Network information theory for classical-quantum channels

classical-quantum channels

Omar Fawzi, Patrick Hayden, Pranab Sen, Ivan Savov^†, Mark Wilde

McGill University School of Computer Science

September 19, 2012

x1

Tx1

x2

Tx2

y Rx

(a) MAC ≡ p(y|x1, x2)

x1

Tx1

x2

Tx2

y1 Rx1

y2 Rx2

(b) IC ≡ p(y1, y2|x1, x2)

x

Tx

y1 Rx1

y2 Rx2 (c) BC ≡ p(y1, y2|x)

x

Tx

y1 Re

x1

y Rx (d) RC ≡ p(y1, y|x, x1)

Classical-quantum channel

(X, N^X→B(x) ≡ ρ^B_x, H^B) ⇔ Tx ^x N^{X→B ρB}x Rx

I Input: x ∈ X (finite set).

I Outputs: conditional density matrices {ρ^B_x} ∈ H^B, hv|ρ^B_x|vi ≥ 0, ∀|vi ∈ H^B, (ρ^B_x)^† =ρ^B_x, Tr[ρ^B_x] = 1.

Each output state can be decomposed as follows:

ρ^B_x =X

y

λρ_x;y|eρ_x;yiheρ_x;y| =X

y

pY |X(y|x)|eρ_x;yiheρ_x;y|,

where we identify eigenvalues ofρxwith a conditional probability dist.

B

ya ∈ Y ⇐⇒ |vi^B ∈ H^B

symbol from a finite set vector in a Hilbert space

Y ≡ pY ∈ P(Y) ⇐⇒ ρ^B ≡ ρ^B ∈ D(H^B)

probability distribution density matrix ≡ quantum state pY(y) ≥ 0, ∀y ∈ Y hv|ρ^B|vi ≥ 0, ∀|vi ∈ H^B

P

ypY(y) = 1 Tr[ρ^B] = 1, (ρ^B)^†= ρ^B pY |X ⇐⇒ {ρ^Bx}, x ∈ X

conditional probability distribution conditional states

≡ classical-classical channel ≡ classical-quantum channel pXY(x, y) ≡ pX(x)pY |X(y|x) ⇐⇒ θ^XB≡P

xpX(x) |xihx|^X⊗ ρ^Bx

joint input-output distribution joint input-output state 1ⁿ

yⁿ∈T_δ⁽ⁿ⁾(Y |xⁿ)o ⇐⇒ Πxⁿ≡ Π^B_ρxnⁿ_,δ indicator function for the conditionally typical

conditionally typical set projector for the state ρ^B_xnⁿ

Quantum multi-user scenarios

I Classical multi-user decoding is based on the notion of jointly typical sequences: (xⁿ₁(m1), xⁿ₂(m2), yⁿ) ∈ J_δⁿ(X1, X2, Y ).

I Equivalently, we can use the conditionally-typical sets: T (Y ), T (Y |xⁿ1), T (Y |xⁿ2), T (Y |xⁿ1, xⁿ2):

1{yⁿ∈T (Y )}1{^{yn∈T (Y |xn}₁^(m1))} 1{^{yn∈T (Y |xn}₂^(m2))} 1{^{yn∈T (Y |xn}₁^(m1),xⁿ₂(m₂))}

I The quantum equivalents of _{1{yn ∈T(Y |xn}

1 (m1))}and

1{yn ∈T(Y |xn2 (m2))}are the conditionally typical projectors Πxⁿ₁(m1)

and Π_xⁿ_2(m2), which in generaldo not commute:

Π_xn

1(m1)Π_xn

2(m2)6= Π_xⁿ_2(m2)Π_xn 1(m1).

I Quantum multi-user decodingrequires new techniques.

Outline

I Introduce basic notions:

I Classical-quantum communication model

I Conditionally typical projectors

I Show the proof of simultaneous decoder for the quantum multiple access cannel

I Briefly discuss other problems in network information theory:

I Quantum interference channels

I Quantum broadcast channels

I Quantum relay channels

Ivan Savov QNIT

x1

Tx1

x2

Tx2

ρ^B_x1,x2 Rx

(a) QMAC ≡˘ρ^B_x1,x2¯

x1

Tx1

x2

Tx2

ρB1 x1,x2 Rx1

ρB2 x1,x2 Rx2

(b) QIC ≡ n

ρ^Bx₁¹,x^B₂²

o

Tx x

ρ^B1_x Rx1

ρ^B2_x Rx2 n B₁B₂o

Tx x

ρB1 x,x1

Re x1

ρBx,x1 Rx

n B₁Bo

Classical-quantum channel coding

Consider first thepoint-to-pointcommunication scenario.

