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Recent Developments on the Additivity Battlefront

Recent Developments on the Additivity Battlefront. or does 2 + 2 = 3?. Andreas Winter (arXiv:0707.0402) Patrick Hayden (arXiv: 0707.3291 ) Andreas Winter (private communication)

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Recent Developments on the Additivity Battlefront

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  1. Recent Developments on the Additivity Battlefront or does 2 + 2 = 3? • Andreas Winter (arXiv:0707.0402) • Patrick Hayden (arXiv:0707.3291)Andreas Winter (private communication) • T. S. Cubitt, A. Harrow, D. Leung, A. Montanaro, A. Winter (schizophrenic communication)T. S Cubitt, A. Montanaro, A. Winter (arXiv:0706.0705, to appear in J. Math. Phys.)

  2. Outline (with apologies to Sir Winston Churchill) • The Gathering Storm: why is the battle of additivity so important? • The Hinge of Fate: lieutenant Winter’s plan of attack • Closing the Ring: the p > 2 campaign • Triumph and Tragedy: the p > 1 campaign • The Grand Alliance: the p = 0 campaign • Their Finest Hour: the battle to come?

  3. k bits k bits … n • Mutual informationCapacity • Rate: • Shannon’s noisy coding theorem: The Gathering Storm:Classical Channel Capacity

  4. … k bits k bits n • Holevo  quantityHolevo capacity • Holevo–Schumacher–Westmorland: The Gathering Storm:Quantum Channel Capacity

  5. ● ● The Gathering Storm:Additivity of the Quantum Capacity

  6. ● ● ● ● ● ● ● ● LOCC … ● ● ● ● ● ● ● ● • Entanglement Cost The Gathering Storm:Entanglement of Formation … • State decomposition: Entanglement of formation

  7. The Gathering Storm:Additivity of Entanglement of Formation

  8. ● ● • Minimum output entropy The Gathering Storm:Minimum Output Entropy

  9. p – Renyi entropy • p – norm • maximum output p–norm The Gathering Storm:Minimum Output p – Renyi Entropy

  10. Entanglement of formation Holevo capacity Minimum output entropy (for p=1) The Gathering Storm:Equivalence of Additiviy Conjectures P. W. Shor, “Equivalence of Additivity Questions in Quantum Information Theory”Comm. Math. Phys. 246(3):453-472 (2004)   (also strong subadditivity of entanglement of formation)

  11. Kraus decomposition ● ● Choi-Jamiołkowski state ● ● ● ● Steinspring representation ● |0iE AB U ● S The Hinge of Fate:Representations of Quantum Channels

  12. Outline (with apologies to Sir Winston Churchill) • The Gathering Storm: why is the battle of additivity so important? • The Hinge of Fate: lieutenant Winter’s plan of attack • Closing the Ring: the p > 2 campaign • Triumph and Tragedy: the p > 1 campaign • The Grand Alliance: the p = 0 campaign • Their Finest Hour: the battle to come?

  13. ● ● The Hinge of Fate:Lieutenant Winter’s Plan of Attack • Pick a pair of channels such that: • Individually they have almost maximum output entropy (equivalently, almost minimum output p – norm). • A “conspiracy” occurs when the maximally entangled state is fed into the tensor product channel, suppressing the output entropy.

  14. Outline (with apologies to Sir Winston Churchill) • The Gathering Storm: why is the battle of additivity so important? • The Hinge of Fate: lieutenant Winter’s plan of attack • Closing the Ring: the p > 2 campaign • Triumph and Tragedy: the p > 1 campaign • The Grand Alliance: the p = 0 campaign • Their Finest Hour: the battle to come?

  15. We call a channel “ – randomising” if i.e. if all inputs are mapped to something close to the maximally mixed state.(Reminder: ) • Lemma: With high probability, a random unitary channel with sufficiently large n is  – randomising: Closing the Ring: p > 2 P. Hayden, D. Leung, P. W. Shor, A. Winter, “Randomising Quantum States: Constructions and Applications”, Comm. Math. Phys. 250(2):371 (2004)

  16. Choose a pair of random unitary channels: • – randomising ) all eigs. of bounded: Convexity )p–norm max. when eigs. as non-uniform as possible ) Closing the Ring: p > 2 A. Winter, “The maximum output p-norm of quantum channels is not multiplicative for any p > 2”, arXiv:0707.0402 • Individually they have small maximum output p – norm: Proof:

  17. Closing the Ring: p > 2 • However, the tensor product channel hashigh maximum output p – norm: Proof:

  18. ● ● Closing the Ring: p > 2 Multiplicativity Violation!

  19. Outline (with apologies to Sir Winston Churchill) • The Gathering Storm: why is the battle of additivity so important? • The Hinge of Fate: lieutenant Winter’s plan of attack • Closing the Ring: the p > 2 campaign • Triumph and Tragedy: the p > 1 campaign • The Grand Alliance: the p = 0 campaign • Their Finest Hour: the battle to come?

