Bound entangled states with secret key and their classical counterpart

Transcription

1 Bound entangled states with secret key and their classical counterpart Māris Ozols Graeme Smith John Smolin February 6, 2014

2 A brief summary Main result A new construction of bound entangled states with secret key Steps involved 1. Understand what is the classical analogue of this 2. Construct a probability distribution P ABE that has the desired properties 3. Set ψ ABE = P ABE

3 A brief summary Main result A new construction of bound entangled states with secret key Steps involved 1. Understand what is the classical analogue of this 2. Construct a probability distribution P ABE that has the desired properties 3. Set ψ ABE = P ABE The significance of trash in cryptography

4 Shared entanglement and randomness Cryptographic and computational resource Alice Bob ρ AB

5 Shared entanglement and randomness Cryptographic and computational resource Joint state: ψ ABE or P ABE Eve Alice Bob ψ ABE

6 Shared entanglement and randomness Cryptographic and computational resource Joint state: ψ ABE or P ABE When is such resource useful? Eve Alice Bob ψ ABE

7 Perfect resources When is P ABE a perfect resource? 1. Identical for A and B 2. Uniformly random 3. Private from E

8 Perfect resources When is P ABE a perfect resource? 1. Identical for A and B 2. Uniformly random 3. Private from E P ABE = { KEY AB TRASH E 0 A 0 B w.p. 1/2 KEY AB = 1 A 1 B w.p. 1/2

9 Perfect resources When is P ABE a perfect resource? 1. Identical for A and B 2. Uniformly random 3. Private from E P ABE = { KEY AB TRASH E 0 A 0 B w.p. 1/2 KEY AB = 1 A 1 B w.p. 1/2 ψ ABE = KEY AB TRASH E KEY AB = 1 2 ( 00 AB + 11 AB

10 Distillation Eve Alice Bob ψ ABE

11 Distillation Eve LOCC.. Alice Bob ψ ABE

12 Distillation Eve LOCC.. Alice ψ ABE KEY k AB TRASH E Bob

13 Distillation Eve LOCC.. Alice KEY k ψ AB ABE TRASH E Bob Entanglement distillation rate 1 ( D( ψ ABE := lim # of KEY AB from ψ n n n ABE via LOCC

14 More distillation rates ψ ABE LOCC KEY AB P ABE LOPC KEY AB D( ψ ABE entanglement distillation rate K(P ABE key distillation rate

15 LOCC More distillation rates ψ ABE LOCC KEY AB P ABE LOPC KEY AB D( ψ ABE entanglement distillation rate K(P ABE key distillation rate K( ψ ABE key distillation rate

16 More distillation rates ψ ABE LOCC KEY AB = ( LOCC measure P ABE LOPC KEY AB = ( D( ψ ABE entanglement distillation rate K(P ABE key distillation rate K( ψ ABE key distillation rate

17 More distillation rates ψ ABE LOCC KEY AB = ( LOCC measure P ABE LOPC KEY AB = ( D( ψ ABE entanglement distillation rate K(P ABE key distillation rate K( ψ ABE key distillation rate K( ψ ABE D( ψ ABE

18 More distillation rates ψ ABE LOCC KEY AB = ( LOCC measure P ABE LOPC KEY AB = ( D( ψ ABE entanglement distillation rate K(P ABE key distillation rate K( ψ ABE key distillation rate K( ψ ABE D( ψ ABE Bound entanglement ψ ABE is entangled but D( ψ ABE = 0

19 More distillation rates ψ ABE LOCC KEY AB = ( LOCC measure P ABE LOPC KEY AB = ( D( ψ ABE entanglement distillation rate K(P ABE key distillation rate K( ψ ABE key distillation rate K( ψ ABE D( ψ ABE Bound entanglement ψ ABE is entangled but D( ψ ABE = 0 Private bound entanglement ψ ABE is bound entangled but K( ψ ABE > 0 [HHHO05]

20 Entanglement vs classical key A E B Compare T A ( Ψ 1 := AB + 11 AB ϕ TA T B φ E ( Ψ 2 := AB ϕ TA T B + 11 AB ϕ TA T B φ E T B

