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MULTIPARTICLE ENTANGLEMENT Sebastian Hartweg, Mario Berta (QSIT Seminar, 10.12.2012) Literature Vol 438|1 December 2005|doi:10.1038/nature04251 LETTERS Creation of a six-atom ‘Schrödinger cat’ state D. Leibfried1, E. Knill1, S. Seidelin1, J. Britton1, R. B. Blakestad1, J. Chiaverini1†, D. B. Hume1, W. M. Itano1, J. D. Jost1, C. Langer1, R. Ozeri1, R. Reichle1 & D. J. Wineland1 Among the classes of highly entangled states of multiple quantum systems, the so-called ‘Schrödinger cat’ states are particularly useful. Cat states are equal superpositions of two maximally different quantum states. They are a fundamental resource in fault-tolerant quantum computing1–3 and quantum communication, where they can enable protocols such as open-destination teleportation4 and secret sharing5. They play a role in fundamental tests of quantum mechanics6 and enable improved signal-to-noise ratios in interferometry7. Cat states are very sensitive to decoherence, and as a result their preparation is challenging and can serve as a demonstration of good quantum control. Here we report the creation of cat states of up to six atomic qubits. Each qubit’s state space is defined by two hyperfine ground states of a beryllium ion; the cat state corresponds to an entangled equal superposition of all the atoms in one hyperfine state and all atoms in the other hyperfine state. In our experiments, the cat states are prepared in a three-step process, irrespective of the number of entangled atoms. Together with entangled states of a different class created in Innsbruck8, this work represents the current state-of-the-art for large entangled states in any qubit system. One promising candidate system for scalable universal quantum information processing (QIP) consists of atomic ions that are confined in electromagnetic traps and manipulated with laser beams9. Most of the basic ingredients for QIP10 have been demonstrated separately in the last few years in this system. Furthermore, some simple algorithms that could serve as primitives for larger scale QIP, including quantum error correction, teleportation, and the S~ z j " l ¼ 12 j " l and S~ z j # l ¼ 2 12 j # l (for simplicity we set h! ¼ 1). We define j " , Nl ; j " l1j " l2…j " lN and j # , Nl ; j # l1j # l2…j # lN. In this notation, prototypical cat states of N qubits can be written as: 1 ð1Þ jN Catl ¼ pffiffi ðj "; Nl þ eiv j #; NlÞ 2 To generate such states we initially prepare the ions in state j # , Nl and then apply the following unitary operation to transform the initial state into jN Catl (ref. 7): $ h " #%& h p i yp p i'& h p i' exp i J 2z U N ¼ exp i J x exp i J z exp i J x ð2Þ 2 2 2 2 The operators in the left and right pairs of parentheses represent a common rotation by angle p2 of all N qubits, written in terms of the global angular momentum operators !composed P ! of the sum of the N individual spin-1/2 operators J ¼ N j¼1 Sj (Dicke operators). The operator in the middle pair of parentheses represents a global entangling interaction that is diagonal in the measurement basis spanned by all product states of N qubits, each in either j " l or j # l, and can be implemented by generalizing the phase-gate mechanism described in ref. 14 (see also below). If N is odd, y ¼ 1; y ¼ 0 otherwise. Because of experimental imperfections, we need a measure to indicate how close the generated state jW Nl is to the ideal state jN Catl. The simplest measure, called the fidelity, is the square modulus of the overlap of these two states: 2 Literature Vol 438|1 December 2005|doi:10.1038/nature04251 LETTERS Creation of a six-atom ‘Schrödinger cat’ state D. Leibfried1, E. Knill1, S. Seidelin1, J. Britton1, R. B. Blakestad1, J. Chiaverini1†, D. B. Hume1, W. M. Itano1, J. D. Jost1, C. Langer1, R. Ozeri1, R. Reichle1 & D. J. Wineland1 PRL Among the classes of highly entangled states of multiple quantum S~ z j " l ¼ 12 j " l and S~ z j # l ¼ 2 12 j # l (for simplicity we set h! ¼ 1). We systems, the so-called ‘Schrödinger cat’ states are particularly define j " , Nl ; j " l1j " l2…j " lN and j # , Nl ; j # l1j # l2…j # lN. In this notation, prototypical cat states of N qubits can be written useful. Cat states are equal superpositions of two maximally week ending as: different quantum states. They are a fundamental resource in HY S I C AcommuniL R E V I E W L E TjNTCatl E R¼Sp1ffiffi ðj "; Nl þ eiv j #; NlÞ 1 APRIL 2011 106, 130506 quantum (2011) computing1–3 Pand quantum fault-tolerant ð1Þ 2 cation, where they can enable protocols such as open-destination teleportation4 and secret sharing5. They play a role in fundamental To generate such states we initially prepare the ions in state j # , Nl tests of quantum mechanics6 and enable improved signal-to-noise and then apply the following unitary operation to transform the and Coherence states are veryEntanglement: sensitive to decoher- Creation ratios in interferometry7. Cat14-Qubit initial state into jN Catl (ref. 7): $ h " #%& h ence, and as a result their preparation is challenging and can serve i p p 2 i'& h p i' 1 of good quantum control. 