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Classical capacities of bidirectional channels

Classical capacities of bidirectional channels. Thanks to: Andrew Childs Hoi-Kwong Lo Peter Shor. Charles Bennett, IBM Aram Harrow , MIT/IBM, aram@mit.edu Debbie Leung, MSRI/IBM John Smolin, IBM. AMS meeting, Boston, Oct 5, 2002. quant-ph/0205057. Outline.

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Classical capacities of bidirectional channels

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  1. Classical capacities of bidirectional channels Thanks to: Andrew Childs Hoi-Kwong Lo Peter Shor Charles Bennett, IBM Aram Harrow, MIT/IBM, aram@mit.edu Debbie Leung, MSRI/IBM John Smolin, IBM AMS meeting, Boston, Oct 5, 2002 quant-ph/0205057

  2. Outline • Background: bidirectional channel capacities and mutual information. • Example. • Main result: determining the entanglement-assisted one-way capacity. • Upper bound. • Remote state preparation and a protocol for achieving the capacity. • Plenty of open questions…

  3. One-way channels A channel can achieve a rate Rif nuses of the channel can transmit n(R-dn) bits with error en, where dn,en!0 as n!1. The (classical) capacity is the largest rate achievable by the channel.

  4. Bidirectional Channels • A pair of rates (RÃ,R!) is achievable if n uses of the channel can transmit ¼nRÃ bits from Bob to Alice and ¼nR! bits from Alice to Bob. Result: A zoo of different capacities. Our approach: Specialize to entanglement-assisted one-way capacity.

  5. Mutual Information: c For an ensemble E={pi, si}, the mutual information is For pure states E={pi, |yiiAB}, we use Bob’s reduced density matrix.

  6. Example: CNOT Hamiltonian Applying H for time t yields the unitary gate U=e-iHt. Goal: Send the maximum number of bits from Alice to Bob per unit time.

  7. Example protocols 1/2 • Alice begins with either |0i or |1i. • Bob begins with |0i. • After time t, Bob has either |0i or cos t|0i + sin t|1i. • The ensemble is • The mutual information is c(Et) = H2(sin2t), where H2(p)=-plog2p-(1-p)log2(1-p).

  8. Example protocols 2/2 Orthogonal states: c(Et)=1 Optimal chord: max c(Et)/t Optimal slope

  9. What we’d like to do Create n copies of the optimal ensemble E. Apply N to each copy. Measure, obtaining mutual information nc(N(E)). Use nc(E) bits to recreate n copies of E and keep the remaining n(c(N(E))-c(E)) bits as message. Return to step 2 and repeat. Asymptotically c(N(E))-c(E) bits per use of N.

  10. General result Theorem: In English: With free entanglement, the asymptotic capacity of a bidirectional channel N is equal to the maximum increase in mutual information from a single use of N.

  11. Upper bound Claim:n uses of N can increase c by no more than n¢supEc(N(E))-c(E). Proof:The most general n-use protocol looks like: Local operations can never increase c.

  12. Relating c to classical bits (Block coding) For large n, E­n can encode ¼nc(E)bits. (Weak converse) If a measurement on E yields classical mutual information I between outcomes and encoding, then I·c. (Strong converse) With free entanglement, E­n can be prepared by transmitting ¼nc(E) bits.

  13. Remote State Preparation • Given large amounts of shared entanglement, Alice chooses a state to transmit, makes a measurement and sends the classical result to Bob, from which he can reconstruct the state. • With “mixed-state” RSP, E={pi, |yiiAB}can be sent using c(E) cbits and free entanglement. (Shor, unpublished, 2001) • 1 cbit + many ebits ! 1 qubit • (Bennett et al., PRA87 (2001) 077902) • If E={pi, |yiiB}, then Alice can Schumacher compress E and send only S(E) cbits.

  14. Achieving the bound (proof) • Alice breaks up her message into strings M1,…,Mk, each of length n(c(N(E))-c(E)). • She will recursively determine strings R1,…,Rk, each of length nc(N(E)) from RSP measurements. • First let Rk be an arbitrary string. • For i=k, k-1, …, 3, 2 choose |fii2E­n such that N­ n(|fiihfi|) encodes (Mi, Ri). • Perform the RSP measurement for |fii to obtain Ri-1. • Send (M1, R1) inefficiently, with O(n) uses of N. • For i=2…k • Bob uses Ri-1 to construct |fii. • They apply N­n to |fii. • Bob measures N­ n(|fiihfi|) to obtain (Mi, Ri).

  15. nc(E) bits n(c(N(E))-c(E)) bits Rk Mk block decoding N­n(|fihf|) R2 M2 R1 M1 |fi Bob RSP Achieving the bound (Bob)

  16. nc(E) bits n(c(N(E))-c(E)) bits block coding Rk Mk N­n(|fihf|) Rk-1 Mk-1 |fi Alice RSP R1 M1 Achieving the bound (Alice)

  17. More open questions than results… • For entanglement-assisted communication, how many elements are in the optimal ensemble? What dimension ancilla are necessary? Can we ever determine the optimal ensemble exactly? • How are communication capacities related to entanglement generating rates? • How do forward and backward capacities trade off with one another? Are they ever asymmetric for unitary gates? How does entanglement affect this? • Can we define a bidirectional mutual information? Or bidirectional remote state preparation?

  18. RA LA = U V RB LB Symmetry? Two qubit gate capacities are always locally equivalent to symmetric gates due to the decomposition: For d>2, no such decomposition exists, and there may be asymmetric gates.

  19. Asymmetric capacities? Define a gate U acting on a d£d dimensional space by • The forward capacity is at least log d, but the backward capacity is thought to be less than log d. • With free entanglement, the backwards capacity is also log d. • For one use without entanglement, the backwards mutual information is provably less than log d.

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