21 Entanglement vs classical key A E B Compare T A ( Ψ 1 := AB + 11 AB ϕ TA T B φ E ( Ψ 2 := AB ϕ TA T B + 11 AB ϕ TA T B φ E Observe If T A and T B are discarded, Ψ 1 contains a quantum KEY whereas Ψ 2 contains only a classical KEY T B

22 Entanglement vs classical key A E B Compare T A T B ( Ψ 1 := AB + 11 AB ϕ TA T B φ E ( Ψ 2 := AB ϕ TA T B + 11 AB ϕ TA T B φ E Observe If T A and T B are discarded, Ψ 1 contains a quantum KEY whereas Ψ 2 contains only a classical KEY Quantum KEY is immune against Eve accessing T A and T B (due to monogamy of entanglement

23 Distillation with remanent devices ψ ABE.. Alice Bob

24 Distillation with remanent devices ψ ABE.. Alice Bob Extra assumption At the end of the protocol Eve confiscates both devices; she can recover all information that was erased Alice and Bob can keep only the key

25 Distillation with remanent devices φ TA T B E.. Alice Bob EPR AB Extra assumption At the end of the protocol Eve confiscates both devices; she can recover all information that was erased Alice and Bob can keep only the key

26 Distillation with remanent devices φ TA T B E.. Alice Bob EPR AB Extra assumption At the end of the protocol Eve confiscates both devices; she can recover all information that was erased Alice and Bob can keep only the key

27 Quantum distillation ψ ABE Ψ A A E T A LOCC E B B Classical distillation P ABE Q A A E T A LOPC E B B T B ( AB + 11 AB ϕ TA T B φ E ( AB + 11 AB ϕ TA T B φ E entangled key classical key? ( AB ϕ TA T B + 11 AB ϕ TA T B φ E ( AB ϕ TA T B + 11 AB ϕt A T B φ E classical key classical key? T B

28 Quantum distillation ψ ABE Ψ A A E T A LOCC E B B Classical distillation P ABE Q A A E T A LOPC E B B T B ( AB + 11 AB ϕ TA T B φ E ( AB + 11 AB ϕ TA T B φ E entangled key classical key? ( AB ϕ TA T B + 11 AB ϕ TA T B φ E ( AB ϕ TA T B + 11 AB ϕt A T B φ E classical key classical key? T B Private randomness There are two types of private randomness!

29 Quantum distillation ψ ABE Ψ A A E T A LOCC E B B Classical distillation P ABE Q A A E T A LOPC E B B T B ( AB + 11 AB ϕ TA T B φ E ( AB + 11 AB ϕ TA T B φ E entangled key classical key? ( AB ϕ TA T B + 11 AB ϕ TA T B φ E ( AB ϕ TA T B + 11 AB ϕt A T B φ E classical key classical key? T B Private randomness There are two types of private randomness! Classical analog of entanglement [CP02] is private randomness that is distilled on remanent devices

30 Quantum distillation ψ ABE Ψ A A E T A LOCC E B B Classical distillation P ABE Q A A E T A LOPC E B B T B ( AB + 11 AB ϕ TA T B φ E ( AB + 11 AB ϕ TA T B φ E entangled key classical key? ( AB ϕ TA T B + 11 AB ϕ TA T B φ E ( AB ϕ TA T B + 11 AB ϕt A T B φ E classical key classical key? T B Private randomness There are two types of private randomness! Classical analog of entanglement [CP02] is private randomness that is distilled on remanent devices This resource needs a new name...

31 cl[assical] en[t]anglement = enclanglement

32 Remanent devices Regular devices T A T B T A T B KEY AB D ψ ABE K KEY AB KEY AB D P ABE K KEY AB

33 Remanent devices Regular devices T A T B T A T B KEY AB D ψ ABE Private bound entanglement D( ψ ABE = 0 but K( ψ ABE > 0 K KEY AB KEY AB D P ABE Private bound enclanglement D(P ABE = 0 but K(P ABE > 0 K KEY AB

34 Main results Theorem 1 If ψ ABE = P ABE is quantical then D(ψ ABE D(P ABE and K(ψ ABE K(P ABE Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0

