Counterexamples to the maximal p -norm multiplicativity conjecture
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Transcript of Counterexamples to the maximal p -norm multiplicativity conjecture
Counterexamples to the maximal p-norm multiplicativity conjecture
Patrick Hayden (McGill University)
|| ||N(½)p
C&QIC, Santa Fe 2008
A challenge to the physicists
John Pierce [1973]: I think that I have never met a physicist
who understood information theory. I wish that physicists would stop talking about reformulating information theory and would give us a general expression for the capacity of a channel with quantum effects taken into account rather than a number of special cases.
Sending classical information
through noisy quantum channels
Physical model of a noisy channel:(Trace-preserving, completely positive map)
HSW noisy coding theorem: In the limit of many uses, the optimalrate at which Alice can send bits reliably to Bob through N is given by the (regularization of the) formula
where the maximization is over some family of input/output states.
m Encoding( state)
Decoding(measurement)
m’
Sending classical information
through noisy quantum channels
Physical model of a noisy channel:(Trace-preserving, completely positive map)
m Encoding( state)
Decoding(measurement)
m’
HSW noisy coding theorem: In the limit of many uses, the optimalrate at which Alice can send bits reliably to Bob through N is given by the (regularization of the) formula
The additivity conjecture:These two formulas are equal
where
Sustained, heroic, and so far inconclusive efforts by: Datta, Eisert, Fukuda, Holevo, King, Ruskai,
Schumacher, Shirokov, Shor, Werner...
Why do they care so much?
The additivity conjecture:These two formulas are equal
where
Operational interpretation: •Alice doesn’t need to entangle her inputs across multiple uses of the channel.• Codewords look like ¾x1
¾x2 L ¾xn
QMAC solution pre-QIP 2005
Interpretation: Alice and Bob treat each others’ actions as noise. Independent decoding.No-go theorem for use of quantum side information.
[Yard/Devetak/H 05 v1]
QMAC solution post-QIP 2005
Interpretation: Charlie decodes Alice’s quantum data first and uses itto help him decode Bob’s. (Or vice-versa.)Go theorem for use of quantum side information.
[Yard/Devetak/H 05 v2]
Capacity formulas matter
Fair question to throw at the speaker if you’re getting bored in any quantum Shannon theory talk: “Can you describe an effective procedure
for calculating this capacity you claim to have determined?”
If we can’t write down a tractable formula for thesolution to a capacity problem, then we don’t fully understand the structure of the optimal codes.
Lesson:
An (Almost) Equivalent Form:
Minimum Entropy Outputs
• H() = - Tr[ log ] (von Neumann entropy of the density operator )
• N, N1 and N2 are quantum channels. (CPTP)
Notation:
• Hmin(N) = min H(N()) is the minimum output entropy of N.
Conjecture:
The minimum entropy output state for the product channel N1 N2 is attained by a product state input 1 2.
[King-Ruskai 99]
Maximal p-norm multiplicativity conjecture
Conjecture:
The minimum entropy output state for the product channel N1 N2 is attained by a product-state input 1 2.
Maximal p-norm multiplicativity conjecture
Conjecture:
The minimum entropy output state for the product channel N1 N2 is attained by a product-state input 1 2.
Renyi entropy (1 < p ):
(Recover von Neumann entropy as p 1.)
Norm? What norm?
[Amosov-Holevo-Werner 00]
Partial results: Additivity holds if...
One channel is Unitary A unital qubit channel A generalized depolarizing channel A generalized dephasing channel Entanglement-breaking A very noisy channel
Complements of these channels
[Amosov, Devetak, Eisert, Fujiwara, Hashizume, Holevo, King, Matsumoto, Nathanson, Ruskai, Shor, Wolf, Werner]
[See Holevo ICM 2006]
But...
2002: Additivity fails for p > 4.79... [Holevo-Werner]
2007: Additivity fails for p > 2.
[Winter]
Counterexamples for 1<p<2!
