Download Quantum channels and their capacities: An introduction

Quantum channels and their capacities: An introduction
Ana Kontrec
December 2, 2013
Elements of Classical Information Theory.
What is information?
Claude Shannon (1948.) related information with uncertainty: the
more rare the event, the more information we gain if we know that it
happened
Denition
Consider an event described by a random variable X with probability
distribution p (x ). Then we dene the measure of uncertainty of getting the
outcome x as the
surprisal (or self-information)
I (x )
:= − log p (x ).
Denition
The expected amount of surprisal of X is the
H (X )
:= −
X
Shannon entropy H (X ),
p (x ) log p (x ).
Elements of Classical Information Theory.
What is information?
Claude Shannon (1948.) related information with uncertainty: the
more rare the event, the more information we gain if we know that it
happened
Denition
Consider an event described by a random variable X with probability
distribution p (x ). Then we dene the measure of uncertainty of getting the
outcome x as the
surprisal (or self-information)
I (x )
:= − log p (x ).
Denition
The expected amount of surprisal of X is the
H (X )
:= −
X
Shannon entropy H (X ),
p (x ) log p (x ).
Elements of Classical Information Theory.
What is information?
Claude Shannon (1948.) related information with uncertainty: the
more rare the event, the more information we gain if we know that it
happened
Denition
Consider an event described by a random variable X with probability
distribution p (x ). Then we dene the measure of uncertainty of getting the
outcome x as the
surprisal (or self-information)
I (x )
:= − log p (x ).
Denition
The expected amount of surprisal of X is the
H (X )
:= −
X
Shannon entropy H (X ),
p (x ) log p (x ).
Elements of Classical Information Theory.
Shannon's model for a classical channel
both the
input X and the output Y
(with probability distributions
{πi }
are modeled as random variables
and
{qj },
respectively); we assume
output is correlated with input
Denition
The
classical channel is a linear map N
π = (π1 , π2 , ...)
and q
= (q1 , q1 , ...).
between probability distributions
It acts independently on each input
letter, according to the xed stochastic matrix
pij
= P (Y = j | X = i )):
N (π) = q ,
N (π)j =
{pij },(where
X
i
pij πi .
Elements of Classical Information Theory.
Shannon's model for a classical channel
both the
input X and the output Y
(with probability distributions
{πi }
are modeled as random variables
and
{qj },
respectively); we assume
output is correlated with input
Denition
The
classical channel is a linear map N
π = (π1 , π2 , ...)
and q
= (q1 , q1 , ...).
between probability distributions
It acts independently on each input
letter, according to the xed stochastic matrix
pij
= P (Y = j | X = i )):
N (π) = q ,
N (π)j =
{pij },(where
X
i
pij πi .
Elements of Classical Information Theory.
Shannon channel capacity
(memoryless) channel capacity: maximal number of classical bits
which can be transmitted per channel use, in a reliable manner (i.e.
with arbitrarily small error at the receiver)
Theorem
The channel capacity of a discrete memoryless channel
C
where I
(X : Y )
(N ) =
max I
{π(x )}
N
is given by
(X : Y ) ,
is the mutual information of the random variables X and
Y , and the maximum is taken over all possible input distributions
{π(x )} .
Mutual information
I
(X : Y ) :=
P
p
i ,j πi pij log q =⇒
about X when we know Y
ij
j
amount of information that we gain
Elements of Classical Information Theory.
Shannon channel capacity
(memoryless) channel capacity: maximal number of classical bits
which can be transmitted per channel use, in a reliable manner (i.e.
with arbitrarily small error at the receiver)
Theorem
The channel capacity of a discrete memoryless channel
C
where I
(X : Y )
(N ) =
max I
{π(x )}
N
is given by
(X : Y ) ,
is the mutual information of the random variables X and
Y , and the maximum is taken over all possible input distributions
{π(x )} .
Mutual information
I
(X : Y ) :=
P
p
i ,j πi pij log q =⇒
about X when we know Y
ij
j
amount of information that we gain
Elements of Classical Information Theory.
