quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf ·...
TRANSCRIPT
![Page 1: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/1.jpg)
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
![Page 2: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/2.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Outline for section 1
1 Rejection sampling approach
2 A source coding scheme
3 A coherent (quantum) rejection sampling
4 Applications
5 Conclusion
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 3: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/3.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 4: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/4.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 5: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/5.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Outline for section 2
1 Rejection sampling approach
2 A source coding scheme
3 A coherent (quantum) rejection sampling
4 Applications
5 Conclusion
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 6: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/6.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The communication task
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 7: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/7.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The communication task
{p(x), x}, x ∈ {1, 2, . . .N}
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 8: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/8.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The communication task
{p(x), x}
x
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 9: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/9.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The communication task
{p(x), x}
x
x
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 10: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/10.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x xp(x)
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 11: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/11.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 12: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/12.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x x ′
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 13: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/13.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x
x ′′
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 14: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/14.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x
x
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 15: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/15.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x
x
3
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 16: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/16.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x
x
3
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 17: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/17.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
A rejection sampling approach
{p(x), x}
x
x
3
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 18: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/18.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Communication cost
Worst case cost: For error ε, Hε0(X ).
Average cost:∑
x p(x) log 1p(x) + 1.
Another scheme for Huffman coding!
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 19: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/19.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 20: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/20.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Outline for section 3
1 Rejection sampling approach
2 A source coding scheme
3 A coherent (quantum) rejection sampling
4 Applications
5 Conclusion
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 21: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/21.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Task: Quantum state transfer
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 22: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/22.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Task: Quantum state transfer
R
A
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Task: Quantum state transfer
R
A |Ψ〉RA=∑
x
√p(x) |x〉R |x〉A
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 24: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/24.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Task: Quantum state transfer
R
A|Ψ〉RA
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 25: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/25.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Trouble: monogamy of entanglement
R
A
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 26: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/26.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Trouble: monogamy of entanglement
R
A
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 27: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/27.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Trouble: monogamy of entanglement
R
A
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 28: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/28.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 29: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/29.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The distributions with Bob
{p(x), x}
x ′
x ′′
x
3
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 30: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/30.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The distributions with Bob
{p(x), x}
x ′
x
2
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 31: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/31.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The desired quantum state with Bob and Reference
R
A
3
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 32: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/32.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The desired quantum state with Bob and Reference
R
A
2
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 33: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/33.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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 .
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 34: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/34.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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 .
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 35: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/35.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
The desired quantum state with Bob and Reference
R
A
R
A
R
A
p1 + p2 + p3 + . . .
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 36: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/36.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Trouble: Global quantum state doesn’t look the same
R
A
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 37: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/37.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Resolution: Uhlmann’s theorem
R
A
R
A
R
p1 + p2 + . . . ε≈
Alice can apply an isometry guaranteed by Uhlmann’s theorem.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 38: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/38.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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
).
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 39: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/39.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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
).
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 40: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/40.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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
).
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Convex-split Lemma: In pictures
R
B1
B2
BN
ε≈ 1
N + 1N + 1
N
If logN ≥ Dmax(ΨRB‖ΨR ⊗ σB) + log 1ε .
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 43: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/43.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 44: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/44.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) =1N
∑i (D(ΨRBj ⊗ σ−j‖ΨR ⊗ σ)−D(ΨRBj ⊗ σ−j‖τ))
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) =1N
∑i (D(ΨRBj‖ΨR ⊗ σBj )−D(ΨRBj ⊗ σ−j‖τ))
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) =
D(ΨRB‖ΨR ⊗ σB)− 1N
∑i D(ΨRBj ⊗ σ−j‖τ)
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)− 1
N
∑i D(ΨRBj‖τRBj )
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)−D(ΨRB1‖τRB1)
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)−D(ΨRB1‖τRB1)
τRB1 =1N ΨRB1 + (1− 1
N )ΨR ⊗ σB1 .
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)−D(ΨRB1‖τRB1)
τRB1 �2k
N ΨR ⊗ σB1 + (1− 1N )ΨR ⊗ σB1 .
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 51: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/51.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)−D(ΨRB1‖τRB1)
τRB1 �(1 + 2k
N )ΨR ⊗ σB1 .
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 52: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/52.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Proof
D(τ‖ΨR ⊗ σ) ≤D(ΨRB‖ΨR ⊗ σB)−D(ΨRB1‖ΨR ⊗ σB) + log(1 + 2k
N )
Done.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state splitting and merging
R
A
C
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
![Page 54: Quantum message compression with applicationsims.nus.edu.sg/events/2017/wbey/files/anurag.pdf · Rejection sampling approach A source coding scheme A coherent (quantum) rejection](https://reader034.vdocuments.us/reader034/viewer/2022050103/5f423f803c458055fa0694dd/html5/thumbnails/54.jpg)
Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Outline for section 4
1 Rejection sampling approach
2 A source coding scheme
3 A coherent (quantum) rejection sampling
4 Applications
5 Conclusion
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state splitting and merging
R
AC
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state splitting and merging
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state splitting and merging
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 ).
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state splitting and merging
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 ).
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state redistribution
R
A
C
B
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state redistribution
R
A
C
B
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state redistribution
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].
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state redistribution
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].
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Application: Quantum state redistribution
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].
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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.
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
Outline for section 5
1 Rejection sampling approach
2 A source coding scheme
3 A coherent (quantum) rejection sampling
4 Applications
5 Conclusion
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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].
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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].
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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].
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression
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Rejection sampling approachA source coding scheme
A coherent (quantum) rejection samplingApplicationsConclusion
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].
Anurag Anshu1, Vamsi Krishna Devabathini1, Rahul Jain1,2, Naqueeb Ahmad Warsi1,3Quantum message compression