quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf ·...

Rejection sampling approachA source coding scheme

A coherent (quantum) rejection samplingApplicationsConclusion

Quantum message compression with applications

Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2,Naqueeb Ahmad Warsi1,3

1. Centre for Quantum Technologies, NUS, Singapore2. MajuLab, UMI 3654, Singapore.

3. School of Physical and Mathematical Sciences, Nanyang TechnologicalUniversity, Singapore and IIITD, Delhi.

arXiv:1410.3031arXiv:1702.02396

July 25, 2017

Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression



Outline for section 1

1 Rejection sampling approach

2 A source coding scheme

3 A coherent (quantum) rejection sampling

4 Applications

5 Conclusion




Simple example

Given samples from a distribution p(x), x ∈ {1, 2, . . .N}.You want to see 7.

Procedure: Keep sampling till you see 7.

Number of samples required ≈ 10p(7) to get error down to e−10.




General case

Given samples from p(x).

You want to produce samples from q(x).

Similar procedure that looks at profile of p and q.








4 Applications

5 Conclusion




The communication task





{p(x), x}, x ∈ {1, 2, . . .N}





{p(x), x}

x





{p(x), x}

x

x




A rejection sampling approach

{p(x), x}

x xp(x)





{p(x), x}

x





{p(x), x}

x x ′





{p(x), x}

x

x ′′





{p(x), x}

x

x





{p(x), x}

x

x

3




Communication cost

Worst case cost: For error ε, Hε0(X ).

Average cost:∑

x p(x) log 1p(x) + 1.

Another scheme for Huffman coding!




Generalizations and applications

Considered in [Jain,Radhakrishnan, Sen:2003,2005] (classicaland classical-quantum), [Harsha, Jain, McAllester,Radhakrishnan, 2010] (classical), [Braverman and Rao, 2011](classical).

These works solve a more general task where Alice receives Xand wishes to send Mx , and Bob may have side information Y .

Application to direct-sum problem in communicationcomplexity.








4 Applications

5 Conclusion




Task: Quantum state transfer





R

A





R

A |Ψ〉RA=∑

x

√p(x) |x〉R |x〉A





R

A|Ψ〉RA




Trouble: monogamy of entanglement

R

A




Resolution: Look back at rejection sampling approach

After succeeding at some index, Alice has induced the desireddistribution on Bob’s share of random variable at that index.

After receiving the message, Bob simply picks up the randomvariable at that index.




The distributions with Bob

{p(x), x}

x ′

x ′′

x

3




The distributions with Bob

{p(x), x}

x ′

x

2




The desired quantum state with Bob and Reference

R

A

3





R

A

2




Resolution: Add steering

Steering: Alice can induce desired ensemble on Bob andReference’s registers if she has an appropriate purification.

Suppose desired ensemble is {pi ,ΦiRB}i .

If Alice had the following purification:∑

i

√pi |i〉I

∣∣Φi⟩RAB

:

Measure I .Obtain i with probability pi .Conditioned on i , Bob and Reference have Φi

RB .





R

A

R

A

R

A

p1 + p2 + p3 + . . .




Trouble: Global quantum state doesn’t look the same

R

A




Resolution: Uhlmann’s theorem

R

A

R

A

R

p1 + p2 + . . . ε≈

Alice can apply an isometry guaranteed by Uhlmann’s theorem.




Convex-split Lemma

Let ΨRB , σB be quantum states, k = Dmax(ΨRB‖ΨR ⊗ σB).

Consider the following quantum state

τRB1B2...BN=

1

N

N∑j=1

ΨRBj⊗ σB1 ⊗ σB2 . . .⊗ σBj−1

⊗ σBj+1. . .⊗ σBN

Then,

D(τRB1B2...BN‖ΨR ⊗ σB1 ⊗ σB2 . . .⊗ σBN

) ≤ log

(1 +

2k

N

).




Convex-split Lemma: In pictures

R

B1

B2

BN

ε≈ 1

N + 1N + 1

N

If logN ≥ Dmax(ΨRB‖ΨR ⊗ σB) + log 1ε .




Proof

A simple fact:

Let ρ =∑

i piρi . Then

D(ρ‖θ) =∑i

pi (D(ρi‖θ)−D(ρi‖ρ)).