35 Main results Theorem 1 If ψ ABE = P ABE is quantical then D(ψ ABE D(P ABE and K(ψ ABE K(P ABE Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0 Corollaries 1. New construction of private bound entanglement 2. Noise helps in one-way classical key distillation

36 quant[um] + [class]ical = quantical

37 Quantical distributions / states P ABE is quantical if any single party s state can be unambiguously determined by the rest of the parties: b, e : {a : p(a, b, e = 0} 1 a, b : {e : p(a, b, e = 0} 1 a, e : {b : p(a, b, e = 0} 1 Similar distributions have appeared in [OSW05, CEH + 07]

38 Quantical distributions / states P ABE is quantical if any single party s state can be unambiguously determined by the rest of the parties: b, e : {a : p(a, b, e = 0} 1 a, b : {e : p(a, b, e = 0} 1 a, e : {b : p(a, b, e = 0} 1 If P ABE is quantical, we call ψ ABE := P ABE quantical too Similar distributions have appeared in [OSW05, CEH + 07]

39 Quantical distributions / states P ABE is quantical if any single party s state can be unambiguously determined by the rest of the parties: b, e : {a : p(a, b, e = 0} 1 a, b : {e : p(a, b, e = 0} 1 a, e : {b : p(a, b, e = 0} 1 If P ABE is quantical, we call ψ ABE := P ABE quantical too Dual nature Quantical P ABE and ψ ABE describe the same entity Entropic quantities for P ABE and ψ ABE agree Quantical P ABE has a classical Schmidt decomposition w.r.t. any bipartition Similar distributions have appeared in [OSW05, CEH + 07]

40 Theorem 1 If ψ ABE = P ABE is quantical then D(ψ ABE D(P ABE and K(ψ ABE K(P ABE

41 Theorem 1 If ψ ABE = P ABE is quantical then D(ψ ABE D(P ABE and K(ψ ABE K(P ABE Proof idea 1. Classical protocol can be promoted to a quantum one ψ ( 2 P U U ψ 2 U ( 2???

42 Theorem 1 If ψ ABE = P ABE is quantical then D(ψ ABE D(P ABE and K(ψ ABE K(P ABE Proof idea 1. Classical protocol can be promoted to a quantum one 2. P ABE remains quantical throughout the protocol ψ ( 2 P U U ψ 2 U ( 2???

43 Theorem 1 If ψ ABE = P ABE is quantical then D(ψ ABE D(P ABE and K(ψ ABE K(P ABE Proof idea 1. Classical protocol can be promoted to a quantum one 2. P ABE remains quantical throughout the protocol 3. Entropic quantities for P ABE and ψ ABE agree ψ ( 2 P U U ψ 2 U ( 2???

44 Theorem 1 If ψ ABE = P ABE is quantical then D(ψ ABE D(P ABE and K(ψ ABE K(P ABE Proof idea 1. Classical protocol can be promoted to a quantum one 2. P ABE remains quantical throughout the protocol 3. Entropic quantities for P ABE and ψ ABE agree 4. Promoted protocol achieves the same rate ψ ( 2 P U U ψ 2 U ( 2???

45 Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0

46 Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0 Proof idea Choose P ABE so that ρ AB := Tr E ( ψ ψ ABE is PT-invariant

47 Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0 Proof idea Choose P ABE so that ρ AB := Tr E ( ψ ψ ABE is PT-invariant ρ Γ AB = ρ AB 0, hence ρ AB is PPT

48 Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0 Proof idea ( Choose P ABE so that ρ AB := Tr E ψ ψ ABE is PT-invariant ρab Γ = ρ AB 0, hence ρ AB is PPT D(P ABE = 0, since D(P ABE D(ψ ABE = 0

49 Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0 Proof idea Choose P ABE so that ρ AB := Tr E ( ψ ψ ABE is PT-invariant ρ Γ AB = ρ AB 0, hence ρ AB is PPT D(P ABE = 0, since D(P ABE D(ψ ABE = 0 One-way distillable key: K(P ABE max A X [ I(X; B I(X; E ]

