coherent classical communication aram harrow (mit) quant-ph/0307091
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CoherentClassicalCommunication
Aram Harrow (MIT)quant-ph/0307091
Outline
• What is coherent classical communication (CCC)?
• Where does CCC come from?• What is CCC good for? • Remote state preparation with CCC• Noisy CCC and applications
beyond qubits and cbits
Let {|xi}x=0,1 be a basis for C2.• [q!q]: |xiA!|xiB (qubit)• [c!c]: |xiA!|xiB|xiE (cbit)
• [qq]: |i=2-1/2x|xiA|xiB (ebit)• [cc]: 2-1/2x|xiA|xiB|xiE (rbit)
• [[c!c]]: |xiA!|xiA|xiB (coherent cbit)
(notation due to Devetak and Winter, quant-ph/0304196)
simple resource relations
Trivial relations:
[q!q] > [[c!c]] > [c!c] > [cc]
[q!q] > [[c!c]] > [qq] > [cc]
Teleportation (TP):
2[c!c] + [qq] > [q!q]
Super-dense coding (SDC):
[q!q] + [qq ] > 2[[c!c]] (coherent output!)
distributed unitary gatesTheorem:
If U is a unitary gate on HA HB such that
U + e [qq] > C![c!c] + CÃ[cÃc] (A)
then U + e [qq] > C![[c!c]] + CÃ[[cÃc]] (A).
Examples:CNOTAB|xiA|0iB=|xiA|xiB(HZa I)CNOTAB(Xa Zb)2-1/2x|xiA|xiB=|biA|aiB
Note:1. The proof requires careful accounting of ancillas.
2. It also holds for isometries (e.g. |xiA!|xiA|xiB)
Teleportation
H
X Z
2 [c!c] + 1 [qq] > 1 [q!q]
+ 2 [cc]
uniformly
random
Before measuring, the state is 2-1ab|ai|biAZaXb|iB.
Teleportation with coherent communication
H
X Z
2 [[c!c]] + 1 [qq] > 1 [q!q]+ 2 [qq]
coherent
classical
comm.
2-1ab|abiAZaXb|iB
2-1ab|abiA|abiBZaXb|iB
the power of coherent cbits
Teleportation with recycling:2 [[c!c]] + 1 [qq] > 1 [q!q]+ 2 [qq]2 [[c!c]] > 1 [q!q]+ 1 [qq] (C)
Super-dense coding:1 [q!q]+ 1 [qq] > 2 [[c!c]] (C)
Therefore:2 [[c!c]] = 1 [q!q]+ 1 [qq] (C)Teleportation and super-dense coding
are no longer irreversible.
Recycling in the remote CNOT
H
=
[c!c] + [cÃc] + [qq] > CNOT [Gottesman, quant-ph/9807006][[c!c]] + [[cÃc]] + [qq] > CNOT + 2 [qq][[c!c]] + [[cÃc]] > CNOT + [qq] (C)
the power of a CNOT
Making a remote CNOT coherent:[[c!c]] + [[cÃc]] > CNOT + [qq] (C)
Using a CNOT for bidirectional communication:(HZa I)CNOTAB(Xa Zb)2-1/2x|xiA|xiB=|biA|aiBCNOT + [qq] > [[c!c]] + [[cÃc]]
Combined: CNOT + [qq] = [[c!c]] + [[cÃc]] (C)2 CNOT = 2 [[c!c]] + 2 [[cÃc]] – 2 [qq]
= [q!q] + [qÃq] = SWAP (C)
Remote State Preparation
1 cbit + 1 ebit > 1 remote qubit
Given |di and a description of 2Cd, Alice can prepare in Bob’s lab with error by sending him log d + O(log (log d)/2) bits.
[Bennett, Hayden, Leung, Shor and Winter, quant-ph/0307100]
definitions of remote qubits
What does it mean for Alice to send Bob n remote qubits?
She can remotely prepare one of
RSP lemmaFor any d and any >0, there exists n=O(d log d/2) and a set of d x d unitary gates R1,…,Rn such that for any ,
Use this to define a POVM:
RSP protocol
k
Neumark’s theorem:any measurement can be made
unitary
k
UM
Entanglement recycling in RSP
UA
discard
coherent
classical
communication
of log n bits
Implications of recycled RSP
1 coherent cbit > 1 remote qubit (with catalysis)
Corollary 1: The remote state capacity of a unitary gate equals its classical capacity.
Corollary 2: Super-dense coding of quantum states (SDCQS)
1 qubit + 1 ebit > 2 remote qubits (with catalysis)
(Note: [Harrow, Hayden, Leung; quant-ph/0307221] have a direct proof of SDCQS.)