1 Here we report the 1 1 i yp J exp i1 William U exp exp iA.J x Coish, ð2Þ 2,3 ¼ exp i J J as a demonstration N x z Thomas Monz, Philipp Schindler, Julio T. Barreiro, Michael Chwalla, Daniel Nigg, 2 2 2 z 2 creation of cat states of up to six atomic qubits. Each qubit’s state 4 1 1,* 1,4 Blatt Harlander, Hänsel, Markus Hennrich, space is definedMaximilian by two hyperfine ground statesWolfgang of a beryllium ion; The operators in the left and and right Rainer pairs of parentheses represent a 1 corresponds to an entangled equal superposition of all p the cat state common rotation by angle of all N qubits, Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, written Austriain terms 2 2 in one hyperfine state and all atoms in the other the atoms of Physics the globaland angular momentum operators composed Institute for Quantum Computing and Department of Astronomy, University of!Waterloo, P ! of the hyperfine state. In our experiments, the cat states are prepared sum of the N individual spin-1/2 operators J ¼ N j¼1 Sj (Dicke Waterloo, ON, N2L 3G1, Canada in a three-step process, operators). The operator in the middle pair of parentheses 3 irrespective of the number of entangled of ofPhysics, McGill University, Montreal, Quebec, Canada H3A 2T8 atoms. Together with Department entangled states a different class created represents a global entangling interaction that is diagonal in the 4 8 für represents Quantenoptik und state-of-the-art Quanteninformation, Österreichische Akademie Wissenschaften, , this work the current for in InnsbruckInstitut measurement basis spanned by all der product states of N qubits, each large entangled states in any qubit system. in either j " l or j # l, and can be implemented by generalizing the Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria One promising candidate system(Received for scalable 30 universal quantum mechanism described September 2010;phase-gate published 31 March 2011)in ref. 14 (see also below). If N is information processing (QIP) consists of atomic ions that are odd, y ¼ 1; y ¼ 0 otherwise. confined inWe electromagnetic traps and manipulated with laser Because experimental we need a measure report the creation of Greenberger-Horne-Zeilinger statesofwith up to 14 imperfections, qubits. By investigating the to beams9. Most of the basic ingredients for QIP10 have been demonindicate how close the generated state jW Nl is to the ideal state coherence of up to 8 ions over time, we observe a decay proportional to the square of the number of qubits. strated separately in the last few years in this system. Furthermore, jN Catl. The simplest measure, called the fidelity, is the square Thealgorithms observedthat decay a theoretical modelmodulus whichofassumes a of system affected some simple couldagrees serve as with primitives for larger scale the overlap these two states: by correlated, 2 QIP, including quantum error correction, teleportation, and the Literature Vol 438|1 December 2005|doi:10.1038/nature04251 LETTERS Creation of a six-atom ‘Schrödinger cat’ state D. Leibfried1, E. Knill1, S. Seidelin1, J. Britton1, R. B. Blakestad1, J. Chiaverini1†, D. B. Hume1, W. M. Itano1, J. D. Jost1, C. Langer1, R. Ozeri1, R. Reichle1 & D. J. Wineland1 Nobel Prize in Physics 2012 PRL Among the classes of highly entangled states of multiple quantum S~ z j " l ¼ 12 j " l and S~ z j # l ¼ 2 12 j # l (for simplicity we set h! ¼ 1). We systems, the so-called ‘Schrödinger cat’ states are particularly define j " , Nl ; j " l1j " l2…j " lN and j # , Nl ; j # l1j # l2…j # lN. In this notation, prototypical cat states of N qubits can be written useful. Cat states are equal superpositions of two maximally week ending as: different quantum states. They are a fundamental resource in HY S I C AcommuniL R E V I E W L E TjNTCatl E R¼Sp1ffiffi ðj "; Nl þ eiv j #; NlÞ 1 APRIL 2011 106, 130506 quantum (2011) computing1–3 Pand quantum fault-tolerant ð1Þ 2 cation, where they can enable protocols such as open-destination teleportation4 and secret sharing5. They play a role in fundamental To generate such states we initially prepare the ions in state j # , Nl tests of quantum mechanics6 and enable improved signal-to-noise and then apply the following unitary operation to transform the and Coherence states are veryEntanglement: sensitive to decoher- Creation ratios in interferometry7. Cat14-Qubit initial state into jN Catl (ref. 7): $ h " #%& h ence, and as a result their preparation is challenging and can serve i p p 2 i'& h p i' 1 of good quantum control. 