1. J. Eisert University of Potsdam, Germany Entanglement and transfer of quantum information Cambridge, September 2004 Optimizing linear optics quantum gates

2. Quantum computation with linear optics

3. Effective non-linearities Output Optical network Input  Photons are relatively prone to decoherence, precise state control is possible with linear optical elements  Universal quantum computation can be done using optical systems only  The required non-linearities can be effectively obtained …

4. Effective non-linearities Output Optical network Input  Photons are relatively prone to decoherence, precise state control is possible with linear optical elements  Universal quantum computation can be done using optical systems only  The required non-linearities can be effectively obtained …

5. Effective non-linearities Output Optical network Input ?  Photons are relatively prone to decoherence, precise state control is possible with linear optical elements  Universal quantum computation can be done using optical systems only  The required non-linearities can be effectively obtained …

6. Effective non-linearities Output Auxiliary modes, Auxiliary photons Linear optics network Input  by employing appropriate measurements Measurements  Photons are relatively prone to decoherence, precise state control is possible with linear optical elements  Universal quantum computation can be done using optical systems only  The required non-linearities can be effectively obtained …

7. KLM scheme Knill, Laflamme, Milburn (2001): Universal quantum computation is possible with  Single photon sources  linear optical networks  photon counters, followed by postselection and feedforward E Knill, R Laflamme, GJ Milburn, Nature 409 (2001) TB Pittman, BC Jacobs, JD Franson, Phys Rev A 64 (2001) JL O’ Brien, GJ Pryde, AG White, TC Ralph, D Branning, Nature 426 (2003) Output Auxiliary modes, Auxiliary photons Linear optics network Input Measurements

8. Non-linear sign shifts  At the foundation of the KLM contruction is a non-deterministic gate, - the non-linear sign shift gate, acting as NSS  Using two such non-linear sign shifts, one can construct a control-sign and a control-not gate

9. Success probabilities  At the foundation of the KLM contruction is a non-deterministic gate, - the non-linear sign shift gate, acting as  Using teleportation, the overall scheme can be uplifted to a scalable scheme with close-to-unity success probability, using a significant overhead in resources  To efficiently use the gates, one would like to implement them with as high a probability as possible

10. Central question of the talk  How well can the elementary gates be performed with - static networks of arbitrary size, - using any number of auxiliary modes and photons, - making use of linear optics and photon counters, followed by postselection?  Meaning, what are the optimal success probabilities of elementary gates?

11. Central question of the talk Seems a key question for two reasons:  Quantity that determines the necessary overhead in resources  For small-scale applications such as quantum repeaters, high fidelity of the quantum gates may often be the demanding requirement of salient interest (abandon some of the feed-forward but rather postselect)  How well can the elementary gates be performed with - static networks of arbitrary size, - using any number of auxiliary modes and photons, - making use of linear optics and photon counters, followed by postselection?  Meaning, what are the optimal success probabilities of elementary gates?

12. Networks for the non-linear sign shift Input: Output:

13. Networks for the non-linear sign shift Input: Output: Network of linear optics elements  Success probability (obviously, as the non-linearity is not available)

14. Networks for the non-linear sign shift Input: Output: Auxiliary mode  Success probability (the relevant constraints cannot be fulfilled) Photon counter Network of linear optics elements

15. Networks for the non-linear sign shift Input: Output: Auxiliary modes  Success probability (the best known scheme has this success probability Photon counters Network of linear optics elements

16. Networks for the non-linear sign shift Input: Output:  Success probability (the best known scheme has this success probability E Knill, R Laflamme, GJ Milburn, Nature 409 (2001) Alternative schemes: S Scheel, K Nemoto, WJ Munro, PL Knight, Phys Rev A 68 (2003) TC Ralph, AG White, WJ Munro, GJ Milburn, Phys Rev A 65 (2001)