WANTED Noiseless classical communication:

[c → c]

Ivan Savov QNIT

Classical-quantum channel coding

AVAILABLE Noisy classical-quantum channel:

(X, N^X→B(x) ≡ ρ^B_x, H^B)

Tx x N^X→B ρ^B_x Rx

Classical-quantum channel coding

We are allowed to use the channel many times in parallel:

x1 ρB1

x1 N X→B

x2 ρB2

x2 N X→B

x3 ρB3

x3 N X→B

x4 ρB4

x4 N X→B

Ivan Savov QNIT

Classical-quantum channel coding

We are allowed to use the channel many times in parallel:

x1

m E

∈ [1 : 2^nR]

ρB1 x1 N X→B

x2 ρB2

x2 N X→B

x3 ρB3

x3 N X→B

x4 ρB4

x4 N X→B

{Λ_M⁰} M⁰

∈ [1 : 2^nR]

Decoding POVM : {Λm} X

Λ_m=I and Λ_m≥ 0, ∀m.

Classical-quantum channel coding

m

∈ [1 : 2^nR]

E Xⁿ

∈ Xⁿ

ρ^B_Xⁿn

∈ H^Bn

N^⊗n

{ΛM⁰} M⁰

∈ [1 : 2^nR]

p¯e≡ 1 2^nR

X

m∈[1:2^nR]

Pr{M⁰6= m | m is sent} ≤ .

Ivan Savov QNIT

Classical-quantum channel coding

m

∈ [1 : 2^nR]

E Xⁿ

∈ Xⁿ

ρ^B_Xⁿⁿ

∈ H^Bn

N^⊗n

{ΛM⁰} M⁰

∈ [1 : 2^nR]

p¯e≡ 1 2^nR

X

m∈[1:2^nR]

Trn

I − Λ^B_mⁿ ρ^B_xnⁿ(m)

o≤ .

Classical-quantum code

A rateR is achievable over the c-q channel N^X→Bif there exists a codebookxⁿ(m), m ∈ [1 : 2^nR]and a corresponding decoding measurement POVM {Λm}, m ∈ [1 : 2^nR]such that the average probability of error is bounded from above by epsilon: ¯pe≤ .

⇔

n · N^{X→B (1−)}−→ nR · [c → c].

Ivan Savov QNIT

Classical decoding: conditionally typical sets

Consider an input distributionpX(x) and the channel pY |X(y|x).

For any input codewordxⁿ, we define the conditionally typical set:

T_δ⁽ⁿ⁾(Y |xⁿ) ≡



yⁿ ∈ Yⁿ:    

−logpYⁿ|Xⁿ(yⁿ|xⁿ)

n − H(Y |X)

≤ δ

 .

IDEA: An input stringxⁿis passed through the channel is likely to result in a conditionally typical output string.

Quantum decoding: conditionally typical subspaces

We define thexⁿ-conditionally typical projector as follows:

Πⁿ_ρB

xn,δ= X

yⁿ∈T_δ⁽ⁿ⁾(Y |xⁿ)

|eρxn;yⁿiheρxn;yⁿ|,

where:

I The sum is over the set of conditionally typical strings of eigenvalue:

T_δ⁽ⁿ⁾(Y |xⁿ) ≡

 yⁿ:

−logpYⁿ|Xⁿ(yⁿ|xⁿ)

n − H(Y |X)

≤ δ

 , withpYⁿ|Xⁿ(yⁿ|xⁿ) =Qn

i=1pY |X(yi|xi).

I The states |eρxn;yⁿi are built from tensor products of eigenvectors for the individual signal states:

|eρxn;yⁿi = |eρ_x1;y1i ⊗ |eρ_x2;y2i ⊗ · · · ⊗ |eρ_xn;yni.

