Quantum Optics Theory
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1 Quantum Optics Theory PhD ICFO (blue=qoit) Armand Niederberger Christian Trefzger Ania Kubasiak Sibylle Braungardt Ulrich Ebling Alejandro Zamora Postdocs ICFO André Eckardt Omjyoti Dutta Remigiusz Augusiak Pietro Masignan John Lapeyre Gergely Szirmai Edina Szirmai Ex-Icfonians (QOIT external) Ujjwal Sen (faculty, Delhi) Aditi Sen(De) (faculty, Delhi) Géza Tóth (Bilbao) Chiara Menotti (Trento) Bogdan Damski (res. assoc., Los Alamos) Jarosław Korbicz (Gdańsk) Mirta Rodriguez (CSIC, Madrid) Caixa Manresa Fellows Lluis Masanes Fernando Cucchietti Philipp Hauke (PhD)
2 Spintomics Spin based quantum devices SPINTOMICS:The general objective of this subproject is to develop novel spin-based protocols and design novel spin-based devices, considering as physical systems trapped atoms and ions, ultracold atomic gases and disordered systems. QIT Pure: 8+1 (highlight: NJP on Quantum Kinetic Models) Disorder, Frustration, and Control: 4+1 (highlight: Nature Phys. Progress review) Quantum Communications and Networks: 3+2 (highlight: Multipartite percolation pending in PRL) Spintomics Pure: 3 (highlight: PRA on spinor Josephson effect) Dipolar: 2+2 (highlights: PRL on pair supersolid and ROP review) Atoms in Artificial Gauge Fields: 3+1 (highlight: Nature Phys. on quantum vortex nucleation)
3 Entanglement Percolation: Distributing entanglement on quantum networks Presenter: John Lapeyre QOIT Annual Meeting February 4, 2010
4 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
5 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
6 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
7 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
8 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
9 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
10 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
11 Quantum Information: Entanglement is a resource for tasks:teleportation, repeater, key distribution, fault tolerant computation Creating entanglement requires local interaction. Noise increases with distance. Depolarization. Absorption. Can t distribute entanglement over long distance in a single stage! Establish long range entanglement via Network of stations or nodes that store and purify a state. (Many problems: eg, memories in yesterday s session.) Generalization of quantum repeater schemes. Nodes share partially entangled states of qubits Nodes(stations)/channels, Vertices/edges, Sites/bonds Quantum operations probabilistic Large number of random components Statistical physics: Random networks Percolation theory
12 Antonio Acín ICFO Maciej Lewenstein ICFO Ignacio Cirac MPG Quantenoptik Garching A. Acín, J. I. Cirac, and M. Lewenstein Entanglement Percolation in Quantum Networks Nature Physics 3, 256 (2007).
13 Maciej Lewenstein ICFO Antonio Acín ICFO Ignacio Cirac MPG Quantenoptik Garching Jan Wehr Mathematics U of Arizona Sébastien Perseguers MPG Quantenoptik Garching S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr Entanglement distribution in pure-state quantum networks Phys. Rev. A 77, (2008).
14 Maciej Lewenstein Antonio Acín Ignacio Cirac Jan Wehr Sébastien Perseguers Daniel Cavalcanti John Lapeyre ICFO ICFO MPG Quantenoptik Garching Mathematics U of Arizona MPG Quantenoptik Garching ICFO ICFO
15 John Calsamiglia Martí Cuquet Dieter Jaksch Uwe Dorner Stuart Broadfoot Andrzej Grudka Joanna Modławska Universitat Autònoma de Barcelona Universitat Autònoma de Barcelona Clarendon, Oxford Clarendon, Oxford Clarendon, Oxford Adam Mickiewicz University Adam Mickiewicz University
16 Goal of Entanglement Percolation Given a network with a specified amount of quantum and classical resources, and a specific long range entanglement task, design the optimal protocol to acheive the task. E.g. Optimal: Smallest amount of resources (entanglement) that acheives task. Or protocol that acheives task with highest probability for a given amount of resources. E.g. Topology of lattice(network) may be an external constraint. E.g. Task: entangle fixed widely separated nodes A and B.
17 A single qubit A qubit is a two-state quantum system.
18 A single qubit A qubit is a two-state quantum system.
19 A single qubit A qubit is a two-state quantum system. This talk limited to collections of two state systems. Only pure states. I won t address physical embodiment.
20 A single qubit A qubit is a two-state quantum system. This talk limited to collections of two state systems. Only pure states. I won t address physical embodiment. We write basis kets 0 and 1.