17. Networks for the non-linear sign shift Input: Output: Auxiliary modes  Success probability Photon counters Network of linear optics elements

18. Networks for the non-linear sign shift Input: Output: Auxiliary modes  Success probability Photon counters Network of linear optics elements

19. Short history of the problem for the non-linear sign-s  Knill, Laflamme, Milburn/Ralph, White, Munro, Milburn, Scheel, Knight (2001-2003): Construction of schemes that realize a non-linear sign shift with success probability 1/4  Knill (2003): Any scheme with postselected linear optics cannot succeed with a higher success probability than 1/2  Reck, Zeilinger, Bernstein, Bertani (1994)/ Scheel, Lütkenhaus (2004): Network can be written with a single beam splitter communicating with the input Conjectured that probability 1/4 could already be optimal  Aniello (2004) Looked at the problem with exactly one auxiliary photon E Knill, R Laflamme, GJ Milburn, Nature 409 (2001) TC Ralph, AG White, WJ Munro, GJ Milburn, Phys Rev A 65 (2001) S Scheel, K Nemoto, WJ Munro, PL Knight, Phys Rev A 68 (2003) M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994) S Scheel, N Luetkenhaus, New J Phys 3 (2004)

20. (A late) overview over the talk  Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics  Why is this a difficult problem? J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

21. (A late) overview over the talk  Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics  Why is this a difficult problem?  Help from an unexpected side: Methods from semidefinite programming and convex optimization as practical analytical tools J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

22. (A late) overview over the talk  Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics  Why is this a difficult problem?  Help from an unexpected side: Methods from semidefinite programming and convex optimization as practical analytical tools  Formulate strategy: will develop a general recipe to give rigorous bounds on success probabilities  Look at more general settings, work in progress J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

23. (A late) overview over the talk  Finding optimal success probabilities of elementary gates within the paradigm of postselected linear optics  Why is this a difficult problem?  Help from an unexpected side: Methods from semidefinite programming and convex optimization as practical analytical tools  Formulate strategy: will develop a general recipe to give rigorous bounds on success probabilities  Look at more general settings, work in progress  Finally: stretch the developed ideas a bit further:  Experimentally accessible entanglement witnesses for imperfect photon detectors  Complete hierarchies of tests for entanglement J Eisert, quant-ph/0409156 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 J Eisert, M Curty, M Luetkenhaus, work in progress WJ Munro, S Scheel, K Nemoto, J Eisert, work in progress

24. Quantum gates Input:Output:  These are the quantum gates we will be looking at in the following (which include the non-linear sign shift)

25. Quantum gates Arbitrary number of additional field modes auxiliary photons (Potentially complex) networks of linear optics elements Input:Output:

26. Quantum gates Arbitrary number of additional field modes auxiliary photons Input:Output: (Potentially complex) networks of linear optics elements

27. Quantum gates Arbitrary number of additional field modes auxiliary photons Input:Output: (Potentially complex) networks of linear optics elements M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994) S Scheel, N Luetkenhaus, New J Phys 3 (2004)

28. The input is linked only once to the auxiliary modes  State vector of auxiliary modes “preparation” Input:Output:  “measure- ment” (Potentially complex) networks of linear optics elements M Reck, A Zeilinger HJ Bernstein, P Bartani, Phys Rev Lett 73 (1994) S Scheel, N Luetkenhaus, New J Phys 3 (2004)

29. Finding the optimal success probability  State vector of auxiliary modes “preparation”  “measure- ment” Input:Output:

30. Finding the optimal success probability  State vector of auxiliary modes “preparation”  “measure- ment”  Single beam splitter, characterized by complex transmittivity

31. Finding the optimal success probability  Arbitrarily many ( ) states of arbitrary or infinite dimension  State vector of auxiliary modes “preparation”  “measure- ment”

32. Finding the optimal success probability  Arbitrarily many ( ) states of arbitrary or infinite dimension  Non-convex function (exhibiting many local minima)  State vector of auxiliary modes “preparation”  “measure- ment”  Weights

