### Slides - NetCod 2013

```Quantum Network Coding
- How can network coding be applied to
quantum information?
Harumichi Nishimura (Nagoya U.)
June 9th, 2013
NetCod 2013 @Calgary
Quantum Network Coding: Motivations
• Quantum communication
– point-to-point case
– extension to networks
– applications (unconditionally secure key distribution,
powerful verification systems, etc.)
– expensive resource
• Network coding [Ahlswede-Cai-Li-Yeung00]
– efficient communication on networks
– many other merits (wireless networks, secure
communication, etc.)
Butterfly Network
1
1
2
1
2
2

1 +2
1
1 +2
2
2

2
1 +2
1
1
Quantum Butterfly
1
1
2
1
2
2
quantum
bit

1 +2
1
quantum
channel
1 +2
2
2

2
1 +2
1
1
quantum
operation
Outline of This Talk
• Basics on quantum information
• Basic setting
– One-shot case
– Asymptotic case
– Network coding vs. routing
• With extra resources
– Classical communication
– Entanglement
Quantum States
Quantum Mechanics
Mathematical Representation
quantum system
Hilbert space
quantum state
unit vector
time evolution
unitary operator
measurement
projectors
Qubit system 2 := two dim. complex inner product space
with orthonormal basis |0 and |1 ≅ ℂ2
Bloch sphere
|0
=  0 + |1 ≅   |
|
= cos |0 +   sin  |1
≅ (cos  , cos  sin , sin  sin )
|0 +|1
|1
|0 +|1
Multiple Qubits
Two-qubit states can be represented by a unit vector in 2 ⊗
2 spanned by |00 ≔ |0 ⊗ |0 , |01 , |10 , and |11
=  00 +  01 +  10 + |11
( 0 + β 1 ) ⊗ (|0 + |1 )
EPR pair
|Φ+ ≔
EinsteinPodolskiy
-Rosen
1
2
|00 +
1
2
The first and second
qubits are entangled
|11
N-qubit states can be represented by a unit vector
in 2⊗ ≔ 2 ⊗ 2 ⊗ ⋯ ⊗ 2
|Φ +
Measurements
• Physical process to obtain classical information from quantum
states (in general, the states are changed by the measurement)
• Described by projectors corresponding to an orthonormal basis
=  0 + |1
measurement in
basis {|0 , |1 }
obtains 0 with 0
2
= ||2
obtains 1 with 1
2
= ||2
inner product between |1 and |
• Orthogonal states are perfectly distinguishable
|Φ+ = |00 + |11 / 2
|Φ− = |00 − |11 / 2
|Ψ + = |01 + |10 / 2
|Ψ − = |01 − |01 / 2
orthogonal
perfectly distinguishable by the
measurement in the basis (Bell basis)
{|Φ+ , |Φ− , |Ψ + , |Ψ − }
Quantum Operations & Channels
• In a narrow sense, a quantum operation
– transforms N-qubit states into N-qubit states
– is represented by a unitary operator
• In this talk, a quantum channel with capacity N is the identity
operator on the N-qubit states (noiseless)
• More generally, a quantum operation is implemented by 3 steps
1.
2.
3.
Add extra qubits (ancilla) prepared in a fixed state
Apply a unitary to the composite system (input+ancilla)
Take some part of the composite system as output
input |
ancilla
|0000
U
output
NOTE: Even a general operation is represented by a linear operator.
Two Quantum Tools
• Quantum Teleportation [BBCJPW93]
– Alice can send a qubit to Bob using two bits and an EPR
pair between Alice and Bob
|Φ+ = |00 + |11 / 2

=
|
• Dense coding [Bennett-Wiesner92]
– Alice can send two bits to Bob using a qubit and an EPR
pair between Alice and Bob
|Φ+ = |00 + |11 / 2
|
=

Dense Coding
Dense coding [Bennett-Wiesner92] Alice can send two bits to Bob using
a qubit and an EPR pair between Alice and Bob

|Φ+ = |00 + |11 / 2
1st stage: Alice’s local operation
= 1 Alice applies bit-flip
= 1 Alice applies phase-flip
=00
|Φ +
=
= |00 + |11 / 2
=10
|Ψ + = |10 + |01 / 2
=01
|Φ− = |00 − |11 / 2
=11
|Ψ − = −|10 + |01 / 2
0 1
: bit-flip
1 0
|0 + |1
orthogonal
=

|1 + |0
1 0
: phase-flip
0 −1
|0 + |1

2nd stage: Alice sends her part of the pair and Bob measures the pair
in basis {|Φ+ , |Φ− , |Ψ + , |Ψ − }.
