechanics, the Heisenberg
uncertainty principle states that certain
pairs of physical properties, like position
and momentum, cannot both be known to
predicted with arbitrary precision.“
(Wikipedia)
1
any
she chooses,
there is
a measure
quantified by Heisenberg
using the standard
deviation.
rt
of measurement
fferFor
two
observables,
R
and
S,
the
resulting
bound
on
the
1 as
memory
which gives
thestandard
same deviation.
outcome
quantified
by Heisenberg
using the
ton
ofBob’s
o the
uncertainty can be expressed in terms of the commutafferFor two
observables,
R andobservables,
S, the resulting bound
on the
obtains.
Hence,
for
any
R
and
S, th
ciple
tor [5]:
the
uncertainty can be expressed in terms of the commutarcertainties
inH(S)1 vanish, in clear violati
iple
tor Robertson
[5]: H(R) and
form
h be
(1)
∆R generally,
· ∆S ≥ |�[R, S]�|.
r,Equation
in(2).
More
the
violation
depen
21
the
h
be
S]�|. the system
(1) an
∆R · ∆S ≥ |�[R,
servthe
amount
of
entanglement
between
Variance
of
Inspired
by information theory,
uncertainty has since
2
the
ured
measurement result
beenmemory.
quantified using the Shannon Entropy [6]. The
quantum
erved in
Inspired
by information
theory,
uncertainty
has
since
first
uncertainty
relations
of
this
type
were
by
Bia�
l
ynickiured
mutabeen
quantified
using
the
Shannon
Entropy
[6]. The
Birula
and
Mycielski
[7]
and
Deutsch
[8].
Later,
Maassen
dthis
in
Entropic
form
first
uncertainty
relationsDeutsch’s
of this type
were
by
Bia�that
lynickiand
Uffink
[9]
improved
result
to
show
utaeigenbasis of S
the
Birula and Mycielski [7] and Deutsch [8]. Later, Maassen
eigenbasis of R
this
per-Shannonand
entropy
of
1 to show that
Uffink [9] improved Deutsch’s result
1
the
,
(2)
H(R)
+
H(S) ≥its
log2standard
measurement result
HowFor
observable
R,
we
denote
deviation
∆
c
p
perrver �R2 � − �R�2 , where �R� denotes the1 expectation value o
, of the proba(2)
H(R)
+
H(S)
≥
log
Bialynicki-Birula,
Mycielski,
Deutsch,
2
Howwhere
H(R)
denotes
the
Shannon
entropy
unre2
2 For
c max
Kraus,
Maassen and Uffink
non-degenerate
observables,
c
:=
|�ψ
|φ
�|
j
j,k
rver
bility distribution of the outcomes when R is measured. k
logi1are
�
and
|φ
�
the
eigenvectors
of
R
and
S,
respectivel
|ψ
j
k
where
H(R)
denotes
the
Shannon
entropy
of
the
probanrequantifies
complementarity
of
the
observThe
term
n the
c
bility 2distribution of the outcomes when R is measured.
ogi-
The Uncertainty Principle
•
•
1
nbergbymeasured
tified
Heisenberg using
the
standard
deviation.
been quantified using the Shannon Entropy [6]
n be
boundedRinand S, the resulting bound on the
wo
observables,
first uncertainty relations of this type were by Bia
of
the can
commutatainty
be expressed
in terms
of the commutaBirula
and Mycielski
[7] and Deutsch [8]. Later, M
plication of this
]:
and Uffink [9] improved Deutsch’s result to show
nnot predict the
easurements per1
1
(1) ≥ log2 ,
· ∆S ≥ |�[R, S]�|.
H(R) + H(S)
precision. ∆R
How2
c
d if the observer
red
by information
theory,
uncertainty
has
whereonH(R)
theissince
Shannon
entropy of the
emory,
an
unre-depends
Variance
valuedenotes
Entropy
good measure
quantified
using
Shannon
Entropyof[6].