For all 1 < p < 2, there exist channels N1 and N2 to Cd such that:• Hp
min(N1) , Hpmin(N2) log d - O(1)
• Hpmin(N1 N2) p log d + O(1)
Additivity would have implied:
Hpmin(N1 N2) 2 log d -
O(1)Near p=1, minimum output entropy of N1 N2
not significantly greater than that of N1 or N2 alone!
Intuition: Channels that look very noisy (nearly depolarizing)need not be anywhere near depolarizing on entangled input.
2p
1
The counterexamples
U|0
N()R
SA
B TRASHN N()S A
Fix dimensions |R|<<|S|, |A|=|B| and choose U at randomaccording to Haar measure. Demonstrate resulting channelsviolate Renyi additivity with non-zero probability.
Two things to prove:i) Product channel has low minimum output entropy.ii) Individual channels have high minimum output entropies.
N N has low output entropy
The key identity:
N N has low output entropy
The key identity (v1):
The key identity (v2):
U|0
N()R
SA
B TRASH
Easy calculation:
This is BIG if |R| is small! (Compare 1/|A|2 for maximally mixed state.)Choose |R| ~ |A|p-1.
N and N have high output entropy
U|0 N()R
SA
B TRASH|
If U is selected at random, what can be said about U||0?
U||0 is highly entangled between A and B: Hp( N() ) log|A| - O(1)
(Compare maximally mixed state: log|A|.)
N N()S A|
[Lubkin, Lloyd, Page, Foong & Kanno, Sanchez-Ruiz, Sen…]
Is this true simultaneously for all | S with a typical U?
i.e. Is min| S Hp( N() ) log|A| - O(1) ?
Concentration of measure
Sn
LEVY: Given an -Lipschitz function f : Sn ! R with median M, the probability that, for a random x 2R Sn , f (x) is further than from M is bounded above by exp (-n2 C/2) from some C > 0.
An
An < exp[-n g()]for some g() indep. of n
f (x)=x1
Just need a Lipschitz constant: Choosing f the map from | to Hp(N()), can take 2 |A|p-1.
Pr[ Hp(N()) < log|A|- const - ] ~ exp( - const 2|A|3-p )
Connect the dotsU (S |0 ½ A B 1) Choose a fine net F of states on the
unit sphere of S |0.2) P( Not all states in UF highly entangled )
· |F| P( One state isn’t )3) Highly entangled for sufficiently
fine N implies same for all states in S.
THEOREM: If |R|~|A|p-1, then |S| ~ |A|3-p and w.h.p. as|A| ,
min| S Hp( N() ) log|A| - O(1).
N and N have high minimum output entropy.
Done!
For all 1 < p < 2, there exist channels N1 and N2 to Cd such that:• Hp
min(N1) , Hpmin(N2) log d - O(1)
• Hpmin(N1 N2) p log d + O(1)
Additivity would have implied:
Hpmin(N1 N2) 2 log d -
O(1)Near p=1, minimum output entropy of N1 N2
not significantly greater than that of N1 or N2 alone!
What about von Neumann (p=1)???
Method fails: recall |R|~|A|p-1. Constants depend on p and blow up.
Artifact of the analysis or does the conjecture
survive at p=1?
|R|=3 |A|=|B|=24(NN)()
What about von Neumann (p=1)???
Method fails: recall |R|~|A|p-1. Constants depend on p blow up.
Artifact or does the conjecturesurvive at p=1?
Hp for p > 1 very sensitive to a single large eigenvalue, but H1 is not.
Do some calculating
Contribution from eigenvalue ~1/|R|
Contribution from all the others
For Hp, p > 1, first term dominates but second term dominates H1
H1((N N)()) = 2 log|A| - O(1)is BIG not small
No additivity violations. To be sure, can anyone calculate the O(1) terms?
Summary
Additivity fails for 1 < p < 2. Closes main approach to additivity for capacity itself.
Further developments: Winter tightened Lipschitz bound, showing
same examples work for 1 < p < Dupuis showed orthogonal group can
replace unitary group: N1 = N2
Cubitt, Harrow, Leung, Montanaro & Winter have found violations for 0 p 0.12