  20. Triumph and Tragedy: p > 1 • Lemma: With high probability, a random subspaceS½A­ B of dimension |S| » |A|1/p|B| only contains states with high entanglement • proof is a modification of proof of p=1 case in D. Leung, P. Hayden, A. Winter, “Aspects of Generic Entanglement”, Comm. Math. Phys. 265, 95 (2006) • Patrick Hayden, “The Maximal p – norm multiplicativity conjecture is false”, arXiv:0707.3291 (2007) • Andreas Winter, private communication to a large number of people!

  21. Individually, channels have high minimum output entropy with high probability. Proof: The random V maps S into a random subspaceof A­B. Choose |B| = 2|A|,  = 1, and |S| » |A|1/p|B|. Then by Lemma, Triumph and Tragedy: p > 1 Patrick Hayden, “The maximal p – norm multiplicativity conjecture is false” arXiv:0707.3291 (2007) • Construct pair of channels from S to A by picking a random unitary in the Steinspring representation:

  22. ● ● ● ● ● ● ● ● ● ● Triumph and Tragedy: p > 1 • Lemma: Proof:

  23. Triumph and Tragedy: p > 1 • The tensor product channel has low minimum output entropy:

  24. ● ● Triumph and Tragedy: p > 1 Additivity violation!

  25. Outline (with apologies to Sir Winston Churchill) • The Gathering Storm: why is the battle of additivity so important? • The Hinge of Fate: lieutenant Winter’s plan of attack • Closing the Ring: the p > 2 campaign • Triumph and Tragedy: the p > 1 campaign • The Grand Alliance: the p = 0 campaign • Their Finest Hour: the battle to come?

  26. The Grand Alliance: p = 0 Reminder: 0–Renyi entropy is just log(rank) • Pick two channels: Choi-Jamiołkowski states 1 and 2 • Individually, they should have full output rank,i.e. output always has non-zero overlap with all states: • A “conspiracy” occurs when the maximally entangled state is fed into the tensor product channel, that makes the output rank-deficient:

  27. The Grand Alliance: p = 0 T. S. Cubitt, A. Harrow, D. Leung, A. Montanaro, A. Winter“Counterexamples to additivity of minimum output p-Renyi entropy for p close to 0” (to appear in proceedings of QIP 2008, paper in preparation) • Wanted: • Choi-Jamiołkowski states 1, 2 supported on orthogonal subspaces S1, S2 ½ A­B… • …whose orthogonal complements S1?, S2?contain no product states. • Simplify by taking S1 = S2? (so S2 = S1?). • Wanted: • subspaces S, S?½ A­Bneither of which contain any product states.

  28. State product state iff (cf. Schmidt decomposition) iff all order–rmatrix minors = 0 The Grand Alliance: p = 0 • Wanted: • Sets S, S? of dA£dB matrices with all Mi2S orthogonal to all Mk? 2S?, such that any linear combination of Mi has at least one non-zero order–2 matrix minor (similarly for lin. combs. of Mk?).

  29. The Grand Alliance: p = 0 T. S. Cubitt, A. Montanaro, A. Winter, “On the Dimension of Subspaces with Bounded Schmift Rank”, arXiv:0706.0705 (2007), to appear in J. Math. Phys. • 2£2 and 3£3 QFT matrices are both unitary and “totally non-singular”: • For our purposes, “totally non-singular” means any linear combination of the columns can contain at most one 0 entry, e.g.

  30. The Grand Alliance: p = 0 • Construct sets of matrices S, S? by putting columns of QFT matrices down diagonals:

  31. The Grand Alliance: p = 0“The Bristol Channel” • Let Choi-Jamiołkowski states 1, 2 be projectors onto subspaces spanned by S, S?. • Orthogonal complement of each subspace contains no product states)individual channels have maximal output rank. • The two subspaces are orthogonal) input maximally entangled state, output is orthogonal to maximally entangled state on the outputs)tensor product channel doesn’t have maximaloutput rank. Additivity violation!

  32. Their Finest Hour: the battle to come? Conclusions: • p > 1 result kills all hope of proving additivity by the only technique we know of.(Namely, proving additivity for p > 1 then arguing by continuity that additivity holds for p = 1.) • Numerics shows that Bristol channel violates additivity for p < »0.12 • p = 0 result suggests that arguing by continuity from below won’t work either. Future Goals? • Next milestone: p = ½ (negativity) • Extend counter-examples to all p < 1? • Lieutenant Winter’s plan of attack might help again, but require new ideas – need to control all eigenvalues of output state, instead of just the largest/smallest.

  33. Additivity Will be Defeated! Thank you for listening.

  34. The Gathering Storm:The battle so far • 1964: Gordon conjectures that maximum information extractable from a quantum ensemble ·(pi,i). • 1973: Holevo proves conjecture, implying C(N) · (N). • 1996: Holevo, Schumacher–Westmorland prove equality is attainable. • 2002: Holevo–Werner give counterexample to additivity of minimum output p – Renyi entropy for p > 4.79 • 2003: Shor completes equivalence proof of the four “standard” additivity conjectures. • 2007: …

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