Shannon channel capacity
(memoryless) channel capacity: maximal number of classical bits
which can be transmitted per channel use, in a reliable manner (i.e.
with arbitrarily small error at the receiver)
Theorem
The channel capacity of a discrete memoryless channel
C
where I
(X : Y )
(N ) =
max I
{π(x )}
N
is given by
(X : Y ) ,
is the mutual information of the random variables X and
Y , and the maximum is taken over all possible input distributions
{π(x )} .
Mutual information
I
(X : Y ) :=
P
p
i ,j πi pij log q =⇒
about X when we know Y
ij
j
amount of information that we gain
The quantum channel
Denition
Quantum channel is a linear completely positive trace-preserving map
(CPTP) between quantum states.
Why is a quantum channel given by a linear CPTP map
(i) Linearity:
Φ : Mn → Mm ?
Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒
allows for a
probabilistic interpretation of any mixed state
(ii) Trace-preserving (TP): Tr
(iii) Positive: if
=⇒
ρ ≥ 0,
then
(Φ(ρ)) = Tr (ρ)
Φ (ρ) ≥ 0.
these criteria ensure that the image under
Φ
of a quantum state is
indeed a quantum state.
Is this enough to produce a physically realizable transformation?
The quantum channel
Denition
Quantum channel is a linear completely positive trace-preserving map
(CPTP) between quantum states.
Why is a quantum channel given by a linear CPTP map
(i) Linearity:
Φ : Mn → Mm ?
Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒
allows for a
probabilistic interpretation of any mixed state
(ii) Trace-preserving (TP): Tr
(iii) Positive: if
=⇒
ρ ≥ 0,
then
(Φ(ρ)) = Tr (ρ)
Φ (ρ) ≥ 0.
these criteria ensure that the image under
Φ
of a quantum state is
indeed a quantum state.
Is this enough to produce a physically realizable transformation?
The quantum channel
Denition
Quantum channel is a linear completely positive trace-preserving map
(CPTP) between quantum states.
Why is a quantum channel given by a linear CPTP map
(i) Linearity:
Φ : Mn → Mm ?
Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒
allows for a
probabilistic interpretation of any mixed state
(ii) Trace-preserving (TP): Tr
(iii) Positive: if
=⇒
ρ ≥ 0,
then
(Φ(ρ)) = Tr (ρ)
Φ (ρ) ≥ 0.
these criteria ensure that the image under
Φ
of a quantum state is
indeed a quantum state.
Is this enough to produce a physically realizable transformation?
The quantum channel
Denition
Quantum channel is a linear completely positive trace-preserving map
(CPTP) between quantum states.
Why is a quantum channel given by a linear CPTP map
(i) Linearity:
Φ : Mn → Mm ?
Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒
allows for a
probabilistic interpretation of any mixed state
(ii) Trace-preserving (TP): Tr
(iii) Positive: if
=⇒
ρ ≥ 0,
then
(Φ(ρ)) = Tr (ρ)
Φ (ρ) ≥ 0.
these criteria ensure that the image under
Φ
of a quantum state is
indeed a quantum state.
Is this enough to produce a physically realizable transformation?
The quantum channel
Denition
Quantum channel is a linear completely positive trace-preserving map
(CPTP) between quantum states.
Why is a quantum channel given by a linear CPTP map
(i) Linearity:
Φ : Mn → Mm ?
Φ(aρ1 + aρ2 ) = aΦ(ρ1 ) + bΦ(ρ2 ) =⇒
allows for a
probabilistic interpretation of any mixed state
(ii) Trace-preserving (TP): Tr
(iii) Positive: if
=⇒
ρ ≥ 0,
then
(Φ(ρ)) = Tr (ρ)
Φ (ρ) ≥ 0.
these criteria ensure that the image under
Φ
of a quantum state is
indeed a quantum state.
Is this enough to produce a physically realizable transformation?