Ivan Savov QNIT

Conditionally typical sets and subspaces

1ⁿ

yⁿ∈T_δ⁽ⁿ⁾(Y |xⁿ)o

Xⁿ

T_δ⁽ⁿ⁾(X)

Yⁿ

T_δ⁽ⁿ⁾(Y |xⁿ) xⁿ

XnE X

yn ∈Yn

p`yⁿ|Xⁿ´ 1^

yn ∈T(n) δ (Y |Xn)

ff≥ 1 − ,

X

yn ∈Yn

1^

yn ∈T(n) δ (Y |xn)

ff ≤ 2n[H(Y |X)+δ]

Π_xn≡ Π^B_ρⁿ

xn,δ Xⁿ

T_δ⁽ⁿ⁾(X) xⁿ

H^Bn

Πρxn

XEⁿ Trh

ρ^B_XnΠⁿ_ρB Xn,δ

i≥ 1 − ,

Trh Πⁿ_ρB

xn,δ

i≤ 2^{n[H(B|X)ρ+δ]}, Πⁿ_ρB

Xn,δρ^BXⁿΠⁿ_ρB

Xn,δ≤ 2^{−n[H(B|X)ρ−δ]}Πⁿ_ρB Xn,δ

Quantum multiple access channel

Ivan Savov QNIT

Quantum multiple access channel

X1×X2, N^X1X2→B(x1, x2) ≡ρ^B_x1,x2, H^B

⇔

x1

Tx1

x2

Tx2

ρ^B_x1,x2 Rx

Communication task:

n · N^{X1X2→B (1−)}−→ nR1· [c¹→ c] + nR2· [c²→ c]

IDEA: Tradeoff between the ratesR1andR2for the two senders.

QMAC capacity region

Theorem 10 in [Winter2001]: The capacity region for the

classical-quantum multiple access channel (X1× X₂, ρ^B_x1,x2, H^B)is:

CMAC(N ) ≡ [

p_X1,p_X2







(R1, R2) ∈ R²+

R1 ≤ I(X1;B|X2)_θ R2 ≤ I(X2;B|X1)_θ R1+R2 ≤ I(X1X2;B)θ





 ,

where

θ^X1X2B ≡ X

x1,x2

pX1(x1)pX2(x2) |x1ihx1|^X1⊗ |x2ihx2|^X2⊗ ρ^B_x1,x2.

Ivan Savov QNIT

Winter’s achievability proof: sequential decoding

Coding for the corners:

Two codesC_αand Cβwith rates:

αp: (I(X1;B), I(X2;B|X1) ) βp: (I(X1;B|X2), I(X2;B) ).

Time sharing:

Intermediate points can be switching between the two codes, e.g., use Cα70% of the time, and Cβ30% of the time.

Simultaneous decoding

New theorem: Any rate pair (R1, R2) ∈CMAC(N )can be achieved usingsimultaneous decoding. [FHSSW11]

Simultaneous decoding allows us to use asingle codeto achieve any rate pair in the QMAC capacity region.

Ivan Savov QNIT

Proof sketch

We want to show that ∃ codebooksxⁿ₁(m1),xⁿ₂(m2)and a decoding POVM {Λm1,m2} such that the average error probability is small:

p_e≡ 1

|M1||M2| X

m1,m2

Tr(I − Λm1,m2)ρxⁿ₁(m1),xⁿ₂(m2) ≤ .

We will show that the expectation ofp_efor random codebooks X1ⁿ(m1),X2ⁿ(m2)is small:

Xⁿ₁E,X₂ⁿ{ p_e} ≤ .

This implies the existence of a good code.

Random codebook construction

Codebook construction: Randomly and independently generate 2^nRi sequencesxⁿ_i(mi),mi∈1 : 2^nRi, according toQ pⁿ Xi. Transmission: When the message pair (m1, m2)is transmitted, the channel output will be

ρ^B_mⁿ_1,m2 ≡ ρ^B_xnⁿ

1(m1),xⁿ₂(m2).

Define the conditionally typical projector for this output state:

Πⁿ_m1,m2 ≡ Π^B_ρBnⁿ xn1 (m1),xn

2 (m2)

.

