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Reversibility of entanglement assisted coding |
| contact: |
P. Shor |
date: |
31 Jan 2003 |
last progress: |
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solved by: |
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For any two quantum channels S and T, define the entanglement
assisted capacity CE(T,S) of T for S-messages as the supremum of
all rates r such that, for large n, rn parallel copies of T may be
simulated by n copies of S, where the simulation involves arbitrary coding
and decoding operations using (if necessary) arbitrarily many entangled pairs
between sender and receiver, and where the errors go to zero as n\to\infty.
Show that CE(T,S)=CE(S,T)-1.
As for other capacities, the two-step coding inequality CE(T,S)
CE(S,R)\leq CE(T,R) is easy to show. Hence CE(T,S)
CE(S,T)\leq1. Equality means here, that the two channels are
essentially equivalent as a resource for simulating other channels R (apart
from a constant factor): CE(R,S)=const CE(R,T) (with
const=CE(T,S)). In this case we call S and T reversible
for entanglement assisted coding.
For ordinary capacity C(T,S) (without entanglement assistance) reversibility
fails in general: When S is an ideal classical 1 bit channel, and T is an
ideal 1 qubit quantum channel, we have C(S,T)=1, but C(T,S)=0, because
quantum information cannot be sent on classical channels. On the other hand,
with entanglement assistance we have C(S,T)=2 by superdense coding and
C(T,S)=1/2 by teleportation.
Because all ideal channels S are equivalent as reference channels, we can
define CE(T)=CE(T,S1), with S1 the ideal classical 1 bit
channel as the entanglement assisted capacity of T. For this quantity there
is an explicit formula (coding theorem) by [BSST1]. The problem stated
above appears in [BSST2] as the "Reverse Shannon Theorem''.
The problem is solved for the special case of a known "tensor power source'',
i. e. a source emitting the same, known, density matrix at each time step.
Recent efforts by P. Shor focus on the unknown tensor power source and the
known "tensor product source'' where the density matrix of the source is a
tensor product [SH].
| [BSST1] | C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V.
Thapliyal, »Entanglement-assisted classical capacity of noisy
quantum channels«, Phys. Rev. Lett. 83, 3081 (1999) and
quant-ph/9904023 (1999). |
| [BSST2] | C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V.
Thapliyal, »Entanglement-assisted capacity of a quantum channel and
the reverse Shannon theorem«,
quant-ph/0106052
(2001). |
| [SH] | P. W. Shor, private communication (2003). |
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