The
bility
distribution
the
outcomes
when R is me
cent
technologiof the
observable
of
ignorance
1
uncertainty
relations
of
this
type
by Bia�the
lynickiquantifies
complementarity of the
The term cwere
e strengthen the
2
at and
Mycielski
Deutsch
[8]. Later,
Maassen
applies
even[7]
if andables,
similarly
to
the
commutator
in
Equation
(
State independent lower
Uffink
improved
resulttotothink
showabout
that uncertainty relations
BoundDeutsch’s
can be
zero
mory.[9]QuantifyOne
way
bound
ropy, we provide
the following game (the uncertainty game) betwe
1 and Bob. Before the game comm
nty of the
outplayers,
Alice
(2)
H(R) + H(S) ≥ log2 ,
Based
on measurements
advanced
c
hich depends
on
Alice
and
Bob
agree
on
two
R and
Elementary to prove
ween the system
game proceeds astheorem
follows: from
Bob analysis
creates a quantum
eart
H(R)
denotes
the
Shannon
entropy
of
the
probafrom its sigof his choosing and sends it to Alice. Alice then pe
distribution
of
the
outcomes
when
R
is
measured.
Goal
of
this
talk:
uantum
physics,
one of the two measurements and announces her ch
1
quantifies
the
complementarity
of
the
observerm
c
generalisation
of
the
entropic
form
ted
the
first
promeasurement
to
Bob.
Bob’s
task
is
then
to
minim
2
similarly to the commutator in Equation (1).
The Uncertainty Principle
•
I.
FORMULAE
R
=
σ
Examplez
RR==σσz|ψz0� = |0�
• Measurement of spin 1/2 particle
2
|ψ1 � = |1�
S
=
σ
S=σ x
R = σz
S = σxx
1
√
|φ0 � =
(|0� + |1�)
2
1
|φ1 � = √ (|0� − |1�)
2
|ψ0 � = |0�
|ψ1 � = |1�
S = σx
H(R)
H(S)
≥
1
1+H(S)
1
H(R)
+
≥
1
√ (|0�
|φ0 � = √ (|0�
|1�) |φ1 � =≥
H(R)
++ H(S)
1 − |1�)
1
2
2
√
|φ0 � =
(|0� + |1�)
2
ForII. mutually
bases UNCERTAINTY
(e.g. Fourier)
OF THE ENTROPIC
RELATION
1STATEMENTunbiased
|φ1 � = √ (|0� − |1�)
H(R)
H(S)≥≥log
logdd
H(R)
H(R) + 2H(S)
≥ 1++H(S)
•
H(R)
+
H(S)
≥
log
d
The statement of our uncertainty relation involves two measurements described by orth
**insert introduction**
mal bases {|ψj �} and {|φk �} in a d-dimensional Hilbert space (note that the observables ar
uter Science, ETH Zurich, 8092 Zurich, Switzerland.
nische Universität Darmstadt, 64289 Darmstadt, Germany.
Dated: October 31, 2009)
The Uncertainty Game
Alice‘s
quantified by Heisenberg using the standard deviation.1
result?
For two observables, R and S, the resulting bound
on the
2
heart of
c differk to the
uncertainty can be expressed in terms of the commutaprinciple
tor [5]:
, for in1
both be
(1)
∆R · ∆S ≥ |�[R, S]�|.
2
sely, the
observInspired by information theory, uncertainty has since
easured
been quantified using the Shannon Entropy
[6]. The
Minimise
nded in
first uncertainty relations of this type were by Bia�lynickimmuta-FIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general,uncertainty
be entangled
with his
Birula
and
Mycielski
[7]
and
Deutsch
[8].
Later,
Maassen
memory. (2) Alice measures either R or S and notes her outcome. (3) Alice announces her measurement choice to
of thisquantum
Bob. Bob’s goal is then to minimize his uncertainty about Alice’s measurement outcome.
her
and Uffink [9] improved Deutsch’s result to about
show that
dict the
result
We now proceed to state our uncertainty relation. It
given the quantum memory is always
greater than
nts per-holds in the presence of quantum memory and provides a
0. As discussed above,
1 Bob can guess both R and
on the uncertainties of the measurement
outcomes
S perfectly
with such a, strategy.