Ivan Savov QNIT

POVM construction

Define the averaged output states:

ρ¯m1 ≡ E

X₂ⁿρxⁿ₁(m1),Xⁿ₂, ρ¯m2 ≡ E

X₁ⁿρX₁ⁿ,xⁿ₂(m2), ρ ≡¯¯ E

Xⁿ₁,X₂ⁿρX₁ⁿ,X₂ⁿ, and the corresponding conditionally typical projectors:

Πⁿ_m1, Πⁿ_m2, Πⁿ_ρ¯¯. POVM construction: Construct the “projector sandwich”

Pm1,m2≡ Πⁿρ¯¯ Πⁿ_m1 Πⁿ_m1,m2 Πⁿ_m1Πⁿρ¯¯, and normalize these to form a valid POVM:

Λ_m1,m2≡



 X

m⁰₁,m⁰₂

Pm⁰₁,m⁰₂





−¹₂

Pm1,m2



 X

m⁰₁,m⁰₂

Pm⁰₁,m⁰₂





−¹₂

.

Proof (1): State smoothing trick

Recall the definition of the average probability of error:

p_e≤ 1

|M1||M2| X

m1,m2

Tr[(I − Λm1,m2)ρm1,m2].

We make a substitution of the output stateρm1,m2 with a Πⁿ_m2-smoothed version:

ρ˜m₁,m₂ ≡ Πⁿ_m2ρm₁,m₂Πⁿ_m2.

This substitution introduces a“smoothing penalty”term:

p_e≤ 1

|M1||M2| X

m1,m2

"

Tr[(I − Λm1,m2) ˜ρm1,m2] + k˜ρm1,m2− ρm1,m2k₁

# .

Ivan Savov QNIT

Proof (2): Hayashi-Nagaoka inequality

Next we use the Hayashi-Nagaoka operator inequality [HN03]:

I − (S + T )⁻¹²S (S + T )⁻¹² ≤ 2 (I − S) + 4T, to obtain:

Tr[(I − Λm1,m2) ˜ρm1,m2] ≤

2Tr[(I − Pm₁,m₂) ˜ρm₁,m₂] + 4 X (^m01,m⁰₂)^6=(m1,m2)

TrP_m⁰_1,m⁰₂ρ˜m₁,m₂ .

Proof (3): Epsilon terms

First term:

X₁ⁿE,X₂ⁿ2Tr[(I − Pm1,m2) ˜ρm1,m2] ≤ 2( + 6√

).

Smoothing-penalty:

EXⁿ₁,X₂ⁿk˜ρm1,m2− ρm1,m2k1= EX₁ⁿ,X₂ⁿkΠⁿ_m2ρm1,m2Πⁿ_m2 − ρm1,m2k1

≤ 2√

.

Therefore we are left with:

EXⁿ₁,X₂ⁿ{p_e} ≤

"

2( + 6√

) + 4X (^m01,m⁰₂)^6=(m1,m2)

TrPm⁰₁,m⁰₂ρ˜m1,m2 + 2√



# .

Ivan Savov QNIT

Proof (4): Split into three error terms

Split the “wrong message” error term into three parts:

X (^m01,m⁰₂)^6=(m1,m2)

TrPm⁰₁,m⁰₂ρ˜m1,m2 =

= X

m⁰₁6=m1

TrP_m⁰_1,m2ρ˜m₁,m₂

(E1)

+ X

m⁰₂6=m2

TrP_m1,m⁰₂ρ˜m1,m2

 (E2)

+ X

m⁰₁6=m₁,m⁰₂6=m₂

TrP_m⁰_1,m⁰₂ρ˜m1,m2 . (E12)

We will attack each of these in turn.