|0 − |1
No-Cloning Theorem [Wootters-Zurek82]
• There is no quantum operation that clones a given state.
input |
output | ⊗ |
U
ancilla
|
• We can copy a bit like
|0 ↦ 00 , 1 ↦ |11
but this is not quantum copying:
| =
1
|0
2
+
1
|1
2
↦ |Φ+ =
1
|00
2
+
1
|11
2
Basic Setting
Before Basic Setting
• This talk focuses on multiple unicast networks
– By the non-cloning theorem, quantum analogue of
multicast networks is not straightforward [Shi-Soljanin06,
Kobayashi-Le Gall-N-Roetteler10]
our target
|1 , |2
|1
|1 , |2
|1 , |2
| 2
| 2
|1
Basic Setting
[Hayashi-Iwama-N-Raymond-Yamashita07]
• Quantum network N
– directed acyclic graph with source nodes 1 , 2 , … ,  and target nodes
1 , 2 , … ,
–  has a qubit |
– each edge is a quantum channel with capacity 1 (while any two
neighboring nodes u and v may have d edges, in which we often say
edge (u,v) has capacity d)
– each node v can apply any quantum operation  that transforms
input qubits from the incoming edges into output qubits to the
outgoing edges
input
v
ancilla
Uv
• N is solvable if there is a choice of quantum operations
such that every | can be recovered at
– we can define the “approximate” version where each target can
recover the qubit with some closeness factor (fidelity).
output
Is Butterfly Quantumly Solvable?
|1
| 2
1
2
quantum
bit
|0

|
quantum
operation
|1
quantum
channel
2
Bloch sphere

1
Is Butterfly Quantumly Solvable?
|1
| 2
1
1
2

1 +2
1 +2

2
1
1
2
1 +2
2
Is Butterfly Quantumly Solvable?
|1
| 2
1
1
2

1
2
1 +2
1 +2
1 +2
Theorem:
The butterfly network is not quantumly solvable
(in fact, not quantumly solvable within some
approximation factor) [HINRY07]

2
1
2
Ideas: Simplifying the Problem
|1
|2 =|
1
2
|0

|1

2
1
Ideas: Simplifying the Problem
|1
1

|

2
1 |
Ideas: Simplifying the Problem
|1
1

|
If we want to distinguish whether
= 0 or 1 faithfully, we cannot
make the red quantum channel
noiseless any more!

2
1 |
Basic Setting: Asymptotic Case
[Leung-Oppenheim-Winter10]
• The previous setting is one-shot; one qubit at each node must
be sent to the corresponding target node by a single use of
the network
1 N if there is a2
• A rate (1 , 2 , … ,  ) is achievable in a network
choice of quantum operations such that each  can send

3
qubits to  by  uses of N.
– negligible approximation errors are allowed
– time-sharing (or fractional coding) is allowed
Theorem:
• The achievable rate region in the butterfly
network is {(1 , 2 )|1 + 2 ≤ 1} [LOW10,
Hayashi07]
 This is achieved by routing
• The rates in some types of shallow networks
are also achievable by routing [LOW10]
2
2
1
1
3
1
1
achieved by routing
Basic Setting: Asymptotic Case
[Leung-Oppenheim-Winter10]
Theorem:
• The achievable rate region in the butterfly
network is {(1 , 2 )|1 + 2 ≤ 1}
 This is achieved by routing
Reduction to quantum secret sharing
[Gottesman00, IMQNTW05]
n qubits |1
n qubits |2
A
A
C
n uses
B
B
C
cannot recover
quantum secret
|1 |2
can recover
quantum secret
|1 |2
dim(C)≥dim(|1 |2 )
Network Coding vs. Routing
NC rate ≥  × (Routing rate)
1
1
2
2
[Harvey-Kleinberg-Lehman04]
1
2
2
1

2

=

=1

……..
