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

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…

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.

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.

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.

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.

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.

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

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.

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.

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 .

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.

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) )  If E ={p i, |  i i B }, then Alice can Schumacher compress E and send only S( E ) cbits.

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 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 ).

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

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

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?

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 =

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.