Xⁿ₁E,X₂ⁿ{(E1)} = E

X₁ⁿ,X₂ⁿ

 X

m⁰₁6=m1

TrPm⁰₁,m₂ρ˜m1,m2



= X

m⁰₁6=m1

XE₂ⁿ

 Tr



XE₁ⁿPm⁰₁,m2

Πⁿ_m2E

X₁ⁿ{ρm1,m2} Πⁿ_m2



=¬ X

m⁰₁6=m1

X₁ⁿEX₂ⁿTr P_m⁰_1,m2Πⁿ_m2ρ¯m₂Πⁿ_m2

≤­ 2^{−n[H(B|X2)−δ]} X

m⁰₁6=m1

Xⁿ₁EX₂ⁿTr Pm⁰₁,m₂Πⁿ_m2

≤ 2^{−n[H(B|X2)−δ]}X

m⁰₁6=m1

Trh Πⁿ_m0

1,m2

i

≤ 2® ^{−n[H(B|X2)−δ]} X

m⁰₁6=m1

2^{n[H(B|X1X2)+δ]}

≤ |M1| 2^{−n[I(X1;B|X2)−2δ]}. (1)

X₁ⁿE,X₂ⁿ

n(E2)o

= E

X₁ⁿ,X₂ⁿ

8

<

: X

m⁰₂6=m₂

Trh

Pm₁,m⁰₂ρ˜m₁,m₂

i 9

=

;

= X

m⁰₂6=m₂ XE₁ⁿ

( Tr

"

XE₂ⁿ n

Πⁿρ,δ¯ Πⁿm₁Πⁿ_m1,m0

2 Πⁿm₁Πⁿρ,δ¯

o

XE₂ⁿ

{˜ρm1,m2}

#)

≤¬ 2^{n[H(B|X1X2)+δ]} X

m⁰₂6=m₂ XE₁ⁿ

( Tr

"

XEⁿ₂ n

Πⁿ_ρ,δ¯ Πⁿ_m1ρ^B_m1,m0

2Πⁿ_m1Πⁿ_ρ,δ¯ o

XE₂ⁿ{˜ρm₁,m₂}

#)

= 2^{n[H(B|X1X2)+δ]} X

m⁰₂6=m₂ XE₁ⁿ

( Tr

"

Πⁿρ,δ¯ Πⁿm₁ E

X₂ⁿ

n

ρ^B_m1,m⁰₂o

Πⁿm₁Πⁿρ,δ¯ E

Xⁿ₂

{˜ρm₁,m₂}

#)

= 2^{n[H(B|X1X2)+δ]} X

m⁰₂6=m₂ XE₁ⁿ

( Tr

"

Πⁿ_ρ,δ¯ Πⁿ_m1ρ¯m1Πⁿ_m1Πⁿ_ρ,δ¯ E

X₂ⁿ

{˜ρm1,m2}

#)

≤ 2­ ^{n[H(B|X1X2)+δ]}2^{−n[H(B|X1)−δ]} X

m⁰₂6=m₂ XEⁿ₁

( Tr

"

Πⁿ_ρ,δ¯ Πⁿ_m1Πⁿ_ρ,δ¯ E

Xⁿ₂{˜ρm₁,m₂}

#)

®

X₁ⁿE,X₂ⁿ

n(E12)o

= E

X₁ⁿ,Xⁿ₂

<

:

X

m⁰₁6=m₁,m⁰₂6=m₂

Trh

Pm⁰₁,m⁰₂ρ˜m₁,m₂

i=

;

=¬ X

m⁰₁6=m₁,m⁰₂6=m₂

Tr

"

X₁ⁿE,X₂ⁿ

n

P_m⁰_1,m⁰₂o

Xⁿ₁E,X₂ⁿ

{˜ρm1,m2}

#

≤­ X

m⁰₁6=m₁,m⁰₂6=m₂

Tr

"

X₁ⁿE,X₂ⁿ

n P_m0

1,m⁰₂

oρ¯¯^⊗n

#

= X

m⁰₁6=m₁,m⁰₂6=m₂ X₁ⁿE,X₂ⁿ

nTrh

Πⁿ_m⁰₁Πⁿ_m⁰_1,m⁰₂Πⁿ_m⁰₁ Πⁿρ,δ¯¯ ρ¯¯^⊗nΠⁿρ,δ¯¯

io

≤®2^{−n[H(B)−δ]} X

m⁰₁6=m₁,m⁰₂6=m₂ X₁ⁿE,X₂ⁿ

n Trh

Πⁿ_m⁰₁ Πⁿ_m⁰_1,m⁰₂Πⁿ_m⁰₁Πⁿρ,δ¯¯

io

≤ 2¯ ^{−n[H(B)−δ]} X

m⁰₁6=m₁,m⁰₂6=m₂

Trh Πⁿ_m1,m0

2

i

≤ 2° ^{−n[H(B)−δ]}2^{n[H(B|X1X2)+δ]} X

m⁰₁6=m₁,m⁰₂6=m₂

1

≤ 2^{−n[I(X1X2;B)−2δ]}|M1||M2|. (3)

I If we want the error terms (E1), (E2) and (E12) to be small, we have to choose the size of the message sets appropriately.

