1. Detecting Incapacity Graeme Smith IBM Research Joint Work with John Smolin QECC 2011 December 6, 2011 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A A

2. Noisy Channel Capacity N N X Y p(y|x) Capacity: bits per channel use in the limit of many channels C = max X I(X;Y) I(X;Y) is the mutual information

4. PPT Sym. Q=0 ? Q>0 Zero-quantum-capacity channels

5. 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

6. N N N... E Quantum Capacity 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. D

7. Quantum Capacity Coherent Information: Q 1 ( N ) = max S(B)-S(E) (cf Shannon formula) Q( N ) ¸ Q 1 ( N ) (Lloyd-Shor-Devetak) Q( N ) = lim n ! 1 (1/n) Q 1 ( N ­ … ­ N ) Q( N )  Q 1 ( N ) (DiVincenzo-Shor-Smolin ‘98) U Ã 0E B

8. Zero Quantum Capacity Channels: Symmetric Channels Example: 50% attenuation channel U 0 E B Ã Output symmetric in B and E U 0 E B Ã = Input mode Output mode vacuum environment 50:50

9. Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 U 0 E B Ã

10. Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 U nU n 0 EnEn à BnBn

11. Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 UnUn 0 EnEn à D E Ã

12. Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 UnUn 0 Ã D E Ã D Ã

13. Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 UnUn 0 Ã D E Ã D Ã IMPOSSIBLE! 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.

14. Zero Quantum Capacity Channels: Positive Partial Transpose Partial transpose: (|i ih j| A ­ |k ih l| B )  = |i ih j| A ­ |l ih k| 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

15. 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

16. PPT has no quantum capacity Let T( ½ ) = ½ T. N is PPT iff is CP

17. PPT has no quantum capacity Let T( ½ ) = ½ T. N is PPT iff is CP Say such N could send quantum info

18. PPT has no quantum capacity Let T( ½ ) = ½ T. N is PPT iff is CP Say such N could send quantum info Then

19. PPT has no quantum capacity Let T( ½ ) = ½ T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get

20. PPT has no quantum capacity Let T( ½ ) = ½ T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get

21. PPT has no quantum capacity Let T( ½ ) = ½ T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get LHS is transpose. RHS is physical. Can’t be!

22. PPT has no quantum capacity Let T( ½ ) = ½ T. N is PPT iff is CP Say such N could send quantum info Then Acting on both sides with T, we get LHS is transpose. RHS is physical. Can’t be! T is continuous, and is PPT when is, so also for capacity.

23. 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

24. P-commutation Let R be unphysical on a set S Say for any physical map, there’s a physical map with Then, if is physical, can’t transmit S

25. Is this really more general? Lemma: If R is linear, invertible, preserves system dimension and trace, and is p- commutative, it is either of the form R( ½ ) = (1-p) ½ T + p I/d or R( ½ ) = (1-p) ½ + p I/d Proof: Consider conj by unitaries Has to be faithful rep. of proj. unitary gp. We know what these are.

26. Improved P-commutation We want to move to non-linear maps, R, but it gets very hard to make sure they P- commute So, we can generalize the notion of P- commutation:for a family of unphysical maps If then can’t send quantum info

27. Anti-degradable channels Recall that a channel is antidegradable if there’s an with Roughly speaking, let R = This map will clone. For unitary decoder, let and Gives

28. Teleportation M P

29. M P Classical information other information goes back in time, wraps around Unitary rotation recovers the state

30. Teleportation M R Classical information other information goes back in time, wraps around Unitary rotation recovers the state

31. Time-traveling information gets confused M R Classical information Unitary rotation recovers the state Now suppose state is PT invariant---PT on A leaves state alone

32. Time-traveling information gets confused M R Classical information Unitary rotation recovers the state Now suppose state is PT invariant---PT on A leaves state alone Other information gets stuck here because it doesn’t know which direction in time to go---can’t get around the bend!!!

33. 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.

34. 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?