Counterexamples to the maximal p -norm multiplicativity conjecture Patrick Hayden (McGill University) || | | N(½)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 optimal rate 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 optimal rate 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 ¾ x 1 ­ ¾ x 2 ­  ­ ¾ x n

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 it to 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 the solution 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, N 1 and N 2 are quantum channels. (CPTP) Notation: H min ( N ) = min  H( N (  )) is the minimum output entropy of N. Conjecture: The minimum entropy output state for the product channel N 1  N 2 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 N 1  N 2 is attained by a product-state input  1   2.

Maximal p-norm multiplicativity conjecture Conjecture: The minimum entropy output state for the product channel N 1  N 2 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 > [Holevo-Werner]  2007: Additivity fails for p > 2. [Winter]

Counterexamples for 1<p<2! For all 1 < p < 2, there exist channels N 1 and N 2 to C d such that: i)H p min ( N 1 ), H p min ( N 2 )  log d - O(1) ii)H p min ( N 1  N 2 )  p log d + O(1) Additivity would have implied: H p min ( N 1  N 2 )  2 log d - O(1) Near p=1, minimum output entropy of N 1  N 2 not significantly greater than that of N 1 or N 2 alone! Intuition: Channels that look very noisy (nearly depolarizing) need not be anywhere near depolarizing on entangled input. 2 p 1

The counterexamples U |0   N()N() R S A B TRASH N  N()N() S A Fix dimensions |R|<<|S|, |A|=|B| and choose U at random according to Haar measure. Demonstrate resulting channels violate 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()N() R S A 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()N() R S A B TRASH |  If U is selected at random, what can be said about U|  |0  ? U|  |0  is highly entangled between A and B:  H p ( N (  ) )   log|A| - O(1) (Compare maximally mixed state: log|A|.) N 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 H p ( N (  ) )   log|A| - O(1) ?

Concentration of measure SnSn LEVY: Given an  -Lipschitz function f : S n ! R with median M, the probability that, for a random x 2 R S n, f ( x ) is further than  from M is bounded above by exp (- n  2 C /  2 ) from some C > 0.  AnAn A n < exp[- n g(  )] for some g(  ) indep. of n f ( x )= x 1 Just need a Lipschitz constant: Choosing f the map from |  to H p ( N (  )), can take  2  |A| p-1. Pr[ H p ( N (  )) < log|A|- const -  ] ~  exp( - const  2 |A| 3-p )

Connect the dots U (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 H p ( N (  ) )   log|A| - O(1). N and N have high minimum output entropy.

Done! For all 1 < p < 2, there exist channels N 1 and N 2 to C d such that: i)H p min ( N 1 ), H p min ( N 2 )  log d - O(1) ii)H p min ( N 1  N 2 )  p log d + O(1) Additivity would have implied: H p min ( N 1  N 2 )  2 log d - O(1) Near p=1, minimum output entropy of N 1  N 2 not significantly greater than that of N 1 or N 2 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 ( N  N )(  )

What about von Neumann (p=1)??? Method fails: recall |R|~|A| p-1. Constants depend on p blow up. Artifact or does the conjecture survive at p=1? H p for p > 1 very sensitive to a single large eigenvalue, but H 1 is not.

Do some calculating Contribution from eigenvalue ~1/|R| Contribution from all the others For H p, p > 1, first term dominates but second term dominates H 1 H 1 (( 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: N 1 = N 2  Cubitt, Harrow, Leung, Montanaro & Winter have found violations for 0  p  0.12