I For example, if we choose the rateR1=I (X1;B|X2) − 3δ we will have:

n→∞lim E

X₁ⁿ,X₂ⁿ{(E1)} = lim

n→∞|M1| 2^{−n[I(X1;B|X2)−2δ]}

= lim

n→∞2^nR1 2^{−n[I(X1;B|X2)−2δ]}

= lim

n→∞2^{n[I(X1;B|X2)−3δ]}2^{−n[I(X1;B|X2)−2δ]}

= lim

n→∞2^−nδ= 0. Thus, any rate pair (R1, R2)which satisfies:

R1 ≤ I(X1;B|X2)θ

R2 ≤ I(X2;B|X1)θ

R1+R2 ≤ I(X1X2;B)θ

Summary of new techniques

For two-sender QMAC:

I Projector sandwich: Pm1,m2 ≡ Πⁿ_ρ,δ¯ Πⁿ_m1 Πⁿ_m1,m2Πⁿ_m1 Πⁿ_ρ,δ¯ .

I Smoothing trick: ˜ρm1,m2 ≡ Πⁿ_m2 ρm1,m2 Πⁿ_m2. Three-sender simultaneous decoding conjecture:

There exists a simultaneous decoding POVM Λm1,m2,m3 for the three-sender quantum multiple access channel.

Result would follow if we hadsimultaneous smoothing(Omar’s talk):

I Detection POVM:

Λ_m1,m2,m2 ≡ (P Π...)^−1/2Π_m1,m2,m3(P Π...)^−1/2.

I Use smooth state ˜ρm1,m2,m3 which corresponds to channel outputρm₁,m₂,m₃ smoothed by Πⁿ_m1,Πⁿ_m2,Πⁿ_m3,

Πⁿ_m1,m2,Πⁿ_m1,m3,Πⁿ_m2,m3 and Πρ,δ¯¯ .

Ivan Savov QNIT

Quantum interference channel

Quantum interference channel

(X1× X2, N^X1X2→B1B2(x1, x2) ≡ρ^B_x1¹_,x^B₂², H^B1⊗ H^B2),

x1

Tx1

x2

Tx2

ρB1 x1,x2 Rx1

ρB2 x1,x2 Rx2

n · N^{X1X2→B1B2 (1−)}−→ nR1· [c¹→ c¹] + nR2· [c²→ c²]

IDEA: Decoding the interference signals instead of treating them as noise.

IDEA: Use QMAC results.

Ivan Savov QNIT

Special case: IC with very strong interference

Theorem 5.1: The channel’s capacity region is given by:

S

p_QX1X2(R1, R2) ∈ R²+such that:

R1 ≤ I (X1;B1|X2Q)_θ, R2 ≤ I (X2;B2|X1Q)_θ whereθ^QX1X2B is the following state:

X

x₁,x₂,q

pQ(q)pX1|Q(x1|q) pX2|Q(x2|q) |qihq|^Q⊗|x1ihx1|^X1⊗|x2ihx2|^X2⊗ρ^B_x1,x2.

IDEA: Decoding the interfering signal before own signal.

Special case: IC with strong interference

Theorem 5.2: The channel’s capacity region is given by:

S

p_QX1X2(R1, R2) ∈ R²+such that:

R1 ≤ I(X1;B1|X2Q)_θ, R2 ≤ I(X2;B2|X1Q)_θ, R1+R2 ≤ min I(X1X2;B1|Q)_θ

I(X1X2;B2|Q)_θ



whereθ^QX1X2B = X

x1,x2,q

pQ(q)pX1|Q(x1|q) pX2|Q(x2|q) |qihq|^Q⊗|x1ihx1|^X1⊗|x2ihx2|^X2⊗ρ^B_x1,x2.

IDEA: Receivers use QMAC simultaneous decoding.