(2)
H(R)
+
H(S)
≥
log
Limit:
2
n. How-bound
which depends on the amount of entanglement between
c
2. If A and B are not
� entangled, i.e. the state takes
the system, A, and the quantum memory, B. Mathematthe form ρ
=
p ρ ⊗ ρ , for probabilities p
observerically, it is the following relation:
3
AB
j
j
j A
j
B
j
� = |0�
The|ψ Uncertainty
Game
0
|ψ1 � = |1�
Alice‘s
result?
2
max. entangled
1
|φ0 � = √ (|0� + |1�)
2
1
|φ1 � = √ (|0� − |1�)
2
Ui ⊗ Ūi (
�
|i�|i�) =
�
|i�|i�
guesses
correctly
FIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general, be entangled with his
conditional
entropy
quantum memory. (2) Aliceimeasures either R or S iand notes her outcome. (3) Alice announces her measurement choice to
Bob. Bob’s
goal is then to minimize his uncertainty about Alice’s measurement outcome.
Alice-Bob
system
after her measurement
We now
given the quantum memory is always greater than
with
Rproceed to state our uncertainty relation. It
holds in the presence of quantum memory and provides a
bound on the uncertainties of the measurement outcomes
which depends on the amount of entanglement between
the system, A, and the quantum memory, B. Mathematically, it is the following relation:3
H(R|B) + H(S|B) = 0
0. As discussed above, Bob can guess both R and
S perfectly with such a strategy.
2. If A and B are not
i.e. the state takes
� entangled,
j
j
the form ρAB = j pj ρA ⊗ ρB , for probabilities pj
The Uncertainty Game
2
partially entangled
FIG. 1: Illustration of the uncertainty game. (1) Bob sends a sta
quantum memory. (2) Alice measures either R or S and notes her
Bob. Bob’s goal is then to minimize his uncertainty about Alice’s m
We now proceed to state our uncertainty relation. It
holds in the presence of quantum memory and provides a
bound on the uncertainties of the measurement outcomes
which depends on the amount of entanglement between
conditional
entropy
the
system,
A,
and
the
quantum
memory,
B.
MathematFIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general, be entangled with his
Alice-Bob
system
3 Alice announces her measurement
quantum memory.
(2)
Alice
measures
either
R
or
S
and
notes
her
outcome.
(3)
choice
to
ically,
it
is
the
following
relation:
Bob. Bob’s goal is then to minimize his uncertainty about Alice’s measurement outcome.
before her measurement
New Limit:
1
+ H(A|B).
H(R|B) + H(S|B) ≥ log
c
We now proceed to state our uncertainty relation. It
holds in the presence of quantum memory and provides a
bound on the uncertainties of the measurement outcomes
which depends on the amount of entanglement between
the system, A, and the quantum memory, B. Mathematically, it is the following relation:3
given the quantum memory is always greater than
0. As discussed
above, Bob can guess both R and
2
S perfectly with such a strategy.
(3)
2. If A and B are not
i.e. the state takes
� entangled,
j
j
the form ρAB = j pj ρA ⊗ ρB , for probabilities pj
are not
entangled, i.e. the
1 2. If A and B �
both
be memory,
quantum
memory,
B. Mathemat�
m,,e cannot
A,
and the
quantum
B. Mathematj(1) j j j
|�[R,
S]�|.
∆Ramount
· ∆S ≥
The
Uncertainty
Principle
in
the
P
the
form
ρ
=
p
ρ
⊗
ρ
,
for
pro
which
depends
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of
entanglement
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,
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proba
3
3
AB
j
AB
j
2
A
B
j
ore
precisely,
the
A
B
j
Bob’s uncertainty about
the
mea
therelation:
following relation:
ng
2.
j
j
j
j
the system,
A, and the quantum
memory,
B. has
Mathematf any two observand
quantum
states
ρvon
and
ρthen
, then
H
andBerta,
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states
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and
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,
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1the
1
by
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Neumann
e
A
B
Inspired by informationMario
theory,
uncertainty
since
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B
Matthias
Christandl,
Roger
Colbeck
3 Since H(R|B) ≤ H(R) and H(S|B)
1is the following relation:
enberg measured
ically,
it
1
Since
H(R|B)
≤
H(R)
and
H(S|B)
≤
1 directly
been
quantified
using
the
Shannon
Entropy
[6].