21 A single qubit A qubit is a two-state quantum system. This talk limited to collections of two state systems. Only pure states. I won t address physical embodiment. We write basis kets 0 and 1. a = a a 1 1, a a 1 2 = 1
22 Two qubits Two qubits can be entangled if they have interacted; or if entanglement was generated elsewhere and transported to them via another system (qubits).
23 Two qubits Two qubits can be entangled if they have interacted; or if entanglement was generated elsewhere and transported to them via another system (qubits). Otherwise the qubits are in an unentangled state.
24 Two qubits Two qubits can be entangled if they have interacted; or if entanglement was generated elsewhere and transported to them via another system (qubits). Otherwise the qubits are in an unentangled state. (Pure) Product state ab a b
25 Entanglement: Two entangled qubits Two qubits that interacted in the past may be entangled.
26 Entanglement: Two entangled qubits Two qubits that interacted in the past may be entangled.
27 Entanglement: Two entangled qubits Two qubits that interacted in the past may be entangled. Cannot be written as a product state (in any basis).
28 Entanglement: Two entangled qubits Two qubits that interacted in the past may be entangled. Cannot be written as a product state (in any basis). α = α α 1 11 α 0 > α 1 α 0 + α 1 = 1 α 1 [0, 1/2]
29 Entanglement: Two entangled qubits Two qubits that interacted in the past may be entangled. Cannot be written as a product state (in any basis). α = α α 1 11 α 0 > α 1 α 0 + α 1 = 1 α 1 [0, 1/2] Pure, partially entangled, bipartite state
30 Entanglement: Two entangled qubits Two qubits that interacted in the past may be entangled. Cannot be written as a product state (in any basis). α = α α 1 11 α 0 > α 1 α 0 + α 1 = 1 α 1 [0, 1/2] Pure, partially entangled, bipartite state α 1 = 0: no entanglement, α 1 = 1/2: max. entanglement
31 Bell State: Singlet Conversion Partially Entangled: α = α α 1 11
32 Bell State: Singlet Conversion Partially Entangled: α = α α 1 11 Local operations (and classical communication): qubits not allowed to interact
33 Bell State: Singlet Conversion Partially Entangled: α = α α 1 11 Local operations (and classical communication): qubits not allowed to interact Maximally Entangled: Ψ = Conversion Probability p = 2α 1 for α 0 > α 1 Otherwise: product state (failure) Singlet
34 Bell State: Singlet Conversion Partially Entangled: α = α α 1 11 Local operations (and classical communication): qubits not allowed to interact Maximally Entangled: Ψ = Singlet, Bell State, Maximally Entangled State Singlet Conversion Probability p = 2α 1 for α 0 > α 1 Otherwise: product state (failure)
35 Bell State: Singlet Conversion Partially Entangled: α = α α 1 11 Local operations (and classical communication): qubits not allowed to interact Maximally Entangled: Ψ = Singlet, Bell State, Maximally Entangled State Singlet Conversion Probability p = 2α 1, for α 0 > α 1 Otherwise: product state (failure)
36 Bell State: Singlet Conversion Partially Entangled: α = α α 1 11 Local operations (and classical communication): qubits not allowed to interact Maximally Entangled: Ψ = Singlet, Bell State, Maximally Entangled State Singlet Conversion Probability p = 2α 1, for α 0 > α 1 Otherwise: product state (failure)
37 Distributing Entanglement α α Get Bell state with same probability as in singlet conversion p = 2α 1! (product state otherwise) Note: if α 1 = 1/2, then p = 1.
38 Distributing Entanglement α α Can we entangle the two outermost qubits? Using only local operations and classical communication. Get Bell state with same probability as in singlet conversion p = 2α 1! (product state otherwise) Note: if α 1 = 1/2, then p = 1.
39 Distributing Entanglement α α Yes. Entanglement Swapping. Get Bell state with same probability as in singlet conversion p = 2α 1! (product state otherwise) Note: if α 1 = 1/2, then p = 1. Using only local operations and classical communication.
40 Distributing Entanglement Ψ Yes. Entanglement Swapping.
41 Distributing Entanglement Ψ Yes. Entanglement Swapping. Get Bell state with same probability as in singlet conversion p = 2α 1! (product state otherwise)
42 Distributing Entanglement Ψ Yes. Entanglement Swapping. Get Bell state with same probability as in singlet conversion p = 2α 1! (product state otherwise) Note: if α 1 = 1/2, then p = 1. Using only local operations and classical communication.