50 Theorem 2 There exist quantical distributions P ABE with D(P ABE = 0 and K(P ABE > 0 Proof idea Choose P ABE so that ρ AB := Tr E ( ψ ψ ABE is PT-invariant ρ Γ AB = ρ AB 0, hence ρ AB is PPT D(P ABE = 0, since D(P ABE D(ψ ABE = 0 One-way distillable key: K(P ABE max A X Choose X = 2 and do numerics [ I(X; B I(X; E ]

51 Recipe for quanticality and PT-invariance B 0 1 A 0 1 o ( a 00 AB + b 01 AB x E + c 11 AB y E

52 Recipe for quanticality and PT-invariance B 0 1 A 0 1 o ( a 00 AB + b 01 AB x E + c 11 AB y E Quantical Union of disjoint cliques Each clique is diagonal (no repeated rows or columns

53 Recipe for quanticality and PT-invariance B 0 1 A 0 1 o ( a 00 AB + b 11 AB x E + c 01 AB y E Quantical Union of disjoint cliques Each clique is diagonal (no repeated rows or columns

54 Recipe for quanticality and PT-invariance B 0 1 A 0 1 o ( a 00 AB + b 11 AB x E + c 01 AB y E Quantical Union of disjoint cliques Each clique is diagonal (no repeated rows or columns PT-invariant Union of crosses Each cross has zero determinant

55 Recipe for quanticality and PT-invariance A 0 1 B 0 1 ( a 00 AB + b 11 AB x E + ( c 01 AB + d 10 AB y E Quantical Union of disjoint cliques Each clique is diagonal (no repeated rows or columns PT-invariant Union of crosses Each cross has zero determinant

56 Example in B A P AB = D(P ABE = 0 but K(P ABE

57 Example in 4 4 A B o o K(P ABE Better than [HPHH08] K(P ABE

58 Example in B A K(P ABE

59 Conclusions Results New construction of private bound entanglement Adding noise can help in one-way classical key distillation

60 Conclusions Results New construction of private bound entanglement Adding noise can help in one-way classical key distillation Open questions How does our construction relate to [HHHO05]? Is the optimal protocol for distilling entanglement or key from a quantical state also quantical?

61 Conclusions Results New construction of private bound entanglement Adding noise can help in one-way classical key distillation Open questions How does our construction relate to [HHHO05]? Is the optimal protocol for distilling entanglement or key from a quantical state also quantical? Work in progress... Quantical mechanics and a classical analogue of superactivation (of the quantical capacity

62 That s it! Image: destroyed Guardian computer with Snowden files

63 Bibliography I [CEH + 07] [CP02] Matthias Christandl, Artur Ekert, Michał Horodecki, Paweł Horodecki, Jonathan Oppenheim, and Renato Renner. Unifying classical and quantum key distillation. In Salil P. Vadhan, editor, Theory of Cryptography, volume 4392 of Lecture Notes in Computer Science, pages Springer, arxiv:quant-ph/ , doi: / _25. Daniel Collins and Sandu Popescu. Classical analog of entanglement. Phys. Rev. A, 65(3:032321, Feb arxiv:quant-ph/ , doi: /physreva [HHHO05] Karol Horodecki, Michał Horodecki, Paweł Horodecki, and Jonathan Oppenheim. Secure key from bound entanglement. Phys. Rev. Lett., 94(16:160502, Apr arxiv:quant-ph/ , doi: /physrevlett

64 Bibliography II [HPHH08] [OSW05] [PBR12] [Spe07] Karol Horodecki, Łukasz Pankowski, Michał Horodecki, and Paweł Horodecki. Low-dimensional bound entanglement with one-way distillable cryptographic key. Information Theory, IEEE Transactions on, 54(6: , arxiv:quant-ph/ , doi: /tit Jonathan Oppenheim, Robert W. Spekkens, and Andreas Winter. A classical analogue of negative information arxiv:quant-ph/ Matthew F. Pusey, Jonathan Barrett, and Terry Rudolph. On the reality of the quantum state. Nature Physics, 8(6: , arxiv: , doi: /nphys2309. Robert W. Spekkens. Evidence for the epistemic view of quantum staappendices oftes: A toy theory. Phys. Rev. A, 75(3:032110, Mar arxiv:quant-ph/ , doi: /physreva