Sending Quantum Information with Zero Capacity Channels

Transcription

1 Sending Quantum Information with Zero Capacity Channels Graeme Smith IM Research KITP Quantum Information Science Program November 5, 2009

2 Joint work with Jon and John

3 Noisy Channel Capacity X N 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 Quantum Communication Quantum Capacity: the rate, in qubits per channel use, at which A can send high-fidelity quantum information to.

5

6 Superactivation of Capacities There are N 1, N 2 with zero quantum capacity but Q(N 1 N 2 ) > 0. G. Smith and J. Yard, Science, 321, (2008)

7 Outline Quantum and Private Capacities Channels with Zero Quantum Capacity Superactivation of Quantum Capacity Superadditivity of Private Capacity Applications

8 Quantum Capacity N. N D 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 a formula for Q(N ) in N terms of N.

9 Quantum Capacity 0 φ U Coherent Information: Q 1 (N ) = max S()-S() (cf Shannon formula) Q(N ) Q 1 (N ) (Lloyd-Shor-Devetak) Q(N ) = lim n (1/n) Q 1 (N N ) Q(N ) Q 1 (N ) (DiVincenzo-Shor-Smolin 98)

10 Coherent Information and no-cloning No cloning: there is no physical operation that copies an unknown quantum state. asically, because ψi ψi ψi isn t linear S() is how much information has S() is how much information has Q 1 = S() S() is how much more ob knows than ve. how much secret information we can send to ob

11 {p x,φ x } X φ x Private Capacity Mutual Information: I(X;) = S(X)+S()- S(X) Private Information: P 1 (N ) = max I(X;)-I(X;) P(N ) P 1 (N ) (Devetak) P(N ) = lim n (1/n) P 1 (N N ) P(N ) P 1 (N ) (Smith, Renes, Smolin 08) P(N ) Q(N) 0 U

12 Outline Quantum and Private Capacities Channels with Zero Quantum Capacity Superactivation of Quantum Capacity Superadditivity of Private Capacity Applications

13 Zero Quantum Capacity Channels: Symmetric Channels Output symmetric in and 0 U ψi

14 Zero Quantum Capacity Channels: Symmetric Channels Output symmetric in and 0 U ψi = 0 ψi U xample: 50% erasure channel: Gives

15 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 0 U ψi

16 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 U n 0 n ψi n

17 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 ψi 0 U n n D ψi

18 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 D ψi ψi 0 U n D ψi

19 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 D ψi ψi 0 U n D ψi IMPOSSIL!

20 Zero Quantum Capacity Channels: Symmetric Channels Suppose a symmetric channel had Q >0 ψi 0 U n D D ψi ψi So, symmetric channels must have zero quantum capacity. Specifically, the 50% erasure channel has zero capacity. It will be one of our two zero quantum capacity channels. IMPOSSIL!

21 Zero Quantum Capacity Channels: Positive Partial Transpose Partial transpose: ( iihj A kihl ) Γ = iihj A lihk If ρ A Γ is not positive, then the state is entangled If ρ A Γ 0, it may be entangled, but then it is very noisy. ound 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

22 Conditions for Private Capacity Channel with private classical capacity lets you make a private state. Straight line indicates Maximally ntangled state A key A 0 0 shield

23 Conditions for Private Capacity Channel with private classical capacity lets you make a private state. A A 0 0 key U U Twisting operation shield U= i,j ii jihi hj A U ij A 0 0

24 Conditions for Private Capacity Channel with private classical capacity lets you make a private state. A key Private State A 0 0 shield

25 Conditions for Private Capacity Channel with private classical capacity lets you make a private state. A key Private State A 0 0 shield Purifying environment

26 Conditions for Private Capacity Channel with private classical capacity lets you make a private state. A key Private State A 0 0 shield Purifying environment A is conditionally independent of : Quantum Markov Chain: A A

27 Zero Quantum Capacity Channels: PPT but Private Construction due to Oppenheim and several Horodeckis Let and are data-hiding states --- separable, orthogonal but indistinguishable via LOCC ρ AA gives a full bit of secret key, but gives substantially less pure entanglement Proof: C mon. To get at the PRs, you d have to distinguish the τ s, but you cant.

