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

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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…

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

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

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Mutual Information: For an ensemble E ={p i, i }, the mutual information is For pure states E ={p i, | i i AB }, we use Bob’s reduced density matrix.

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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.

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Example protocols 1/2 Alice begins with either |0 i or |1 i. Bob begins with |0 i. The mutual information is ( E t ) = H 2 (sin 2 t), where H 2 (p)=-plog 2 p-(1-p)log 2 (1-p). The ensemble is After time t, Bob has either |0 i or cos t|0 i + sin t|1 i.

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Example protocols 2/2 Orthogonal states: ( E t )=1 Optimal chord: max ( E t )/t Optimal slope

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What we’d like to do 1. Create n copies of the optimal ensemble E. 2. Apply N to each copy. 3. Measure, obtaining mutual information n ( N ( E )). 4. Use n ( E ) bits to recreate n copies of E and keep the remaining n( ( N ( E ))- ( E )) bits as message. 5. Return to step 2 and repeat. Asymptotically ( N ( E ))- ( E ) bits per use of N.

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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.

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

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

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Remote State Preparation With “mixed-state” RSP, E ={p i, | i i AB } can be sent using ( E ) cbits and free entanglement. (Shor, unpublished, 2001) 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. 1 cbit + many ebits ! 1 qubit (Bennett et al., PRA 87 (2001) 077902) If E ={p i, | i i B }, then Alice can Schumacher compress E and send only S( E ) cbits.

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Achieving the bound (proof) 1. Alice breaks up her message into strings M 1,…,M k, each of length n( ( N ( E ))- ( E )). 2. She will recursively determine strings R 1,…,R k, each of length n ( N ( E )) from RSP measurements. 1. First let R k be an arbitrary string. 2. For i=k, k-1, …, 3, 2 choose | i i2E n such that N n (| i ih i |) encodes (M i, R i ). 3. Perform the RSP measurement for | i i to obtain R i-1. 3. Send (M 1, R 1 ) inefficiently, with O (n) uses of N. 4. For i=2…k 1. Bob uses R i-1 to construct | i i. 2. They apply N n to | i i. 3. Bob measures N n (| i ih i |) to obtain (M i, R i ).

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Achieving the bound (Bob) n ( E ) bits n( ( N ( E ))- ( E )) bits M1M1 R1R1 RkRk MkMk |i|i Bob RSP N n (| ih |) block decoding M2M2 R2R2

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Achieving the bound (Alice) N n (| ih |) block coding |i|i RkRk n ( E ) bits MkMk M k-1 M1M1 n( ( N ( E ))- ( E )) bits R1R1 R k-1 Alice RSP

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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?

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Symmetry? For d>2, no such decomposition exists, and there may be asymmetric gates. Two qubit gate capacities are always locally equivalent to symmetric gates due to the decomposition: LALA LBLB RARA RBRB UV =

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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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