33. The problem with non-convex problems  This innocent-looking problem of finding the optimal success probability may be conceived as an optimization problem, but one which is - non-convex and - infinite dimensional, as we do not wish to restrict the number of - photons in the auxiliary modes - auxiliary modes - linear optical elements

34. The problem with non-convex problems Infinitely many local maxima  This innocent-looking problem of finding the optimal success probability may be conceived as an optimization problem, but one which is - non-convex and - infinite dimensional, as we do not wish to restrict the number of - photons in the auxiliary modes - auxiliary modes - linear optical elements

35. The problem with non-convex problems Infinitely many local maxima

36. The problem with non-convex problems Infinitely many local maxima

37. The problem with non-convex problems Infinitely many local maxima

38. The problem with non-convex problems Infinitely many local maxima  Somehow, it would be good to arrive from the “other side”

39. The problem with non-convex problems Infinitely many local maxima  Somehow, it would be good to arrive from the “other side”

40. The problem with non-convex problems Infinitely many local maxima  Somehow, it would be good to arrive from the “other side”

41. The problem with non-convex problems Infinitely many local maxima  Somehow, it would be good to arrive from the “other side”  This is what we will be trying to do…

42. Convex optimization? Can it help?

43. Convex optimization problems  Find the minimum of a convex function over a convex set  What is a convex optimization problem again?

44. Convex optimization problems  Find the minimum of a convex function over a convex set  What is a convex optimization problem again? Function Set

45. Convex optimization problems Function Set  Find the minimum of a convex function over a convex set  What is a convex optimization problem again?

46. Semidefinite programs Function Set  Class of convex optimization problems that we will make use of - is efficiently solvable (but we are now not primarily dealing with numerics), - and is a powerful analytical tool:  So-called semidefinite programs

47. Semidefinite programs Set  Class of convex optimization problems that we will make use of - is efficiently solvable (but we are now not primarily dealing with numerics), - and is a powerful analytical tool:  So-called semidefinite programs Minimize the linear multivariate function subject to the constraint Linear function Matrices Vector  We will see in a second why they are so helpful

48. Yes, ok, … … but why should this help us to assess the performance of quantum gates in the context of linear optics?

49. 1. Recasting the problem  Again, the output of the quantum network, depending on preparations and measurements, can be written as for all  Here J Eisert, quant-ph/0409156  Functioning of the gate requires that

50. 1. Recasting the problem for all  Here J Eisert, quant-ph/0409156  Functioning of the gate requires that  After all, the (i) success probability should be maximized, (ii) provided that the gate works

51. 1. Recasting the problem for all  Here J Eisert, quant-ph/0409156  Functioning of the gate requires that  But then, the problem is a non-convex infinite dimensional problem, involving polynomials of arbitrary order in the transmittivity  The strategy is now the following… (one which can be applied to a number of contexts)

52. The strategy Maximize over all - transmittivities - complex scalar products - weights 1.

53. The strategy Maximize over all - transmittivities - complex scalar products - weights - transmittivities - complex scalar products Maximize over all - weights Isolate very difficult part of the problem 1. 2.

54. The strategy Maximize over all - transmittivities - complex scalar products - weights - transmittivities - complex scalar products Maximize over all - weights Isolate very difficult part of the problem Relax to make convex 1. 2.

55. 3. Writing it as a semidefinite problem  Then for each the problem is found to be one with matrix constraints the elements of which are polynomials of arbitrary degree in  The resulting problem may look strange, but it is actually a semidefinite program  Why does this help us?