43 Quantum Network Concrete: Square lattice. Each bond is an entangled pair with amount of entanglement α 1.
44 Quantum Network Concrete: Square lattice. Each bond is an entangled pair with amount of entanglement α 1.
45 Quantum Network How to treat a network larger than two pairs. Naive method: Borrow ideas from one-dimensional quantum repeaters. 1 Attempt to put each pair in a Bell state. Here: Singlet conversion with probability of success p = 2α 1. 2 Entanglement swappings between pairs of these Bell states. Result: New Bell state between outermost qubits, one from each of the pairs. 3 Repeat swappings, entangling ever more distant qubits.
46 Classical Entanglement Percolation B A
47 Classical Entanglement Percolation B A
48 Classical Entanglement Percolation B A
49 Classical Entanglement Percolation B A
50 Classical Entanglement Percolation B A
51 Classical Entanglement Percolation B A
52 Classical Entanglement Percolation B A
53 Classical Entanglement Percolation B A
54 Classical Entanglement Percolation B A
55 Classical Entanglement Percolation B A
56 Classical Entanglement Percolation B A
57 Classical Entanglement Percolation B A
58 Classical Entanglement Percolation B A
59 Classical Entanglement Percolation B A
60 Classical Entanglement Percolation B A
61 Classical Entanglement Percolation B A
62 Classical Entanglement Percolation B A Done!
63 Open question We want to entangle two nodes A and B at opposite ends of a very large network via quantum operations local to each node (swapping, other measurements?) Is there a minimum amount of entanglement (smallest value of α 1 ) required on each link in order that A and B can be entangled?
64 Big Network: α 1 = p = 0.35
65 Big Network: α 1 = p = 0.35
66 Big Network: α 1 = p = 0.35
67 Big Network: α 1 = p = 0.35
68 Big Network: α 1 = p = 0.35
69 Big Network: α 1 = p = 0.35
70 Words for state of two qubits Try a singlet conversion on a partially entangled state. Two outcomes. Maximally entangled state = Bell state = open bond = present bond Product state = non-entangled state = closed bond = absent bond Singlet conversion probability p Open bond density.
71 bond density p = 0 sizes:
72 bond density p = 0.01 sizes: 3,2,1
73 bond density p = 0.05 sizes: 5,4,3,2,1
74 bond density p = 0.1 sizes: 9,7,6,5,4,3,2
75 bond density p = 0.15 sizes: 13,11,10,9,8,7,6
76 bond density p = 0.2 sizes: 30,25,18,16,15,14,13
77 bond density p = 0.25 sizes: 34,27,26,25,22,21,20
78 bond density p = 0.3 sizes: 67,51,48,43,40,39,38
79 bond density p = 0.35 sizes: 105,96,94,87,82,80,71
80 bond density p = 0.4 sizes: 203,194,190,188,185,167,166
81 bond density p = 0.45 sizes: 978,870,594,507,444,416,390
82 bond density p = sizes: 4697,1246,1221,1061,1012,959,932
83 bond density p = 0.48 sizes: 5594,2035,1369,1290,1076,990,985
84 bond density p = 0.49 sizes: 7858,3205,2231,1909,1716,1547,1467
85 bond density p = sizes: 12652,3507,2162,2061,1829,1589,1512
86 bond density p = 0.5 sizes: 15802,3571,2543,1948,1911,1618,725
87 bond density p = sizes: 23293,5730,1761,476,448,356,306
88 bond density p = 0.51 sizes: 30600,1785,486,450,359,268,258
89 bond density p = 0.52 sizes: 35231,561,456,366,250,204,128
90 bond density p = sizes: 36188,561,462,371,253,205,130
91 bond density p = 0.55 sizes: 42433,57,56,51,45,37,36
92 bond density p = 0.6 sizes: 48659,31,28,24,18,17,16
93 bond density p = 0.65 sizes: 53575,17,7,6,5,4,3
94 bond density p = 0.7 sizes: 57953,7,4,3,2,1
95 bond density p = 0.75 sizes: 62256,7,4,2,1
96 bond density p = 0.8 sizes: 66439,4,2,1
97 bond density p = 0.85 sizes: 70525,1
98 bond density p = 0.9 sizes: 74699
99 bond density p = 0.95 sizes: 78710
100 bond density p = 1 sizes: 82780
101 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
102 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
103 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
104 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
105 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
106 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
107 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
108 Percolation theory Bonds are open (present) with probability p; or else closed (absent). p is called the bond density. For large lattices there is a threshold value of the bond density p c. For p > p c there is a single cluster that spans the whole lattice. p c depends on the structure of the lattice. In order for A and B to be connected if they are very far apart, 1 Must have p > p c 2 Both A and B must be in the huge cluster Let θ(p) = Prob(A is in huge cluster) A and B are connected with Prob = θ 2 (p)