RSP of entangled states (eRSP)
Let E={pi,i} be an ensemble of bipartite pure states. Define S(E)=S(ipiTrAi), E(E)=ipiS(TrAi), (E)=S(E)-E(E).
eRSP: (E) [c!c] + S(E) [qq] > E (A) [BHLSW]
make it coherent: (E) [[c!c]] + E(E) [qq] > E (A)
use super-dense coding:
(E)/2 [q!q] + (E(E)+ (E)/2) [qq] > E (A)
Unitary gate capacities
Define Ce to be the forward classical capacity of U assisted by e ebits of entanglement per use, so that
1 use of U + e [qq] > Ce [c!c] (A)
(In [BHLS; quant-ph/0205057], this was proved for e=1.)
Solution:
Ce=supE {(UE) - (E) : E(E) - E(UE)6e}
Warmup: entanglement capacity
Define E(U) to be the largest number satisfying
U > E(U) [qq] (A).
Claim: E(U) = sup|i E(U|i) – E(|i)
Proof: [BHLS; quant-ph/0205057]
|i + U > U|i
> E(U|i) [qq] (concentration)
>|i + E(U|i)-E(|i) [qq] (dilution)
Thus: U > E(U|i)-E(|i) [qq] (A)
Coherent HSW coding
Lemma: Let E={pi,i} be an ensemble of bipartite pure states that Alice can prepare in superposition. Then
E > (E) [[c!c]] + E(E) [qq] (A)
Proof: Choose a good code on E n. Bob’s measurement obtains ¼n(E) bits of Alice’s message and determines the codeword with high probability, causing little disturbance. Thus, this measurement can be made coherent. Since Alice and Bob know the codeword, they can then do entanglement concentration to get ¼nE(E) ebits.
Protocol achieving Ce
E + U > UE
> (UE) [[c!c]] + E(UE) [qq] (coherent HSW)
> E + ((UE)-(E)) [[c!c]] + (E(UE)-E(E)) [qq]
(coherent RSP)
Thus, U + (E(E)-E(UE)) [qq] > ((UE)-(E)) [[c!c]] (A)
Quantum capacities of unitary gates
Define Qe(U) to be the largest number satisfying
U + e [qq] > Qe [q!q].
Using 2[[c!c] = 1[q!q] + 1[qq], we find
Summary
• 2 coherent cbits = 1 qubit + 1 ebit• 2 CNOT = SWAP (catalysis)• 1 qubit + 1 ebit > 2 remote qubits
(catalysis)• eSDCQS using /2 qubits and S-/2 ebits.
• Single-letter expressions for Ce and Qe.
• Remote state capacities and classical capacities are equal for unitary gates.
Noisy CCC[Devetak, Harrow, Winter; quant-
ph/0308044]• Two minute proofs of the hashing
inequality and the quantum channel capacity.
• Generalizations of these protocols to obtain the full trade-off curves for quantum channels assisted by a limited amount of entanglement and entanglement distillation with a limited amount of communication.
Noisy CCC: definitionsLet AB be a bipartite state and |iABE its purification.
I(A:B) = H(A) + H(B) – H(E)
I(A:E) = H(A) + H(E) – H(B)
Ic = H(B) – H(E) = ½ (I(A:B) – I(A:E))
If N is a noisy channel, then evaluate the above quantities on (I N)|i, where |i is a purification of Alice’s input .
{qq} = one copy of AB
{q!q} = one use of N
Noisy CCC: applications
Old results:
S(A) [qq] + {q!q} > I(A:B) [[c!c]] [BSST; q-ph/0106052]
{q!q} > Ic [q!q] [Shor; unpublished]
S(A) [q!q] + {qq} > I(A:B) [[c!c]] [HHHLT; q-ph/0106080]
I(A:E) [c!c] + {qq} > Ic [qq] [DW; q-ph/0306078]
New results:
I(A:E)/2 [qq] + {q!q} > I(A:B)/2 [q!q][father]
I(A:E)/2 [q!q] + {qq} > I(A:B)/2 [qq][mother]
A family of quantum protocols
fathermother
hashing inequality [DW]
I(A:B) [c!c] + {qq} > Ic[q!q] [HDW/Burkard]
CE [BSST]noisy SDC [HHHLT]
{q!q}>Ic[q!q]
[Shor]
TP
TP
TP
SDC
[q!q]
>[qq]
SDC
TP= teleportation
SDC = super-dense coding
A gate with asymmetric capacities?
x=0,…,d-1, U2Cd Cd
U |x0i = |xxiU |xxi = |x0iU |xyi = |xyi for xy0.
C1 = log d
C2 > (log d)/2