56. 4. Lagrange duality  Primal problem  Dual problem  That is, for each problem, one can construct a so-called “dual problem”  Because we can exploit the (very helpful) idea of Lagrange duality Lagrange duality  Both are semidefinite problems

57. 4. Lagrange duality Globally optimal point Original (primal) problem

58. 4. Lagrange duality Globally optimal point Original (primal) problem Dual problem

59. 4. Lagrange duality Globally optimal point Original (primal) problem Dual problem  Every solution (!) of the dual problem (any educated guess) is a bound for the optimal solution of the primal problem Educated guess “Approaching the problem from the other side”

60. 4. Lagrange duality Globally optimal point Original (primal) problem Dual problem  Every solution (!) of the dual problem (any educated guess) is a bound for the optimal solution of the primal problem “Approaching the problem from the other side” Educated guess

61. 4. Lagrange duality Globally optimal point Original (primal) problem Dual problem  Every solution (!) of the dual problem (any educated guess) is a bound for the optimal solution of the primal problem “Approaching the problem from the other side” Educated guess

62. The strategy Maximize over all - transmittivities - complex scalar products - weights - transmittivities - complex scalar products Maximize over all - weights Isolate very difficult part of the problem - transmittivities - complex scalar products Minimize dual over all - matrices Make use of idea of Lagrange duality “approaching from the other side” “ Semidefinite program in - matrix 1. 2. 3. 4.

63.  The dual can be shown to be of the form as optimization problem (still infinite dimensional) in vectors and 5. Construction of a solution for the dual minimize subject to The entries of the matrices are polynomials of degree in the transmittivity

64.  The dual can be shown to be of the form 5. Construction of a solution for the dual minimize subject to  Finding a solution now means “guessing” a matrix  Can be done, even in a way such that the unwanted dependence on preparation and measurement can be eliminated  Instance of a problem that can explicitly solved

65. The strategy Maximize over all - transmittivities - complex scalar products - weights - transmittivities - complex scalar products Maximize over all - weights Isolate very difficult part of the problem - transmittivities - complex scalar products Minimize dual over all - matrices Make use of idea of Lagrange duality “approaching from the other side” “ Semidefinite program in - matrix 1. 2. 3. 4.

66. The strategy Maximize over all - transmittivities - complex scalar products - weights - transmittivities - complex scalar products Maximize over all - weights Isolate very difficult part of the problem - transmittivities - complex scalar products Minimize dual over all - matrices Make use of idea of Lagrange duality “approaching from the other side” Construct explicit solution, independent from “ Semidefinite program in - matrix 1. 2. 3. 4. 5.

67. The strategy Maximize over all - transmittivities - complex scalar products - weights - transmittivities - complex scalar products Maximize over all - weights Isolate very difficult part of the problem - transmittivities - complex scalar products Minimize dual over all - matrices Make use of idea of Lagrange duality “approaching from the other side” Construct explicit solution, independent from This gives a general bound for the original problem (optimal success probability) “ Semidefinite program in - matrix 1. 2. 3. 4. 5.

68.  For the non-linear sign shift, e.g., one can construct a solution for the dual problem for each  This solution delivers in each case 6. Done! Educated guess Optimal point 0.25 0 0.5 0.75 1.0

69.  For the non-linear sign shift, e.g., one can construct a solution for the dual problem for each  This solution delivers in each case using the argument of Lagrange duality, we are done! 6. Done! Educated guess Optimal point 0.25 0 0.5 0.75 1.0

70. 6. Done! Educated guess Optimal point 0.25 0 0.5  For the non-linear sign shift, e.g., one can construct a solution for the dual problem for each  This solution delivers in each case using the argument of Lagrange duality, we are done!  So, this gives a bound for the original problem … … and one of which we know it is optimal

71. Realizing a non-linear sign shift gate Auxiliary modes Photon counters J Eisert, quant-ph/0409156

72. Realizing a non-linear sign shift gate Auxiliary modes Photon counters  No matter how hard we try, there is within the paradigm of linear optics, photon counting, followed by postselection, no way to go beyond the optimal success probability of J Eisert, quant-ph/0409156

