1 / 27

Counterexamples to the maximal p -norm multiplicativity conjecture

This talk discusses counterexamples to the conjecture that the minimum entropy output state for a product channel is attained by a product-state input. It explores the implications of this counterexample and challenges it poses for physicists.

Download Presentation

Counterexamples to the maximal p -norm multiplicativity conjecture

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Counterexamples to the maximal p-norm multiplicativity conjecture | | | | N(½) Patrick Hayden (McGill University) p C&QIC, Santa Fe 2008

  2. 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.

  3. Encoding ( state) Decoding (measurement) m’ m 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.

  4. Sending classical information through noisy quantum channels Physical model of a noisy channel: (Trace-preserving, completely positive map) Encoding ( state) Decoding (measurement) m’ 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

  5. 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?

  6. 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 ¾x1­¾x2­L ­¾xn

  7. 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]

  8. 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]

  9. Lesson: Capacity formulas matter 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. • 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?”

  10. An (Almost) Equivalent Form:Minimum Entropy Outputs Notation: • H() = - Tr[  log  ] (von Neumann entropy of the density operator ) • N, N1 and N2 are quantum channels. (CPTP) • 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]

  11. 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.

  12. 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]

  13. 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]

  14. But... • 2002: Additivity fails for p > 4.79... [Holevo-Werner] • 2007: Additivity fails for p > 2. [Winter]

  15. p 1 Counterexamples for 1<p<2! • For all 1 < p < 2, there exist channels N1 and N2 to Cd such that: • Hpmin(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) 2 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.

  16. The counterexamples |0 R N() A S N() A N  U S  B TRASH 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: Product channel has low minimum output entropy. Individual channels have high minimum output entropies.

  17. NN has low output entropy The key identity: U I U* = I

  18. NN has low output entropy The key identity (v1): The key identity (v2): |0 R N() A U S  B TRASH Easy calculation: This is BIG if |R| is small! (Compare 1/|A|2 for maximally mixed state.) Choose |R| ~ |A|p-1.

  19. N and N have high output entropy |0 R N() A | S A N() N U | S 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|.) Is this true simultaneously for all | S with a typical U? i.e. Is  min| S Hp( N() )  log|A| - O(1) ? [Lubkin, Lloyd, Page, Foong & Kanno, Sanchez-Ruiz, Sen…]

  20. An < exp[-n g()] for some g() indep. of n  An f (x)=x1 Concentration of measure Sn LEVY: Given an -Lipschitz function f : Sn!R with median M, the probability that, for a random x2RSn , f (x) is further than  from M is bounded above by exp (-n2C/2) from some C > 0. 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 )

  21. Connect the dots U (S |0½ A  B • Choose a fine net F of states on theunit sphere of S |0. • P( Not all states in UF highly entangled )· |F| P( One state isn’t ) • 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.

  22. Done! • For all 1 < p < 2, there exist channels N1 and N2 to Cd such that: • Hpmin(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!

  23. 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?

  24. |R|=3 |A|=|B|=24 (NN)()

  25. 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? Hp for p > 1 very sensitive to a single large eigenvalue, but H1 is not.

  26. 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?

  27. 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

More Related