Detecting Incapacity
Graeme Smith
IBM Research
Joint Work with John Smolin
QECC 2011
December 6, 2011
Noisy Channel Capacity
X
N
Y
p(y|x)
Capacity: bits per channel use in the limit of many channels
C = maxX I(X;Y)
I(X;Y) is the mutual information
Zero-quantum-capacity channels
Sym.
PPT
Q=0
?
Q>0
Outline
• Zero-capacity channels
• Anti-degradable
• PPT
• Proof that PPT has no capacity
• General Incapacity criterion
• Linear Maps
Motivation: unify thinking about incapacity,
appeal to physical principles, apply to
generalized prob. Models or mobits
Quantum Capacity
N
.
.
.
E
D
N
N
• Want to encode qubits so
they can be recovered
after experiencing noise.
• Quantum capacity is the
maximum rate, in qubits
per channel use, at which
this can be done.
• We’d like to know when
Q(N ) > 0.
Quantum Capacity
0
Ã
E
U
B
• Coherent Information:
Q1 (N ) = max S(B)-S(E) (cf Shannon formula)
• Q(N ) ¸ Q1(N ) (Lloyd-Shor-Devetak)
• Q(N ) = limn ! 1(1/n) Q1(N … N )
• Q(N )  Q1(N ) (DiVincenzo-Shor-Smolin ‘98)
Zero Quantum Capacity Channels:
Symmetric Channels
Output
symmetric
in B and E
0
Ã
U
E
=
0
B
U
Ã
Example: 50% attenuation channel
vacuum
Input
mode
50:50
environment
Output
mode
B
E
Zero Quantum Capacity Channels:
Symmetric Channels
Suppose a symmetric channel had Q >0
0
Ã
U
E
B
Zero Quantum Capacity Channels:
Symmetric Channels
Suppose a symmetric channel had Q >0
0
Ã
Un
En
Bn
Zero Quantum Capacity Channels:
Symmetric Channels
Suppose a symmetric channel had Q >0
0
Ã
E
Un
En
D
Ã
Zero Quantum Capacity Channels:
Symmetric Channels
Suppose a symmetric channel had Q >0
0
Ã
E
D
Ã
D
Ã
Un
Zero Quantum Capacity Channels:
Symmetric Channels
Suppose a symmetric channel had Q >0
0
Ã
E
D
Ã
D
Ã
Un
So, symmetric channels
must have zero quantum
capacity. Specifically, the
50% attenuation channel
has zero capacity. It will be
one of our two zero
quantum capacity channels.
IMPOSSIBLE!
Zero Quantum Capacity Channels:
Positive Partial Transpose
• Partial transpose:
(|iihj|A|kihl|B) = |iihj|A|lihk|B
• If AB is not positive, then the state is entangled
• If AB ¸ 0, it may be entangled, but then it is
very noisy. Bound entanglement---can’t get any
pure entanglement from it.
• A PPT-channel enforces PPT between output
and purification of the input:
is PPT
• Implies Q(N ) = 0, but can have P(N ) > 0
Outline
• Zero-capacity channels
• Anti-degradable
• PPT
• Proof that PPT has no capacity
• General Incapacity criterion
• Linear Maps
Motivation: unify thinking about incapacity,
appeal to physical principles, apply to
generalized prob. Models or mobits
PPT has no quantum capacity
• Let T(½) = ½T. N is PPT iff T ± N is CP
PPT has no quantum capacity
• Let T(½) = ½T. N is PPT iff T ± N is CP
• Say such N could send quantum info
PPT has no quantum capacity
• Let T(½) = ½T. N is PPT iff T ± N is CP
• Say such N could send quantum info
• Then à = D ± N ± E(Ã)
PPT has no quantum capacity
•
•
•
•
Let T(½) = ½T. N is PPT iff T ± N is CP
Say such N could send quantum info
Then à = D ± N ± E(Ã)
Acting on both sides with T, we get
T (Ã) = T ± D ± N ± E(Ã)
PPT has no quantum capacity
•
•
•
•
Let T(½) = ½T. N is PPT iff T ± N is CP
Say such N could send quantum info
Then à = D ± N ± E(Ã)
Acting on both sides with T, we get
T (Ã) = T ± D ± N ± E(Ã) = D ¤ ± T ± N ± E(Ã)
PPT has no quantum capacity
•
•
•
•
Let T(½) = ½T. N is PPT iff T ± N is CP
Say such N could send quantum info
Then à = D ± N ± E(Ã)
Acting on both sides with T, we get
T (Ã) = T ± D ± N ± E(Ã) = D ¤ ± T ± N ± E(Ã)
• LHS is transpose. RHS is physical. Can’t be!
