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Rahul Jain, Yaoyun Shi, Zhaohui Wei, Shengyu Zhang

Efficient protocols for generating bipartite classical distributions and quantum states. Rahul Jain, Yaoyun Shi, Zhaohui Wei, Shengyu Zhang. randomness/entanglement. Shared randomness/entanglement is an important resource in distributive settings.

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Rahul Jain, Yaoyun Shi, Zhaohui Wei, Shengyu Zhang

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  1. Efficient protocols for generating bipartite classical distributions and quantum states Rahul Jain, Yaoyun Shi, Zhaohui Wei, ShengyuZhang

  2. randomness/entanglement • Shared randomness/entanglement is an important resource in distributive settings. • Question: How hard to generate a bipartiteclassical distribution or quantum state?

  3. 3 scenarios • Distribution generation • Distribution approximation • Quantum state approximation

  4. Target distribution: r • Available resource: seed correlation r. • Correlation complexity: Corr(p) = min size(r) • classical: is classical correlation. -RCorr() • quantum: is quantum state. - QCorr() (,)

  5. Target distribution: • Alternative resource: communication • Communication complexity: Comm(p) = min size(message) • classical: number of bits. - RComm(p) • quantum: number of qubits. - QComm(p) (,)

  6. Target distribution: • Question: What are these measures? • [Z’12] RCorr() = RComm(). QCorr() = QComm(), • [Z’12]

  7. For (entry-wise) nonnegative matrices, we can define more variants. • Nonnegative rank • Extensively-studied in linear algebra and engineering. Many connections to (T)CS.

  8. Quantum complexity • [Z’12] • [Z’12] . [LKZ’11,FMPTW’12] . • [This paper] • Improve previous lower bound to .

  9. r psd-rank n m r • Defined in [FMPTW’12], and showed to characterize the communication complexity of the following task: Requirement:

  10. Proof sketch • [Fact] • S-rank: Schmidt-rank. • Now given , let A and B share • -row of, :-column of. • Measuring registers : • Similar for the other direction.

  11. Scenario 2: Distribution approximation • . • A related measure: common information. • “Asymptotic correlation complexity”. (Our : one-shot) • (where . • [This paper] s.t.

  12. Proof sketch • Idea: find a small number (in terms of ) of ’s s.t.. • Random selection suffices? • : distribution of • Chernoff’s bound needs to be bounded. • Apply Markov on • Destroy independence… but can be fixed…

  13. Scenario 3: quantumstate approximation • Approximation by pure state: • [ASTSVW’03] , where ] ]

  14. Pure quantum state • Issue 1: Both bounds can be arbitrarily loose. • Possible to be 1 vs. N/2. (N: dim of A) • Issue 2: Why not allow mixed states to approximate? • [This paper] where

  15. Proof sketch Recall: Uhlmann: purification of , purification ofs.t. B B1 A1 A : purifies Intuition: It’s harder to generate larger pure state, even with approximations.

  16. Proof sketch (cont.) • , • Lem. ’s Schmidt coefficients:, = min with . • [Eckart-Young’36] : singular values of A, . • eigenvalues of A = Schmidt coeff. of

  17. Thanks

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