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generalizes
the
Shannon
+
H(A|B).
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+
H(S|B)
≥
log
Faculty
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Physics,
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(S|B)
≥ log2 +
an
be bounded
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all states,
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recover
Maassen
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c
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first
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ynickiall
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Maassen
and
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Institute
for
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ETH
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case
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Equation
Institute
of Theoretical
Computer Science, E
1
Birula and Mycielski [7] and
Deutsch
[8].
Maassen
4 Later,
Equation
(2).
The
additional
term
H(A
for
S.
ertainty about
the measurement
R+
is H(S|B)
denoted4 Perimeter
mplication
of this
+
H(A|B).
(3)
H(R|B)
≥
log
Institute
for
Theoretical
Physics,
31
Caroline
2Inresult
and
Uffink
[9] improved Deutsch’s
to show that
out
the
measurement
R
is
denoted
3.
the
absence
of quantifies
theTechnische
quantum
mem
5
c
ditional
von Neumann
nnot predict
the entropy, H(R|B), which 3. In Institute
for hand
Applied side
Physics,
Univers
right
the
am
the
absence
of
the
quantum
memor
neneralizes
Neumann
can 1reduce the bound (3)
to H(R)
(Dated:
June 2
theentropy,
Shannon
entropy, which
H(R), to the
easurements
per- H(R|B),
between
the
system
and
the
mem
1
can
reduce
the
bound
(3)
to
H(R)
+
log
+H(A).
If
the
state
of
the
system
,
(2)
H(R)
+
H(S)
≥
he
Shannon
entropy,
H(R),
to
the
2
yBob
precision.
Howhas a quantum
B, andabout
likewisethe measurement
Bob’smemory,
uncertainty
Rexamples:
is denoted
cc
1 instructive
log2 c then
+H(A).
state
the system,
H(A)If=the
0 and
weofagain
recover3.tA
quantum
memory,
B,
and
likewise
id if
the observer
he
additional
term
H(A|B)
appearing
on
the
by the conditional von Neumann entropy,
H(R|B),
which
Maassen
Uffink,
Equation
(2). the
How
then H(A)
= 0and
we
again
recover
t
where of
H(R)
denotes the Shannon
entropy
ofand
the
probamemory,
an
unredalside
quantifies
theappearing
amount
entanglement
term
H(A|B)
on
the
directly
generalizes
the
Shannon
entropy,
H(R),
to
the
1.
If
the
state,
A,
and
memo
system,
A,
is
in
a
mixed
state
then
H(I
bility
distribution
of
the
outcomes
when
R
is
measured.
Maassen
and
Uffink,
Equation
(2).
Howe
ecent
technologihe
system
and
the
memory.
We
discuss
some
1
tifies the amount
of
entanglement
1
case
that
a quantum
memory,
B,
and
p
H(R|B)
+
H(S|B)
≥
loglikewise
theA,
resulting
bound
isstate
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than
E
quantifies
the complementarity
of athe
observTheBob
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entangled,
H(A|B)
=
we
strengthen
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eand
examples:
system,
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then
H(A
c
c
the memory. We
some
4 discuss
t
2
even
when
there
is
no
quantum
memor
The similarly
additional
term
H(A|B)
appearing
on
the
forifS. ables,
at applies even
to the
commutator
in
Equation
(1).
dimension
of the
system
the resultingthe
bound
is stronger
than
Equ
c
1
0
emory.