PPT has no quantum capacity
•
•
•
•
Let T(½) = ½T. N is PPT iff T ± N is CP
Say such N could send quantum info
Then à = D ± N ± E(Ã)
Acting on both sides with T, we get
T (Ã) = T ± D ± N ± E(Ã) = D ¤ ± T ± N ± E(Ã)
• LHS is transpose. RHS is physical. Can’t be!
• T is continuous, and N - n is PPT when N is,
so also for capacity.
Outline
• Zero-capacity channels
• Anti-degradable
• PPT
• Proof that PPT has no capacity
• General Incapacity criterion
• Linear Maps
Motivation: unify thinking about incapacity,
appeal to physical principles, apply to
generalized prob. Models or mobits
P-commutation
• Let R be unphysical on a set S
• Say for any physical map D,
?
D
there’s a physical map
with R ± D = D ? ± R
• Then, if R ± N is physical,
N can’t transmit S
Is this really more general?
Lemma: If R is linear, invertible, preserves
system dimension and trace, and is pcommutative, it is either of the form R(½) =
(1-p)½T + p I/d or R(½) = (1-p)½ + p I/d
Proof: D ? = R ± D ± R¡ 1 Consider conj by
unitaries N U (½) = U½U y
N U?
Has to be faithful rep. of proj.
unitary gp. We know what these are.
Improved P-commutation
• We want to move to non-linear maps, R,
but it gets very hard to make sure they Pcommute
• So, we can generalize the notion of Pcommutation: RD ± D = D ? ± R for a
family of unphysical maps f R D gD
• If R ± N then N can’t send quantum info
Anti-degradable channels
• Recall that a channel N is antidegradable
if there’s an N 12 with Tr 1 N 12 = Tr 2 N 12 = N
• Roughly speaking, let R = N 12 ± N ¡ 1
• This map will clone.
• For unitary decoder, let U ? = U - U and
RU (Á) = (U - U)R(U y ÁU)U y - U y
?
R
±
D
=
D
• Gives U
U
U ±R
Teleportation
jÃi
P
M
jÃi
jÁd i
Teleportation
jÃi
P
Classical
information
Unitary rotation
recovers the state
M
jÃi
other information
goes back in time,
wraps around
jÁd i
Teleportation
jÃi
R
Classical
information
Unitary rotation
recovers the state
M
jÃi
other information
goes back in time,
wraps around
½A B
Time-traveling information gets
confused
jÃi
R
Classical
information
Unitary rotation
recovers the state
M
jÃi
Now suppose state is
PT invariant---PT on A
leaves state alone
½A B
Time-traveling information gets
confused
jÃi
R
Classical
information
Unitary rotation
recovers the state
M
jÃi
Now suppose state is
PT invariant---PT on A
leaves state alone
Other information gets stuck
here because it doesn’t
AB
know which direction in time
to go---can’t get around the
bend!!!
½
Summary
• Channels with zero classical capacity are trivial, but
there’s lots of structure in zero quantum capacity
channels
• Two known tests for incapacity---symmetric extension
and PPT
• Both can be understood as specal cases of the House
diagram, P-commutation, etc.
• Gives operational proof the PPT channels have zero
quantum capacity---otherwise we could implement the
unphysical time-reversal operation.
• To go beyond PPT, need nonlinear R.
Questions
• Are there other non-linear forbidden operations
that give interesting new channels with no
capacity?
• Can we apply this to generalized probabilistic
theories, mobits, etc. ? Should be yes for
mobits.
• Given a zero-capacity channel, can we find a
“reason” for its incapacity?
• Sensible classification of unphysical maps?
• Can we make the time-travel story more
rigorous?