……..
1

Two Quantum Tools
• Quantum Teleportation [BBCJPW93]
– Alice can send a qubit to Bob using two bits and an EPR
pair between Alice and Bob
|Φ+ = |00 + |11 / 2

=
|
• Dense coding [Bennett-Wiesner92]
– Alice can send two bits to Bob using a qubit and an EPR
pair between Alice and Bob
|Φ+ = |00 + |11 / 2
|
=

Quantum Teleportation
1st Stage: Alice measures her two qubits
(after a unitary operation)
|Φ+ = |00 + |11 / 2
|

2nd Stage: Alice sends her measurement results
and Bob corrects the error

0 1
: bit error
1 0

|0 + |1
|1 + |0
=
1 0
: phase error
0 −1

|0 + |1
|0 − |1
=
|
|
Coding vs. Routing (Quantum)
QNC rate ≥  × (QRouting rate)
[Jain-Franceschetti-Meyer11]

1. Send |Φ+ from p
|1 1
1
| 2
2
2
……..
……..
|

Coding vs. Routing (Quantum)
2. At  , run 1st step of
quantum teleportation

|Φ+ shared
|1 1
1
| 2
2
2
……..
……..
|

Coding vs. Routing (Quantum)
3. Run 2nd step of
quantum teleportation

1 1 1
1  1 1 |1
2 2
2
2  2 2 |2
……..
……..

|

Coding
vs.
Routing
(Quantum)

QNC
) ×
(QRouting
rate)
QNC rate
rate ≥
≥ (2×
(QRouting
rate)
1

1 1
2
2 2
1 1
2  2 2 |2

1 1

……..
……..
2 2
1  1 1 |1
2 2

|

= Σ
= Σ

Coding vs. Routing (Quantum)
QNC rate ≥  × (QRouting rate)

capacity 2
1
1 1
2
2 2
1 1
2  2 2 |2

1 1

……..
……..
2 2
1  1 1 |1
2 2

dense coding
|
= Σ
= Σ

Summary for Basic Setting
• The butterfly network is classically solvable but
not quantumly solvable (in both of the one-shot &
asymptotic cases)
– optimal quantum rate is achieved by routing
– seems to be typical
• There is an example such that the rate for
quantum network coding is unboundedly better
than that for quantum routing (using quantum
teleportation & dense coding)
– seems to be exceptional
• OPEN: achievable rates for general graphs
With Extra Resources
With Free Classical Communication
• This is the second-best for implementation:
– Classical communication is much more “effective” than
quantum (speed, noise-tolerance, money, etc.)
• Setting
– Free two-way classical communication: classical
communication is freely available between any two nodes
– Free one-way (forward/backward) classical communication:
classical communication is freely available according to (or
in the reverse direction to) the directed edges of the
underlying graph
Free Two-way Classical Communication
 Important fact: The underlying graph becomes “undirected”
quantum
teleportation
2
2
2
1
From Classical NC to Quantum NC
Theorem:
If rate (1 , 2 , … ,  ) is achievable in a classical network, then the same
rate is also achievable in the corresponding quantum network under
free classical communication [Kobayashi-Le Gall-N-Roetteler 09, 11]
|Φ+ shared
KLNR
From Classical NC to Quantum NC
|0 + |1
|0 + |1
|Φ+ =|00 +|11
shared
KLNR
=
1st step: Simulate classical protocol at each
node introducing ancilla
|
|
|
,
|
|
| +
|
|
| +
|
| +
|
|
,
|
|
From Classical NC to Quantum NC
|0 + |1
|0 + |1
|Φ+ =|00 +|11
shared
KLNR
|
,
|
| +
|
|
| +
|
=
2nd step: Remove the ancilla
• measurement in Fourier basis
• correction of phase errors using classical communication
|
|
|
| +
|
|
=
,
|
|
From Classical NC to Quantum NC
Theorem:
If rate (1 , 2 , … ,  ) is achievable in a classical network, then the same
rate is also achievable in the corresponding quantum network under
free classical communication [Kobayashi-Le Gall-N-Roetteler 09, 11]
• available to share EPR pairs or more general entangled states
• the amount of classical communication is not much: at most 3
(which can be improved to 2) factors of quantum communication.