The quantum channel
mathematical formalism of quantum channels describes the dynamics
of
open quantum systems =⇒ systems which have interactions with
an environment system
The quantum channel
If we extend the primary space of states
HA ⊗ Henv
HA
to the tensor product
(by adding an ancilla), then the quantum channel
Φ
must take
density matrices of this joint system again to density matrices.
(iv) Completely positive (CP): a linear map
called
completely positive if
Φ : Mn → Mm
Φ ⊗ Ik : Mn ⊗ Mk → Mm ⊗ Ik
is positivity preserving for every k
≥ 1.
is
The quantum channel
If we extend the primary space of states
HA ⊗ Henv
HA
to the tensor product
(by adding an ancilla), then the quantum channel
Φ
must take
density matrices of this joint system again to density matrices.
(iv) Completely positive (CP): a linear map
called
completely positive if
Φ : Mn → Mm
Φ ⊗ Ik : Mn ⊗ Mk → Mm ⊗ Ik
is positivity preserving for every k
≥ 1.
is
The quantum channel
If we extend the primary space of states
HA ⊗ Henv
HA
to the tensor product
(by adding an ancilla), then the quantum channel
Φ
must take
density matrices of this joint system again to density matrices.
(iv) Completely positive (CP): a linear map
called
completely positive if
Φ : Mn → Mm
Φ ⊗ Ik : Mn ⊗ Mk → Mm ⊗ Ik
is positivity preserving for every k
≥ 1.
is
Examples of single qubit channels
Depolarising channel
Denition
A
depolarising channel is a channel which:
leaves the qubit unaected with probability q and replaces the state of the
qubit with a completely mixed state
I
2 with probability 1 − q,
I
Φ (ρ) := (1 − q ) ρ + q .
2
Examples of single qubit channels
Amplitude-damping channel
Amplitude damping channel describes energy dissipation of a quantum
system (for example, the dynamics of a 2-level atom which is
spontaneuosly emmiting a photon).
Unital channels
We say that a channel is
unital if Φ (I ) = I .
Amplitude damping channel is not unital!
Examples of single qubit channels
Amplitude-damping channel
Amplitude damping channel describes energy dissipation of a quantum
system (for example, the dynamics of a 2-level atom which is
spontaneuosly emmiting a photon).
Unital channels
We say that a channel is
unital if Φ (I ) = I .
Amplitude damping channel is not unital!
Classical capacities of quantum channels
Shannon capacity of quantum channel
the most classical of quantum channel capacities
=⇒
no entangled
inputs, no joint measurements
Protocol
1
Each input letter (from a nite alphabet) is encoded in a quantum
state
ρi
Φ (ρi )
2
The channel
3
At the output a measurement (POVM) is performed
Φ
maps it to an output state
Classical capacities of quantum channels
Shannon capacity of quantum channel
the most classical of quantum channel capacities
=⇒
no entangled
inputs, no joint measurements
Protocol
1
Each input letter (from a nite alphabet) is encoded in a quantum
state
ρi
Φ (ρi )
2
The channel
3
At the output a measurement (POVM) is performed
Φ
maps it to an output state
Classical capacities of quantum channels
Shannon capacity of quantum channel
This is actually a particular realization of a classical channel:
Alice chooses a set of states
chooses a POVM
{Ej }
the stochastic matrix
{ρi }
to encode the input letters, and Bob
to measure the output
{pij }
of the channel is given by
pij
= Tr (Ej Φ (ρi )).
Denition
The
Shannon capacity of a quantum channel is dened as
CShan
(Φ) :=
max I
{ρ ,E }
i
(X : Y ) ,
j
where X and Y are classical random variables corresponding to the source
(which emmits input letters) and the outcome of Bob's POVM,
respectively.
Classical capacities of quantum channels
Holevo capacity of quantum channels
What if Alice encodes her messages using product states only, but Bob is
allowed to decode them using entangled measurements? (:= product-state
capacity)
Theorem (Holevo-Schumacher-Westmoreland (HSW) theorem)
Dene the
Holevo capacity of the channel Φ as
"
χ (Φ) :=
where S
χ (Φ)
sup
{pi ,ρi }
(ρ) = −Tr ρ log ρ
!!