109 Fraction of Bonds in largest cluster θ(p) bonds p c p
110 How to beat Classical entanglement percolation Classical entanglement percolation. Naive method. 1 Convert each partially entangled pair to Bell state with probability p. 2 Perform entanglement swappings on a chain of adjacent Bell states starting at node A and ending at node B Quantum entanglement percolation 1 Perform some quantum operations to transform lattice structure to one with better connectivity properties 2 Proceed with classical entanglement percolation on new lattice
111 How to beat Classical entanglement percolation Classical entanglement percolation. Naive method. 1 Convert each partially entangled pair to Bell state with probability p. 2 Perform entanglement swappings on a chain of adjacent Bell states starting at node A and ending at node B Quantum entanglement percolation 1 Perform some quantum operations to transform lattice structure to one with better connectivity properties 2 Proceed with classical entanglement percolation on new lattice
112 How to beat Classical entanglement percolation Classical entanglement percolation. Naive method. 1 Convert each partially entangled pair to Bell state with probability p. 2 Perform entanglement swappings on a chain of adjacent Bell states starting at node A and ending at node B Quantum entanglement percolation 1 Perform some quantum operations to transform lattice structure to one with better connectivity properties 2 Proceed with classical entanglement percolation on new lattice
113 How to beat Classical entanglement percolation Classical entanglement percolation. Naive method. 1 Convert each partially entangled pair to Bell state with probability p. 2 Perform entanglement swappings on a chain of adjacent Bell states starting at node A and ending at node B Quantum entanglement percolation 1 Perform some quantum operations to transform lattice structure to one with better connectivity properties 2 Proceed with classical entanglement percolation on new lattice
114 How to beat Classical entanglement percolation Classical entanglement percolation. Naive method. 1 Convert each partially entangled pair to Bell state with probability p. 2 Perform entanglement swappings on a chain of adjacent Bell states starting at node A and ending at node B Quantum entanglement percolation 1 Perform some quantum operations to transform lattice structure to one with better connectivity properties 2 Proceed with classical entanglement percolation on new lattice
115 How to beat Classical entanglement percolation Classical entanglement percolation. Naive method. 1 Convert each partially entangled pair to Bell state with probability p. 2 Perform entanglement swappings on a chain of adjacent Bell states starting at node A and ending at node B Quantum entanglement percolation 1 Perform some quantum operations to transform lattice structure to one with better connectivity properties 2 Proceed with classical entanglement percolation on new lattice
116 Kagome lattice
117 Kagome lattice
118 Kagome lattice
119 Kagome lattice
120 Kagome lattice
121 Kagome lattice
122 Kagome lattice to Square lattice
123 Kagome lattice to Square lattice p c 0.52 Kagome p c = 0.5 Square lattice
124 Tri to hex a) b) ) d) e) f)
125 Bowtie lattice
126 Bowtie lattice p c 0.40 Bowtie p c 0.35 Triangular
127 Summary and some outlook Long distance entanglement on networks is necessary for quantum information tasks. There are methods for long range entanglement that are more effective than simply performing singlet conversions to get a chain of Bell states. Optimal method is not known: we have success using other methods; eg distillation, multipartite. How to treat mixed states or complex networks?
128 A. Acín, J. I. Cirac, and M. Lewenstein Entanglement Percolation in Quantum Networks Nature Physics 3, 256 (2007). S. Perseguers, J. I. Cirac, A. Acín, M. Lewenstein, and J. Wehr Entanglement distribution in pure-state quantum networks Phys. Rev. A 77, (2008). G. J. Lapeyre Jr., J. Wehr, and M. Lewenstein Enhancement of entanglement percolation in quantum networks via lattice transformations Phys. Rev. A 79, (2009). S. Perseguers, D. Cavalcanti, G. J. Lapeyre Jr, M. Lewenstein, J. I. Cirac, A. Acín Multipartite Entanglement Percolation arxiv: S. Perseguers, M. Lewenstein, A. Acín, J. I. Cirac Quantum Complex Networks arxiv:
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