Recall: τ = 1N

∑Nj=1 ΨRBj

⊗ σ−j .σ−j := σB1 ⊗ σB2 . . .⊗ σBj−1 ⊗ σBj+1 . . .⊗ σBN

.σ := σB1 ⊗ σB2 . . .⊗ σBN




Proof

D(τ‖ΨR ⊗ σ) =1N

∑i (D(ΨRBj ⊗ σ−j‖ΨR ⊗ σ)−D(ΨRBj ⊗ σ−j‖τ))




Proof

D(τ‖ΨR ⊗ σ) =1N

∑i (D(ΨRBj‖ΨR ⊗ σBj )−D(ΨRBj ⊗ σ−j‖τ))




Proof

D(τ‖ΨR ⊗ σ) =

D(ΨRB‖ΨR ⊗ σB)− 1N

∑i D(ΨRBj ⊗ σ−j‖τ)




Proof

D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)− 1

N

∑i D(ΨRBj‖τRBj )




Proof

D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)−D(ΨRB1‖τRB1)




Proof


τRB1 =1N ΨRB1 + (1− 1

N )ΨR ⊗ σB1 .




Proof


τRB1 �2k

N ΨR ⊗ σB1 + (1− 1N )ΨR ⊗ σB1 .




Proof


τRB1 �(1 + 2k

N )ΨR ⊗ σB1 .




Proof

D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)−D(ΨRB1‖ΨR ⊗ σB) + log(1 + 2k

N )

Done.




Application: Quantum state splitting and merging

R

A

C








4 Applications

5 Conclusion





R

AC





Studied in [Horodecki, Oppenheim, Winter 05], [Abeyesinghe,Devetak, Hayden, Winter 06], [Berta 09], [Berta, Christandl,Renner 11],...

Connects to Quantum reverse Shannon theorem, givesmeaning to Quantum conditional entropy.





Achievable in Iεmax(R : C ) + O(log 1ε ), where Iεmax(R : C ) is a

smooth one shot version of mutual information. mutualinformation.

Earlier best known bound, [Berta, Christandl, Renner 11]:Iεmax(R : C ) + log log |C |+ O(log 1

ε )

Converse known: Iεmax(R : C ).




Application: Quantum state redistribution

R

A

C

B





Studied in [Devetak and Yard 06], [Oppenheim 09], [Ye, Bai,Wang 08], [Datta, Hsieh, Oppenheim 16], [Berta, Christandl,Touchette 16], ...

Important application in quantum communication complexity:direct sum theorems [Touchette 16].





We obtain a near optimal quantity that is an optimizationproblem with respect to Bob’s operations.

Further understanding required.

Using position-based decoding [previous talk]:Iεmax(RB : C )−Dε

H(B : C ) + O(log 1ε ).

Improves upon the achievability bound in [Berta, Christandl,Touchette 16].




Entanglement usage

The protocols require a lot of entanglement to begin with.

But entanglement consumed is small:

Quantum state splitting [asymptotic i.i.d.] : Number of ebitsconsumed = H(C )− Q. (Q is communication cost).Quantum state merging [asymptotic i.i.d.] : Number of ebitsconsumed = H(C )− I (B : C )− Q. (Q is communicationcost).

Rest of the entanglement return with fidelity 1− ε.




Application: Port based teleportation

Port-based teleportation: A variant of usual teleportationscheme which used entangled states called ports.

With N ports, average fidelity of transmission is 1− d2

N , whered is dimension of system to be sent.

Using convex-split lemma, reduction in number of ports inpresence of information about state to be sent.




Application: Measurement compression with sideinformation

Alice, Bob, Reference share a quantum state. Alice performsmeasurement and wishes to communicate the outcome toBob. [Winter 04], [Wilde, Hayden, Buscemi, Hsieh 12].

One-shot protocols for this task, asymptotically optimal.

arXiv:1703.02342.




Application: Quantum Slepian-Wolf

Alice, Bob, Charlie, Reference share a joint quantum state.Bob and Charlie with to communicate part of their systems toAlice. Studied in [Abeyesinghe, Devetak, Hayden, Winter 06]when Charlie has no side information and senders send all theregisters.

One shot protocols for this task. Asymptotically optimal whenCharlie has no side information.

arXiv:1703.09961.








4 Applications

5 Conclusion




Conclusion

Convex-split lemma: A quantum version of rejection samplingin communication setting. Helps in encoding.

Position-based decoding: Converse to convex-split lemma.Allows successful decoding.

Applicable to quantum source compression and channelcoding [previous talk] settings.

Other applications in [Majenz, Berta, Dupuis, Renner,Christandl 16], [Wilde 17], [Qi, Wang, Wilde 17].


quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf ·...

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