S
Φ
X
i
pi ρi
#
−
X
i
pi S
(Φ (ρi )) ,
is the Von Neumann entropy. Then the quantity
is the product-state capacity of the channel
Φ.
Holevo capacity is the maximal amount of classical information that can be
transmitted through a noisy quantum channel using product-state inputs.
Classical capacities of quantum channels
Holevo capacity of quantum channels
What if Alice encodes her messages using product states only, but Bob is
allowed to decode them using entangled measurements? (:= product-state
capacity)
Theorem (Holevo-Schumacher-Westmoreland (HSW) theorem)
Dene the
Holevo capacity of the channel Φ as
"
χ (Φ) :=
where S
χ (Φ)
sup
{pi ,ρi }
(ρ) = −Tr ρ log ρ
!!
S
Φ
X
i
pi ρi
#
−
X
i
pi S
(Φ (ρi )) ,
is the Von Neumann entropy. Then the quantity
is the product-state capacity of the channel
Φ.
Holevo capacity is the maximal amount of classical information that can be
transmitted through a noisy quantum channel using product-state inputs.
Classical capacities of quantum channels
Capacities of product channels
If we allow both entangled inputs and entangled measurements, it can be
shown that there exist channels satisfying the superadditivity property
CShan
(Φ ⊗ Φ) > 2CShan (Φ) .
This leads to the question of nding the asymptotic (or ultimate) capacity
Cult
(Φ) =
C
Φ⊗n
lim
n→∞ n Shan
1
Fact
From the HSW theorem it follows that
Cult
(Φ) =
lim
1
n→∞ n
χ Φ⊗n .
Classical capacities of quantum channels
Capacities of product channels
If we allow both entangled inputs and entangled measurements, it can be
shown that there exist channels satisfying the superadditivity property
CShan
(Φ ⊗ Φ) > 2CShan (Φ) .
This leads to the question of nding the asymptotic (or ultimate) capacity
Cult
(Φ) =
C
Φ⊗n
lim
n→∞ n Shan
1
Fact
From the HSW theorem it follows that
Cult
(Φ) =
lim
1
n→∞ n
χ Φ⊗n .
Classical capacities of quantum channels
Capacities of product channels
If we allow both entangled inputs and entangled measurements, it can be
shown that there exist channels satisfying the superadditivity property
CShan
(Φ ⊗ Φ) > 2CShan (Φ) .
This leads to the question of nding the asymptotic (or ultimate) capacity
Cult
(Φ) =
C
Φ⊗n
lim
n→∞ n Shan
1
Fact
From the HSW theorem it follows that
Cult
(Φ) =
lim
1
n→∞ n
χ Φ⊗n .
Additivity conjectures
Conjecture (additivity
of Holevo capacity
For any two quantum channels
Φ
and
Ω,
)
it holds that
χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) .
This would imply that
χ (Φ⊗n ) = nχ (Φ),
Cult
and hence
(Φ) = χ (Φ) .
Physical meaning of additivity conjecture
The channel capacity is achieved by encoding on product states only, i.e.
using entangled inputs does
not increase the capacity.
Additivity conjectures
Conjecture (additivity
of Holevo capacity
For any two quantum channels
Φ
and
Ω,
)
it holds that
χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) .
This would imply that
χ (Φ⊗n ) = nχ (Φ),
Cult
and hence
(Φ) = χ (Φ) .
Physical meaning of additivity conjecture
The channel capacity is achieved by encoding on product states only, i.e.
using entangled inputs does
not increase the capacity.
Additivity conjectures
Conjecture (additivity
of Holevo capacity
For any two quantum channels
Φ
and
Ω,
)
it holds that
χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) .
This would imply that
χ (Φ⊗n ) = nχ (Φ),
Cult
and hence
(Φ) = χ (Φ) .