OPEN (“converse” of Theorem):
If rate (1 , 2 , … ,  ) is achievable in a quantum network under free
classical communication, is the same rate also achievable in the
corresponding “undirected” classical network?
cf. [Conjecture: Li-Li04] In multiple unicast classical networks on
undirected graphs, network coding does not allow any advantage
over routing.
Free Forward Classical Communication
[LOW10]
• We cannot reverse the directed edge
– butterfly is not quantum solvable in this case
• Still, we can use quantum teleportation
2
1
1/2
1/2
1
1
With Free Entanglement
– can prepare shared entangled states
offline (i.e., at any time)
• Setting
– Free entanglement between any two
nodes: any two nodes in networks share
any entangled states at will
– Free entanglement between neighboring
nodes: any two neighboring nodes are
allowed to share entanglement
– Free entanglement between source
nodes: any source nodes are allowed to
shared entanglement
Free Entanglement between Any Two Nodes
Two observations by Leung-Oppenheim-Winter [LOW10]
Proposition 1:
Under free entanglement, the achievable rate for “quantum
communication” in a quantum network is exactly 1/2 of that for
“classical communication” in the same network.
Rate for sending qubits
Rate for sending bits
2
2
2
1
1
1
2
1
quantum
channel
Free Entanglement between Any Two Nodes
Proposition 2:
The achievable rate for “quantum communication” in a quantum
network under free entanglement is at least that for “classical
communication” in the corresponding classical network.
2
Rate for sending bits
1
1
2
1
Rate for sending qubits
1
1
classical
channel
1
quantum
channel
Free Entanglement between Any Two Nodes
Proposition 1:
Under free entanglement, the achievable rate for “quantum
communication” in a quantum network is exactly 1/2 of that for
“classical communication” in the same network.
Proposition 2:
OPEN (conjectured in [LOW10]):
The achievable rate for “quantum communication” in a quantum
The achievable rate for “quantum communication” in a quantum
network under free entanglement is at least that for “classical
network under free entanglement is exactly that for “classical
communication” in the corresponding classical network.
communication” in the corresponding classical network.
OPEN (conjectured in [LOW10]):
The achievable rate for “classical communication” in a quantum
network under free entanglement is exactly 1/2 of that for
“classical communication” in the corresponding classical network.
cf. proven for point-to-point channels [Cleve-van Dam-Nielsen-Tapp97]
Other Cases
Free entanglement between neighboring nodes
2
Same as the basic setting
[Hayashi07]
1
1
1
cf. [Satoh-Le Gall-Imai12]
Free entanglement between source nodes
1
?
1
1
cf. [Hayashi07, SoedaKinjo-Tuner-Murao11]
Another Second Best:
Entanglement + Classical Communication
[Satoh-Le Gall-Imai12]
• uses only (single) EPR pairs between neighboring nodes and
free classical communication (motivated by quantum
repeaters); does not use any quantum channel
• does not introduce ancilla at each node
|Φ+ shared
SLI
not quantum channels
Summary & Future Works
• Basic setting
– even the butterfly is not quantum solvable
– seems “liquid flow” different from classical case (while there exists some
exception)
– routing seems optimal but open to show for general graphs
• With extra resources
– under free classical communication,
• the underlying quantum network becomes undirected
• classical network coding can be converted into the quantum case
– sending classical bits on the quantum network under free entanglement
seems equivalent to that on the classical network (up to factor 2)
• Future works
– general theory: complexity, security, lossy channels, …
– application-oriented: quantum networks whose sources are restricted
(say, the Bennett-Brassard quantum key distribution (BB84))
```