Classical capacities of bidirectional channels

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Classical capacities of bidirectional channels
Thanks to Andrew Childs Hoi-Kwong Lo Peter Shor

• Charles Bennett, IBM
• Aram Harrow, MIT/IBM, aram_at_mit.edu
• Debbie Leung, MSRI/IBM
• John Smolin, IBM

AMS meeting, Boston, Oct 5, 2002
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-dn) bits with error en,
where dn,en!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 (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.
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Mutual Information c

• For an ensemble Epi, si, the mutual
  information is

For pure states Epi, yiiAB, we use Bobs
reduced density matrix.
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Example CNOT Hamiltonian
Applying H for time t yields the unitary gate
Ue-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 0i or 1i.
• Bob begins with 0i.

• After time t, Bob has either 0i or cos t0i
  sin t1i.

• The ensemble is

• The mutual information is c(Et) H2(sin2t),
  where H2(p)-plog2p-(1-p)log2(1-p).

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Example protocols 2/2
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What wed like to do

1. Create n copies of the optimal ensemble E.
2. Apply N to each copy.
3. Measure, obtaining mutual information nc(N(E)).
4. Use nc(E) bits to recreate n copies of E and keep
   the remaining n(c(N(E))-c(E)) bits as message.
5. Return to step 2 and repeat.

Asymptotically c(N(E))-c(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 c by no more
than nsupE c(N(E))-c(E).
Proof The most general n-use protocol looks
like
Local operations can never increase c.
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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 Ic.
(Strong converse) With free entanglement, En can
be prepared by transmitting ¼nc(E) bits.
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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.

• 1 cbit many ebits ! 1 qubit
• (Bennett et al., PRA 87 (2001) 077902)

• If Epi, yiiB, then Alice can Schumacher
  compress E and send only S(E) cbits.

• With mixed-state RSP, Epi, yiiAB can be
  sent using c(E) cbits and free entanglement.
• (Shor, unpublished, 2001)

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Achieving the bound (proof)

1. Alice breaks up her message into strings M1,,Mk,
   each of length n(c(N(E))-c(E)).
2. She will recursively determine strings R1,,Rk,
   each of length nc(N(E)) from RSP measurements.
3. First let Rk be an arbitrary string.
4. For ik, k-1, , 3, 2 choose fii2En such that
   N n(fiihfi) encodes (Mi, Ri).
5. Perform the RSP measurement for fii to obtain
   Ri-1.
6. Send (M1, R1) inefficiently, with O(n) uses of N.
7. For i2k
8. Bob uses Ri-1 to construct fii.
9. They apply Nn to fii.
10. Bob measures N n(fiihfi) to obtain (Mi, Ri).

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Achieving the bound (Bob)
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Achieving the bound (Alice)
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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?
Two qubit gate capacities are always locally
equivalent to symmetric gates due to the
decomposition
For dgt2, no such decomposition exists, and there
may be asymmetric gates.
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Asymmetric capacities?
Define a gate U acting on a dd 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.