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Randomizing Quantum States

Randomizing Quantum States. Charles Bennett Patrick Hayden Debbie Leung Peter Shor Andreas Winter RSP: quant-ph/0307100 RAND: quant-ph/0307104. 1 bit of key per bit of message necessary and sufficient [Shannon]. One-time pad. Message. Shared key. Eavesdropper learns nothing:.

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Randomizing Quantum States

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  1. Randomizing Quantum States Charles Bennett Patrick Hayden Debbie Leung Peter Shor Andreas Winter RSP: quant-ph/0307100 RAND: quant-ph/0307104

  2. 1 bit of key per bit of message necessary and sufficient [Shannon] One-time pad Message Shared key

  3. Eavesdropper learns nothing: Lower bound on key length: [BR,AMTW 2000] Private quantum channels

  4. MAIN RESULT: Given >0, there exists a choice of unitaries {Uk}, k=1,…,n such that the map is ε-randomizing, with Relax security criterion A CPTP map is ε-randomizing if for all states ,

  5. Security: 1 bit of key/qubit (1+2α) bits of key/qubit Approximate PQC Can encrypt a quantum state using 1 secret random bit per encrypted qubit asymptotically.

  6. Remote state preparation:Non-oblivious teleportation A circuit that needs no introduction:

  7. Remote state preparation:Non-oblivious teleportation Allow Alice to perform an arbitrary φ-dependent measurement

  8. From randomization to RSP Circuit for teleportation: l qubits k: 2l bits Before receiving k Bob knows nothing: (Private quantum channel!)

  9. From randomization to RSP Circuit for remote state preparation: l qubits k:l+o(l) bits Before receiving k, Bob knows nothing: (Approximate private quantum channel!)

  10. Knowledge is power All you need in this life is ignorance and confidence, and then success is sure. -Mark Twain Oblivious Alice (Teleportation): 1 ebit + 2 cbits per 1 qubit sent There is no knowledge that is not power. -Ralph Waldo Emerson Non-Oblivious Alice (RSP): 1 ebit + 1 cbit per 1 qubit sent Knowledge is power, if you know it about the right person. -Ethel Watts Mumford

  11. Randomization and nonlocality Does an ε-randomizing R destroy all correlations with the environment? True for separable ρ (easy). Not true for entangled inputs!

  12. Fidelity with maximally mixed state small: Rank argument Recall ε-randomizing map: Act on half of a maximally entangled state: has rank no more than n~d log d ~

  13. X Y Alice and Bob implement an LOCC measurement. Characterizing leftover correlations What does randomization map do to entangled inputs? R Charlie prepares maximally entangled state k then randomizes it.

  14. Separable state! k info randomized Conclusion: LOCC cannot distinguish strong and weak randomization Characterizing leftover correlations What does randomization map do to entangled inputs? R X Y Charlie prepares maximally entangled state k then randomizes it. Alice and Bob implement an LOCC measurement.

  15. (conjectured) Separable states Distillable states Product states Non-distillable states Detecting entanglement

  16. Strategy: • Select the unitaries {Uk} at random from Haar measure. • Show they are ε-randomizing with probability >0. Ingredients: • Large deviation estimate • Discretization Randomization proof ideas

  17. Will be concerned with distribution of: where In particular, (Exponential convergence to the mean) Large deviation estimate

  18. Will allow replacement of “all states” by “all states in M” in definition of ε-randomization. Discretization Find anε-net M: For all pure states φ, there exists a state φ’ in M such that Cd

  19. defn ε-net union bound large deviations Combine (Not as bad at it looks)

  20. ? ? Quantum data hiding GOAL: Charlie hides a bit from Alice and Bob, secure against LOCC RESULT: There exist bipartite n-qubit states hiding a bit with security 2-(n-1). [DLT, 2001]

  21. Security: Decodability: Hiding qubits GOAL: Charlie hides qubits from Alice and Bob, secure against LOCC [dV H T, 2002]

  22. Strategy

  23. Strategy

  24. Similar calculation: Choose Security Hinges on showing (Rank one projectors)

  25. Negligible overlaps between subspaces UkS if Decodability

  26. It is possible to hide a space of dimension in a bipartite system where each of the subsystem has dimension d. Asymptotically: 1 hidden qubit per 2 physical qubits. Previous best: ~ 1 hidden qubit per ~75 physical qubits. Data hiding summary

  27. Conclusion • Encryption of quantum states using 1 bit of shared key per encrypted qubit • Remote state preparation using 1 bit of communication and 1 ebit per qubit sent • Physical operation that destroys classical correlations but not quantum • Data hiding of 1 qubit in 2 physical qubits