73. Realizing a non-linear sign shift gate Auxiliary modes Photon counters  Surprisingly: any additional resources in terms of modes/photons than two auxiliary modes/photons do not lift up the success probability at all J Eisert, quant-ph/0409156

74.  The same method can be immediately applied to other quantum gates, e.g., to the sign-shift with phase Success probabilities of other sign gates

75.  The same method can be immediately applied to other quantum gates, e.g., to the sign-shift with phase Success probabilities of other sign gates 0 0.25 0.5 0.75 1 0

76.  The same method can be immediately applied to other quantum gates, e.g., to the sign-shift with phase Success probabilities of other sign gates 0 0.25 0.5 0.75 1 0 “do nothing” Non-linear sign shift gate

77.  The same method can be immediately applied to other quantum gates, such as those involving higher photon numbers Higher photon numbers 0 0.25 0.5 0.75 1 0 “do nothing”

78. Non-linear sign shift with one step of feed-forward Input: Output:  Assessing success probabilities with single rounds of classical feedback Work in progress with WJ Munro, P Kok, K Nemoto, S Scheel

79. incorrectable failures Non-linear sign shift with one step of feed-forward Input: Output:  Assessing success probabilities with single rounds of classical feedback Work in progress with WJ Munro, P Kok, K Nemoto, S Scheel success failure success Potentially correctable failures

80. Feed-forward seems not to help so much  Then, it turns out that whenever we choose an optimal gate in the first run, succeeding with …  … then any classical feedforward follows by a correction network can increase the success probability to at most  That is, single rounds of feed-forward at the level of individual gates do not help very much at all!

82. Finally, … extending these ideas to find other tools relevant to optical settings Joint work with P Hyllus, O Gühne, M Curty, N Lütkenhaus

83. Stretch these ideas further to get practical tools What was the point of the method before?  We developed a strategy to make methods from convex optimization applicable to solve a - non-convex and - infinite-dimensional problem to assess linear optical schemes

84. Stretch these ideas further to get practical tools What was the point of the method before?  We developed a strategy to make methods from convex optimization applicable to solve a - non-convex and - infinite-dimensional problem to assess linear optical schemes Can such strategies also formulated to find  good experimentally accessible witnesses to detect entanglement, which work for - weak pulses and - finite detection efficiencies?  Practical tools to construct complete hierarchies of criteria for multi-particle entanglement?

85. Entanglement witnesses  Scenario 1: (experimentally) detecting entanglement directly Unknown state “Yes, it is entangled!” “Hm, I don’t know” “Yes, it is entangled!” “Hm, I don’t know” Entanglement witness  An entanglement witness is an observable with Experiment: M Barbieri et al, Phys Rev Lett 91 (2003) M Bourennane et al, Phys Rev Lett 92 (2004) Theory: M & P & R Horodecki, Phys Lett A 232 (1996) BM Terhal, Phys Lett A 271 (2000) G Toth, quant-ph/0406061

86. Entanglement witnesses  Entanglement witnesses are important tools - if complete state tomography is inaccessible/expensive - in quantum key distribution: necessary for the positivity of the intrinsic information is that quantum correlations can be detected U Maurer, S Wolf, IEEE Trans Inf Theory 45 (1999) N Gisin, S Wolf, quant-ph/0005042 M Curty, M Lewenstein, N Luetkenhaus, Phys Rev Lett 92 (2004)

87. Entanglement witnesses Problem: a number of entanglement witnesses are known - for the two-qubit case all can be classified - but,  It is difficult (actually NP) to find optimal entanglement witnesses in general  Include non-unit detection efficiencies: detectors do not click although they should have Imperfect detectors modeled by perfect detectors, preceded by beam splitter

88. Entanglement witnesses  Starting from any entanglement witness, one always gets an optimal one if one knows  Then is an optimal witness  But: how does one find this global minimum? We have to be sure, otherwise we do not get an entanglement witness subject to being a product state vector