One
way
to
think
about
uncertainty
relations
is
via
he
state, QuantifyA, and
memory,
B,
are
maximally
right hand side quantifies the
amount
of2h(�)
entanglement
log
exceed
logmost
even
is cannot
noapplications,
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4. when
In terms
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the
= 1there
−
2 cnew
2 d,w
ropy, we
following
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uncertainty
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between
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ngled,
thenprovide
H(A|B)
=the−
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d, where
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isthe
between
the
and
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discuss
case
is
when
A
and
Bsome
are entangled,
bs
to
H(R|B)+H(S|B)
≥
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w
,dimension
and
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are
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inty of the
outplayers,
andSince
Bob.4. Before
theofgame
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of the
system
sent toAlice
Alice.
In
terms
new
applications,
the most
iu
imally
so.
The
resulting
bound
is
then
instructive
examples:
1
conditional
entropy
of
a
sys
K ≥ H(R|E)R−and
H(R|B)
nhich
H(A|B)
= −on
log
d,
where
d(3)isagree
depends
Alice
and
Bob
on
two
measurements
S. are
Theentangled, butq
cannot
exceed
log
d,
the
bound
reduces
2
2
case isand,
when
A
and
B
c
since the conditional entropy H(A
tween
the
system
of
the
system
sent
to
Alice.
Since
game
proceeds
as
follows:
Bob
creates
a quantum state
(R|B)+H(S|B) ≥ 0, which is trivial, since the imallyative,
so. The
resulting
boundfrom
is then
the bound
is different
the ent
part
from
its sigIfofthe
state,
A, sends
and itmemory,
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are
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and
to Alice.
then
performs
a
itional
entropy
of a1.
system
after
measurement
1 Alice
xceed
log
bound
(3)
reduces
2 d, the
4.
sical
ones.
and,
the conditional
entropy H(A|B
≥and
logsince
− H(R|B)
−choice
H(S|B)
2announces
quantum
physics,
one
of
the
two
measurements
her
of
c− log2 d, where d is
entangled,
then H(A|B)
=
S|B) ≥ 0, which is trivial,
since the
ative,
the
bound
isfundamental
different
from
the exis
ated the first promeasurement
to
Bob.
Bob’s
task
is
then
to
minimize
his
Aside
from
its
significance,
3
separable
? system
entangledsent
opy of a system after
measurement
the
dimension
of
the
to Alice.
Since
A related
uncertainty
relation has be
sical
ones.
y [3, 4]. However,
uncertainty
about Alice’s measurement
outcome.
This is to the develop
also has
potential
application
1
1
ature
[10],
which
is �implied
by the re
�
log
cannot
exceed
log
d,
the
bound
(3)
reduces
≥
log
−
H(R|R
)
−
H(S|S
)
2
2
a
quantum
memillustrated
in in
Figure
1.
24
c
entanglement
ture quantum
technologies.
Oneisobvious
can
uncertainty relation has been
conjectured
the literc
More
precisely,
H(R|B)
the
conditi
from
its
fundamental
significance,
o
P
P
to H(R|B)+H(S|B)
≥ the
0,uncertainty
which
is
trivial,
since
the
inciple
bethe relation
Equation
(2) boundsAside
Bob’s
in
the
case
that
, which is cannot
implied by
we derive.
fieldofofthequantum
cryptography.
In the
|ψ
��ψ
|
⊗
1
1
)ρ
state
(
j
j
AB (
j
also
has
potential
application
to
the
developm
isely,
H(R|B)
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the
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proposals
are
sememory.
However,
if hemeasurement
does
have and Brassard
conditional
entropy
of
a
system
after
80s,
Wiesner
[3],
and
Bennett
Phe has no quantum
P
applying
the
completely
positive map
H(A|B) ≈ − log2 d
|ψj ��ψj | ⊗ 11)ρAB (
|ψj ��ψj | ⊗ 11) formed by
te (
The New Entropic Uncertainty Relation
bound on the uncertainties of the measurement outcomes
which depends on the amount of entanglement between
the system, A, and the quantum memory, B. Mathematically, it is the following relation:3
Proof
to show
1
H(R|B) + H(S|B) ≥ log2 + H(A|B).
