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Trading quantum and classical resources in data compression

Trading quantum and classical resources in data compression. Patrick Hayden (Caltech) with Richard Jozsa and Andreas Winter (Bristol). quant-ph/0204038 JMP Special Issue. Why study trade-off?. Bits (versus qubits): Easy to make and stabilize Discrete

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Trading quantum and classical resources in data compression

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  1. Trading quantum and classical resources in data compression Patrick Hayden (Caltech) with Richard Jozsa and Andreas Winter (Bristol) quant-ph/0204038 JMP Special Issue

  2. Why study trade-off? • Bits (versus qubits): • Easy to make and stabilize • Discrete • Bits and qubits are different but (sometimes) interchangeable • Basic questions: • How `quantum’ is an ensemble of states? • How much does a qubit cost in bits? • (Dual to Holevo)

  3. Overview • Ancient history (1994) • Hybrid classical-quantum compression • Lower bounds • Trade-off coding • Arbitrarily varying sources • Remote state preparation

  4. Storing qubits in qubits:Schumacher’s theorem Alice Bob

  5. Storing qubits in qubits:Schumacher’s theorem Alice Bob

  6. Storing qubits in qubits:Schumacher’s theorem Alice Bob

  7. Storing qubits in qubits:Schumacher’s theorem

  8. Data processing inequality Alice Bob time

  9. Swapping bits for qubits Alice Bob Barnum, Hayden, Jozsa, Winter: Proc. R. Soc. Lond. A 457:2019, 2001.

  10. Eve Classical side channel is useless for most sources Swapping bits for qubits Alice Bob Barnum, Hayden, Jozsa, Winter: Proc. R. Soc. Lond. A 457:2019, 2001.

  11. Trade-off exists Visible compression Alice Bob

  12. Trade-off anatomy

  13. Example: BB84 ensemble

  14. R R G G G Example: BB84 ensemble

  15. Example: BB84 ensemble

  16. G R Hybrid encodings and mutual information Registers: State label Quantum Classical

  17. Hybrid encodings and mutual information Registers: State label Quantum Classical Conditional ensembles:

  18. Additive lower bound Decompose index register: Chain rule: Relate to single-copy encoding map

  19. Perfect-fidelity encodings Single-copy No compression Optimization over classical channels

  20. Trade-off coding

  21. 1-p 1 1 p p 2 2 1-p How? An example

  22. How? Some details Note that because B register states are pure: This is an average over von Neumann entropies, conditioned on the value of classical register C. So, given I=i1i2…in Alice generates J=j1j2…jn according to p(J|I)=p(j1|i1)p(j2|i2)…p(jn|in). This partitions 1,2,…,n into blocks sharing the same value of j. Alice then performs Schumacher compression on these separate blocks. Problem: Alice needs to communicate J to Bob.

  23. Er(I) Noiseless channel m shared random bits Dr(Er(I)) Reverse Shannon theorem n uses of Noisy channel I p(j|i) J Bennett, Shor, Smolin, Thapliyal: quant-ph/0106052

  24. Derandomization Decoding fidelity can be written as an average over values of shared bits: So there must be a choice r0 for which More refined analysis: using O(log n) shared random bits, we can achieve high fidelity encoding-decoding for all typical input strings I.

  25. Example:

  26. 1-p 1 1 p p 2 2 1-p Optimal protocol For a fixed R, find p such that the binary symmetric channel has capacity R. Alice sends Bob a degraded version of I that looks like it has gone through n copies of this noisy channel.

  27. Ensemble with orthogonality

  28. Uniform qubit ensemble Devetak and Berger PRL 87(9):197901, 2001

  29. What does a qubit cost in bits? We still don’t know! Must eliminate dependence on input distribution pi

  30. AVS condition: Arbitrarily varying sources Regular fidelity condition:

  31. Observation #2: Don’t need all that many probability distributions # possible values of (N(1|I),N(2|I),…,N(m|I)) is  (n+1)m Alice sends value using O(log n) bits then compresses as if the distribution were given by AVS coding Observation #1: Every input is typical for some probability distribution Is typical for

  32. AVS conclusion

  33. Absolute bit-qubit trade-off Devetak and Berger PRL 87(9):197901, 2001

  34. Measure: Alice need only tell Bob when she has succeeded: Asymptotically requires only 1 bit per signal. (But lots of entanglement.) Remote state preparation BDSSTW Phys. Rev. Lett. 87:077902, 2001

  35. 1 ebit + 1 cbit sufficient to send 1 qubit asymptotically Combine with trade-off coding: R cbits Q*(R) qubits Q*(R) ebits + Q*(R) cbits R cbits Optimal RSP Previous slide: Tons ebits + 1 cbit sufficient to send 1 qubit Teleportation: 1 ebit + 2 cbits sufficient to send 1 qubit Bennett, Hayden, Leung, Shor, Winter (In preparation.)

  36. Entangled RSP Minimal entanglement for preparing

  37. Conclusions • Basic question: how are the basic resources (bits, qubits, ebits) related to each other? • QIT surprise: something we can actually calculate • Future directions: • Three-way trade-off • Study simultaneous CQ capacity of a quantum channel

  38. Bits in qubits: Holevo bound Alice Bob

  39. Beating the adversary Conclusion: Just need to know how to compress the uniform ensemble Shared random bits

  40. Qubits in qubits: Lower bound Alice Bob

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