89. The strategy Minimize over all state vectors 1.

90. The strategy Minimize over all state vectors Equivalently 1. Minimize over all with 2.

91. The strategy Minimize over all state vectors Equivalently 1. Minimize over all with 2.  Any operator that satisfies is necessarily a pure state (polynomial characterization of pure states) NS Jones, N Linden, quant-ph/0407117 J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135

92. The strategy Minimize over all state vectors Equivalently 1. Minimize over all with 2. Write this as a hierarchy of semidefinite programs 3. Use methods from relaxation theory of polynomially constrained problems

93. The strategy Write this as a hierarchy of semidefinite programs 3. Minimize over all state vectors Equivalently 1. Minimize over all with 2.  Any polynomially constrained problem typically computationally hard NP problems can be relaxed to efficiently solvable semidefinite programs  One can even find hierarchies that approximate the solution to arbitrary accuracy J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 JB Lasserre, SIAM J Optimization 11 (2001)

94. The strategy Minimize over all state vectors Equivalently 1. Minimize over all with 2. Write this as a hierarchy of semidefinite programs 3. Since each step gives a lower bound, each step gives an entanglement witness 4. Use methods from relaxation theory of polynomially constrained problems

95. Entanglement witnesses for finite detection efficiencies  In this way, one can obtain good entanglement witnesses for imperfect detectors subject to being a product state vector  Starting from a witness in an error-free setting, one gets a new optimal witness as J Eisert, M Curty, N Luetkenhaus, work in progress

96. Hierarchies of criteria for multi-particle entanglement (e.g., from state tomography) 1st sufficient criterion for entanglement “Yes, it is entangled!” “I don’t know” Known state J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 Compare also AC Doherty, PA Parillo, FM Spedalieri, quant-ph/0407156  Scenario 2: complete hierarchies of sufficient criteria for multi-particle entanglement: Testing whether a known state (i.e., from state tomography) is multi-particle entangled

97. Hierarchies of criteria for multi-particle entanglement (e.g., from state tomography) 1st sufficient criterion for entanglement “Yes, it is entangled!” “I don’t know” Known state 2nd sufficient criterion for entanglement “Yes, it is entangled!” “I don’t know” J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 Compare also AC Doherty, PA Parillo, FM Spedalieri, quant-ph/0407156  Scenario 2: complete hierarchies of sufficient criteria for multi-particle entanglement: Testing whether a known state (i.e., from state tomography) is multi-particle entangled

98. Hierarchies of criteria for multi-particle entanglement (e.g., from state tomography) 1st sufficient criterion for entanglement “Yes, it is entangled!” “I don’t know” Known state 2nd sufficient criterion for entanglement “Yes, it is entangled!” “I don’t know”  Every entangled state is necessarily detected in some step of the hierarchy J Eisert, P Hyllus, O Guehne, M Curty, quant-ph/0407135 Compare also AC Doherty, PA Parillo, FM Spedalieri, quant-ph/0407156  Scenario 2: complete hierarchies of sufficient criteria for multi-particle entanglement: Testing whether a known state (i.e., from state tomography) is multi-particle entangled

99. The lessons to learn Physically:  We have developed a strategy which is applicable to assess/find optimal linear optical schemes  Non-linear sign shift with linear optics, photon counting followed by postselection cannot be implemented with higher success probability than 1/4  First tight general upper bound for success probability - Good news: the most feasible protocol is already the optimal one - Single rounds of feedforward do not help very much - But: also motivates the search for hybrid methods, leaving the strict framework of linear optics (see following talk by Bill Munro)

100. The lessons to learn Formally: the methods of convex optimization  are powerful when one intends to assess the maximum performance of linear optics schemes without restricting the allowed resources  in a situation where it is hard to conceive that one can find an direct solution to the original problem  And finally, we had a look at where very much related ideas can be useful to detect entanglement in optical settings