28 Zero Quantum Capacity Channels: Start with PPT but Private A key ρ AA A 0 0 shield

29 Zero Quantum Capacity Channels: Start with PPT but Private Add noise to the key system until the whole state is PPT A key ρ AA A 0 0 shield Gives Q = 0 but P > 0

30 Outline Quantum and Private Capacities Channels with Zero Quantum Capacity Superactivation of Quantum Capacity Superadditivity of Private Capacity Applications

31 Superactivation of Quantum Capacity Theorem: Let N be any channel and be a 50% erasure channel. Then Q(N ) (1/2)P(N ). Corollary: There are N and with Q(N ) = Q() = 0 but Q(N ) > 0. Reminder: (ρ) = ½(ρ + eihe )

32 Superactivation of Quantum Capacity: Proof Ideas N has private capacity, so we can use it to make private states. Send the A part of the shield system through the 50% erasure channel,. When the shield is transmitted (½ the time) they establish a maximally entangled state. Otherwise, the state is not too noisy. The resulting state has coherent information ½ P(N) + ½ 0 = ½ P(N), so the N and can be used to transmit quantum information at this rate. A A 0 0 ob

33 Superactivation of Quantum Capacity: Proof Ideas N has private capacity, so we can use it to make private states. Send the A part of the shield system through the 50% erasure channel,. When the shield is transmitted (½ the time) they establish a maximally entangled state. Otherwise, the state is not too noisy. The resulting state has coherent information ½ P(N) + ½ 0 = ½ P(N), so the N and can be used to transmit quantum information at this rate. ½ ½ A A 0 0 A A + A ob ob ob A 0 Lost

34 Interpretation This superactivation effect is completely different from what happens in classical theory, and depends crucially on choosing inputs entangled between the channels. Somehow, each channel can transmit different sorts of quantum information---neither sort is sufficient on its own, but together they can be used for the usual noiseless variety. What are the different kinds of information? Is privacy important?

35 Outline Quantum and Private Capacities Channels with Zero Quantum Capacity Superactivation of Quantum Capacity Superadditivity of Private Capacity Applications

36 Rocket Channels R d = ( R U,V d UVihUV ) A 1 U U,V A 2 V U,V This channel has classical capacity 2 Proof: C mon!

37 High Joint Capacity What channel should we use R d with? We ll use a 50% erasure channel: (ρ) = ½ ρ + ½ eraseiherase How should we use it?

38 High Joint Capacity A A 1/2 U V U V = + 0 A 0 1/2 I/2 U V eraseih erase

39 Coherent Information: average of two terms A 1/2 U V ½I coh (not erased) + 1/2 I/2 U V 0 A + ½I coh (erased) eraseih erase

40 Coherent Information: average of two terms A 1/2 U V ½I coh (not erased) + 1/2 I/2 U V 0 A + ½I coh (erased) eraseih erase This term is zero.

41 Coherent Information: average of two terms A 1/2 U V ½I coh (not erased) + 1/2 I/2 U V 0 A + ½I coh (erased) How big is this? eraseih erase This term is zero.

42 A nice identity A = A T A I φ d i = A i,j ii ji I φ d i = j,i ii ji φ d i = ii ii

43 ob can undo interaction A U V = U 0 V T 0

44 ob can undo interaction Coherent information doesn t increase when ob operates, so when there s no erasure, I coh = log d The overall coherent information, and therefore capacity satisfies Q(R d ) I coh ½ log d

45 Superadditivity of Privacy So, what we have is two channels, one with private capacity P() = 0 and the other with P(R d ) C(R d ) 2 The joint quantum capacity is Q(R d ) ½ log d This tells us that the private capacity is strongly superadditive, but also that privacy was not the necessary component for superactivation of quantum capacity. We were beaten to the punch by Carl Li, Andreas Winter, etc.. Consoled ourselves by concentrating on the simplicity of our channels and extensive violations.

46 Outline Quantum and Private Capacities Channels with Zero Quantum Capacity Superactivation of Quantum Capacity Superadditivity of Private Capacity Applications

47 More implications The quantum capacity is not convex: Coherent information is a poor approximation to the quantum capacity (failure of random coding):

48 Application: ig gap between Coherent Information and Capacity R R Q 1 (T ) 2 since it s no more than max (Q 1 (R),Q 1 ()) R Make one channel R and the other to get large Q 1 (T T )

49 Application: ig gap between Coherent Information and Capacity R R Q 1 (T ) 2 since it s no more than max (Q 1 (R),Q 1 ()) R Gives Q 1 (T ) 2 but Make one channel R and the other to get large Q 1 (T T ) Q(T ) 1/8 log D in

50 Summary The quantum capacity characterizes achievable rates of quantum transmission. There are different types of zero quantum capacity channels (PPT, symmetric) Taking one from each class leads us to superactivation of quantum capacity. This nonadditivity shows that the value of a channel for quantum transmission depends on the context in which it is used. ven though it looked like privacy was important, we can still get an effect with channels that have very little.

51 Questions When does superactivation occur? Quantify different types of information? Measure Value Added instead? What are the zero P and zero Q channels? Specifically, is there a zero P channel near the rocket channels? What resource do the rocket channels provide? Three-way interactions?

52