  28. The End

  29. Act I LOCC data hiding for quantum states (quant-ph/0207147) Act II From RSP to PQC to data hiding Overview

  30. A few years ago in a lab moderately far away...

  31. ? ? 0 1 2 0 1 2 Not always possible: Nonlocality without entanglement [BDFMRSSW, 1999]

  32. Prepare superpositions of well hidden states? Hiding a qubit: First attempt TASK: Hide an arbitrary quantum state

  33. Hiding a qubit: First attempt TASK: Hide an arbitrary quantum state PROBLEM: Data hiding with pure states is impossible! (So much for superpositions.) [WSHV,2000]

  34. 2nd simplest idea THE PLAN: Use classical hidden bits as key to randomize a qubit PROPERTIES: 1) can be recovered using quantum communication 2) Naïve attacks fail (AB1 to find key then rotate B2) PROBLEM: Alice and Bob can attack AB1B2

  35. Actually, not a problem Any method to learn about by LOCC will provide a method to defeat the original cbit hiding scheme. Will argue the contrapositive: Assume there is an LOCC operation L (with output on Bob’s system alone) and two input states to the hiding map E such that

  36. CLAIM: Not all Li can be the same TPCP map. If they were, then by linearity: This says that L would never reveal any information about the input state, violating the hypothesis that L defeats the qubit hiding scheme. Minor algebra

  37. The attack: 1) Bob prepares a local maximally entangled state on B2B3 2) Alice and Bob apply L to AB1B2 3) Bob performs a measurement on B2B3 By Choi, there is a measurement that can partially distinguish L0 andLk Defeating the cbit hiding Conclusion from previous slide: there is a k such that L0Lk

  38. Imperfect hiding Wish to limit distinguishability through LOCC: For all input states and attacks. If the original 2n bit hiding scheme has security , then  <2n+1 . Not so bad: security of classical hiding schemes appears to improve exponentially with number of qubits used.

  39.  Authorized sets can recover the secret using quantum communication Multipartite cbit hiding A E B D C  With LOCC alone the five parties cannot learn i All monotonic access structures are possible: [Eggeling, Werner 2002]

  40. This authorization structure is impossible due to no-cloning! Multipartite qubit hiding A E B D C  With LOCC alone the five parties cannot learn  Authorized sets can recover the secret using quantum communication

  41. Problem for generalizing construction: B must be in all authorized sets Multipartite qubit hiding A E B D C  With LOCC alone the five parties cannot learn  Authorized sets can recover the secret using quantum communication

  42. Quantum secret sharing A e.g. ((2,3)) threshold scheme E B D C Secure against quantum communication in unauthorized sets but secret can be recovered by quantum communication in authorized sets. All monotonic threshold schemes not violating no-cloning [CGL,1999] All monotonic schemes not violating no-cloning [G,2000]

  43. Hiding distributed quantum data A Logical Pauli operators Multipartite cbit hiding states E B D C Resulting state provides strengthening of quantum secret sharing:  Secure against classical communication between all parties  Secure against quantum communication in unauthorized sets  Secret can be recovered only by quantum communication in authorized sets. All monotonic schemes not violating no-cloning

  44. Act II Fig. 1: Glimpse of a master magician’s workshop

  45. On the meaning of “ “ For any probability density P() on states in Cd and >0 there exists a choice of unitaries {Us}, s=1,…,S such that and Compare to the perfect private quantum channel: To achieve =0 requires log M = 2 log d.

  46. Another version There exists a choice of unitaries {Ups}, p=1,…,P, s=1,…,S such that for all states  in Cd and Can randomize every n-qubit state using 1 secret random bit and 1 public random bit per qubit.

  47. Consequences • Universal remote state preparation with only 1 ebit + 1 cbit per qubit • (No shared random bits necessary) • Weakly randomized maximally entangled states indistinguishable from maximally mixed states using LOCC • (Not just 1-way LOCC as sketched earlier)

  48. Application to data hiding R Consider ensemble of randomized states with V chosen using Haar measure Rank bound on entropy So we can do coding to get about n hidden bits using nxn bipartite states!

  49. Glyph collection

  50. Faction 1 Destroying classical correlations requires only 1 rbit per qubit Destroying quantum correlations requires 2 rbits per qubit Faction 2 Randomizing an arbitrary pure quantum state requires 1 public rbit and 1 secret rbit per qubit Competing visions

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