Ivan Savov QNIT

IDEA: Partialinterference cancellation.

I Split msg. into “personal” and “common” partsm1= (m1p, m1c).

I Generate separate random codebooks for each message U1ⁿ(m1p)andW1ⁿ(m1c), and mix them: X1i=f1(U1i, W1i).

X1

X2

B1

B2 W1

W2 U1

U2 f1

f2

Ŵ1Û1 Ŵ2 Ŵ2Û2 Ŵ1 ρ^B_x1¹_,x^B₂²

I Receiver 1 will decodem1p,m1c,m2candignoresm2p.

I Achieving the rates of the Han-Kobayashi rate region RHK(N ) requires thethree-sender quantum simultaneous decoder.

I Workaround: Can show that the quantum Chong-Motani-Garg rate region R (N )is achievable [Sen12, Sav12]. Note that:

Quantum broadcast channel

Ivan Savov QNIT

(X, N^X→B1B2 ≡ ρ^B_x^1B2, H^B1B2)

Tx x

ρ^Bx¹

Rx1

ρ^Bx²

Rx2

n·N^{X→B1B2 (1−)}−→ nR1·[c → c]^X→B1

| {z }

for Rx1 only

+nR·[c → cc]^X→B1B2

| {z }

common message

+nR2·[c → c]^X→B2

| {z }

for Rx2 only

Tx x

ρ^Bx¹

Rx1

ρ^Bx²

Rx2

n · N^{X→B1B2 (1−)}−→ nR ·[c → cc]^X→B1B2

| {z }

common message

+ nR1· [c → c]^X→B1

| {z }

superimposed message

Theorem 6.1: A rate pair (R, R1)is achievable for the quantum broadcast channel {ρ^B_x^1B2} if it satisfies the following inequalities:

R1≤ I(X; B1|W )θ, R1+R ≤ I(X; B1)_θ,

R ≤ I(W ; B2)_θ,

where the above information quantities are with respect to a state θ^{W XB1B2} =P

w,xpW(w)pX|W(x|w) |wihw|^W ⊗ |xihx|^X⊗ ρ^B_x^1B2. IDEA: Simultaneous decoding used by Receiver 1.

Quantum relay channel

Quantum relay channel

X × X1, N^XX1→B1B(x, x1) ≡ρ^B_x,x^1B₁, H^B1⊗ H^B

Source x

ρB1 x,x1

Relay x1

ρBx,x1 Destination

n · N^{XX1→B1B (1−)}−→ nR · [c → c]^X→B

IDEA: Combine the information from the Source and the Relay.

IDEA: Transmission of messages can happen during multiple blocks.

Ivan Savov QNIT

Conclusion

Summary

I Showed that we can adapt some classical multiuser coding techniques to the setting of c-q channels.

I Discussed the obstacles that stand in the way of a general theory of a multi-user classical-quantum communication.

I Described new constructions and ad hoc tricks which can be applied for the case of two senders.

I The general problem withn ≥ 3 senders will require different techniques.

Ivan Savov QNIT

Further research

I Simultaneous decoding of three-sender QMAC?

I Simultaneous smoothing approach? (Omar’s talk)

I Auxiliary “tag space” approach of Pranab Sen.

I Extend other problems in network information theory to the classical-quantum setting:

I Joint source-channel coding problem. See M. M. Wilde and I.

Savov. Joint source-channel coding for a quantum multiple access channel. February 2012. arXiv:1202.3467.

I Entanglement-assisted communication?

1. O. Fawzi, P. Hayden, I. Savov, P. Sen, and M. M. Wilde.

Classical communication over a quantum interference channel.

IEEE Transactions on Information Theory, 58(6):3670-3691, June 2012. arXiv:1102.2624.

2. I. Savov and M. M. Wilde. Classical codes for quantum broadcast channels.

IEEE International Symposium on Information Theory, 2012.

arXiv:1111.3645.

3. I. Savov, M. M. Wilde, and M. Vu. Partial decode-forward for quantum relay channels.

IEEE International Symposium on Information Theory, 2012.

arXiv:1201.0011.

4. I. Savov. Network information theory for classical-quantum channels.

PhD Thesis. McGill University. July 2012. arXiv:1208.4188.

The end

Thank you for your attention!