Physical meaning of additivity conjecture
The channel capacity is achieved by encoding on product states only, i.e.
using entangled inputs does
not increase the capacity.
Additivity conjectures
Can we do without entangled inputs?
The following theorem was proved in 2003 by Shor:
Theorem
The following conjectures are equivalent:
(1) additivity of the Holevo capacity of a quantum channel
(2) additivity of the minimal entropy output Smin
(Φ)
of a quantum
channel, where
Smin
(Φ) = inf S (Φ (ρ)) ,
ρ
(3) additivity of the entanglement of formation EF
ρ,
(ρ)
of a bipartite state
where
EF
(ρ) = min
E
X
i
pi S (TrB
(| ρi >< ρi |)) .
In 2008, Hastings proved the non-additivity of the minimal output entropy,
hence disproving all of the above conjectures.
Additivity conjectures
Can we do without entangled inputs?
The following theorem was proved in 2003 by Shor:
Theorem
The following conjectures are equivalent:
(1) additivity of the Holevo capacity of a quantum channel
(2) additivity of the minimal entropy output Smin
(Φ)
of a quantum
channel, where
Smin
(Φ) = inf S (Φ (ρ)) ,
ρ
(3) additivity of the entanglement of formation EF
ρ,
(ρ)
of a bipartite state
where
EF
(ρ) = min
E
X
i
pi S (TrB
(| ρi >< ρi |)) .
In 2008, Hastings proved the non-additivity of the minimal output entropy,
hence disproving all of the above conjectures.
Additivity for unital qubit channels
Theorem (King, 2001)
Let
Φ
be a unital qubit channel. Then for any (!) channel
χ (Φ ⊗ Ω) = χ (Φ) + χ (Ω) .
Ω,
it holds that
What next?
entanglement-assisted classical capacity (what if the sender and
receiver share some prior entanglement?)
I
protocols of quantum teleportation and superdense coding
quantum capacities of quantum channels
What next?
entanglement-assisted classical capacity (what if the sender and
receiver share some prior entanglement?)
I
protocols of quantum teleportation and superdense coding
quantum capacities of quantum channels
Summary
quantum information has features which are distinctly dierent from
classical information, mainly due to the phenomenon of entanglement
this opens the questions not only of quantifying entanglement, but of
nding new mathematical approaches to the quantum case
(geometry?)
a quote by Wojciech Zurek:
Indeed, quantum computing inevitably poses questions that
probe to the very core of the distinction between quantum and
classical. (...) Questions originally asked for the most impractical
of reasons - questions about the EPR paradox, the
quantum-to-classical transition, the role of information, and the
interpretation of the quantum state vector - have become
relevant for practical applications such as quantum criptography
and quantum computation.
Summary
quantum information has features which are distinctly dierent from
classical information, mainly due to the phenomenon of entanglement
this opens the questions not only of quantifying entanglement, but of
nding new mathematical approaches to the quantum case
(geometry?)
a quote by Wojciech Zurek:
Indeed, quantum computing inevitably poses questions that
probe to the very core of the distinction between quantum and
classical. (...) Questions originally asked for the most impractical
of reasons - questions about the EPR paradox, the
quantum-to-classical transition, the role of information, and the
interpretation of the quantum state vector - have become
relevant for practical applications such as quantum criptography
and quantum computation.
Summary
quantum information has features which are distinctly dierent from
classical information, mainly due to the phenomenon of entanglement
this opens the questions not only of quantifying entanglement, but of
nding new mathematical approaches to the quantum case
(geometry?)
a quote by Wojciech Zurek:
Indeed, quantum computing inevitably poses questions that
probe to the very core of the distinction between quantum and
classical. (...) Questions originally asked for the most impractical
of reasons - questions about the EPR paradox, the
quantum-to-classical transition, the role of information, and the
interpretation of the quantum state vector - have become
relevant for practical applications such as quantum criptography
and quantum computation.