c
(3)
method:
smooth
entropy
calculus
(Renner,
Bob’s
uncertainty
about the
measurement
R is 2005)
denoted
eigenvalue
2
by the conditional von Neumann entropy, H(R|B), which
directly generalizes the Shannon entropy, H(R), to the
case that Bob has a quantum memory, B, and likewise
for S.4 The additional term H(A|B) appearing on the
right hand side quantifies the amount of entanglement
between the system and the memory. We discuss some
instructive examples:
1. If the state, A, and memory, discontinuous
B, are maximally
entangled, then H(A|B) = − smooth
log2 d, where
d is
entropies
the dimension of the system sent to Alice. Since
log2 1c cannot exceed log2 d, the bound (3) reduces
to H(R|B)+H(S|B) ≥ 0, which is trivial, since the
3
4
bound on the uncertainties of the measurement outcomes
which depends on the amount of entanglement between
the system, A, and the quantum memory, B. Mathematically, it is the following relation:3
Proof
2
1
H(R|B) + H(S|B) ≥ log2 + H(A|B).
c
(3)
to show
threeBob’s
stepsuncertainty about the measurement R is denoted
by the conditional von Neumann entropy, H(R|B), which
1.Inequality
for min
and max-entropies
directly generalizes
the Shannon
entropy, H(R), to the
case that Bob has a quantum memory, B, and likewise
for S.4 The additional term H(A|B) appearing on the
right hand side quantifies the amount of entanglement
between the system
and theentropies
memory. We discuss some
2.Convert
to smooth
instructive examples:
3
1. If the state, A, and memory, B, are maximally
entangled, then H(A|B) = − log2 d, where d is
3.Asymptotic
limit
von
Neumann
entropy
the dimension of the system sent to Alice. Since
log2 1c cannot exceed log2 d, the bound (3) reduces
to H(R|B)+H(S|B) ≥ 0, which is trivial, since the
4
Proof
to show
proof
Christandl and Winter, 2005; Boileau and Renes, 2009
Application: Entanglement Witness
Alice‘s
result?
2
entangled ?
error
probability
FIG. 1: Illustration of the uncertainty game. (1) Bob sends a state to Alice, which may, in general, be entangled with his
quantum memory. (2) Alice measures either R or S and notes her outcome. (3) Alice announces her measurement choice to
Bob. Bob’s goal is then to minimize his uncertainty about Alice’s measurement outcome.
We now proceed to state our uncertainty relation. It
holds in the presence of quantum memory and provides a
bound on the uncertainties of the measurement outcomes
which depends on the amount of entanglement between
the system, A, and the quantum memory, B. Mathematically, it is the following relation:3
given the quantum memory is always greater than
0. As discussed above, Bob can guess both R and
S perfectly with such a strategy.
2. If A and B are not
i.e. the state takes
� entangled,
j
j
the form ρAB = j pj ρA ⊗ ρB , for probabilities pj
4
Perimeter
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Bob.
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Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario, N2L 2Y5,
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Application: Quantum Key Distribution
Application: Quantum Key Distribution
• Intuition for security of BB84:
uncertainty principle
• Proofs of security:
entanglement distillation
privacy amplification
• Problem:
Eve has quantum memory
• New uncertainty principle
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ically,
it is the following
Summary
•
quantum memory
1
H(R|B) + H(S|B) ≥ log2 + H(A|B).
c
(3)
Fundamental relevance
• Bob’s
uncertainty about the measurement R is denoted
by
the
conditional
von
Neumann
entropy,
H(R|B),
which
Applications:
• directly generalizes the Shannon entropy, H(R), to the
•
Entanglement Witness
case that Bob has a quantum memory, B, and likewise
Key Distribution
The additional
term H(A|B) appearing on the
for S.4Quantum
right
hand side quantifies
the amount
of entanglement
Generalisation
to smooth
entropies
between the system and the memory. We discuss some
operational
meaning, quantum information theory
instructive
examples:
andstate,
Computer
to
1. If the
A, and Science
memory, relative
B, are maximally
• Physics
entangled,
then H(A|B)
quantum
knowledge
= − log2 d, where d is