1 Here we report the 1 1 i yp J exp i1 William U exp exp iA.J x Coish, ð2Þ 2,3 ¼ exp i J J as a demonstration N x z Thomas Monz, Philipp Schindler, Julio T. Barreiro, Michael Chwalla, Daniel Nigg, 2 2 2 z 2 creation of cat states of up to six atomic qubits. Each qubit’s state 4 1 1,* 1,4 Blatt Harlander, Hänsel, Markus Hennrich, space is definedMaximilian by two hyperfine ground statesWolfgang of a beryllium ion; The operators in the left and and right Rainer pairs of parentheses represent a 1 corresponds to an entangled equal superposition of all p the cat state common rotation by angle of all N qubits, Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, written Austriain terms 2 2 in one hyperfine state and all atoms in the other the atoms of Physics the globaland angular momentum operators composed Institute for Quantum Computing and Department of Astronomy, University of!Waterloo, P ! of the hyperfine state. In our experiments, the cat states are prepared sum of the N individual spin-1/2 operators J ¼ N j¼1 Sj (Dicke Waterloo, ON, N2L 3G1, Canada in a three-step process, operators). The operator in the middle pair of parentheses 3 irrespective of the number of entangled of ofPhysics, McGill University, Montreal, Quebec, Canada H3A 2T8 atoms. Together with Department entangled states a different class created represents a global entangling interaction that is diagonal in the 4 8 für represents Quantenoptik und state-of-the-art Quanteninformation, Österreichische Akademie Wissenschaften, , this work the current for in InnsbruckInstitut measurement basis spanned by all der product states of N qubits, each large entangled states in any qubit system. in either j " l or j # l, and can be implemented by generalizing the Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria One promising candidate system(Received for scalable 30 universal quantum mechanism described September 2010;phase-gate published 31 March 2011)in ref. 14 (see also below). If N is information processing (QIP) consists of atomic ions that are odd, y ¼ 1; y ¼ 0 otherwise. confined inWe electromagnetic traps and manipulated with laser Because experimental we need a measure report the creation of Greenberger-Horne-Zeilinger statesofwith up to 14 imperfections, qubits. By investigating the to beams9. Most of the basic ingredients for QIP10 have been demonindicate how close the generated state jW Nl is to the ideal state coherence of up to 8 ions over time, we observe a decay proportional to the square of the number of qubits. strated separately in the last few years in this system. Furthermore, jN Catl. The simplest measure, called the fidelity, is the square Thealgorithms observedthat decay a theoretical modelmodulus whichofassumes a of system affected some simple couldagrees serve as with primitives for larger scale the overlap these two states: by correlated, 2 QIP, including quantum error correction, teleportation, and the Outline Qubits in Ion Traps Multiparticle Entanglement - ‘Cat States’ Creation of ‘Cat States’ Witnessing Multiparticle Entanglement Superdecoherence Conclusions Outline Qubits in Ion Traps Multiparticle Entanglement - ‘Cat States’ Creation of ‘Cat States’ Witnessing Multiparticle Entanglement Superdecoherence Conclusions Qubits in Ion Traps (I) A string of ions in a linear Paul trap forms a quantum register. Qubits in Ion Traps (I) A string of ions in a linear Paul trap forms a quantum register. Laser pulses can manipulate individual ions, or all ions collectively. Qubits in Ion Traps (I) A string of ions in a linear Paul trap forms a quantum register. Laser pulses can manipulate individual ions, or all ions collectively. 2-level-ions can be realized: ● By electronic ground- and excited state. ● By hyperfine states of an electronic ground state. Qubits in Ion Traps (II) Read-out can be performed by fluorescence detection. Qubits in Ion Traps (II) Read-out can be performed by fluorescence detection. Ion-ion interactions are induced by laser excitation of harmonic oscillations of ions in the trap potential. Outline Qubits in Ion Traps Multiparticle Entanglement - ‘Cat States’ Creation of ‘Cat States’ Witnessing Multipartite Entanglement Superdecoherence Conclusions Multiparticle Entanglement - ‘Cat States’ Superpositions of maximally different quantum states: | #ii , | "ii N qubits | #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN Multiparticle Entanglement - ‘Cat States’ Superpositions of maximally different quantum states: | #ii , | "ii N qubits | #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN 1 p |N, Cati = | ", N i + ei✓ | #, N i 2 Cat state Multiparticle Entanglement - ‘Cat States’ Superpositions of maximally different quantum states: | #ii , | "ii N qubits | #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN 1 p |N, Cati = | ", N i + ei✓ | #, N i 2 Cat state Important for fault tolerant quantum computation, quantum communication, quantum simulations, metrology and quantum to classical transition. Multiparticle Entanglement - ‘Cat States’ Superpositions of maximally different quantum states: | #ii , | "ii N qubits | #, N i ⌘ | #i1 ⌦ | #i2 ⌦ . . . |⌦ #iN | ", N i ⌘ | "i1 ⌦ | "i2 ⌦ . . . |⌦ "iN 1 p |N, Cati = | ", N i + ei✓ | #, N i 2 Cat state Important for fault tolerant quantum computation, quantum communication, quantum simulations, metrology and quantum to classical transition. Difficult to create → therefore used as demonstration of good quantum control. Outline Qubits in Ion Traps Multiparticle Entanglement - ‘Cat States’ Creation of ‘Cat States’ Witnessing Multiparticle Entanglement Superdecoherence Conclusions Creation of Cat States - Blatt’s Group [2] Optical transition in 40Ca+: |S1/2 (m = 1/2)i ⌘ | #i |D5/2 (m = 1/2)i ⌘ | "i [2] Phys. Rev. Lett. 106, 130506 (2011) Creation of Cat States - Blatt’s Group [2] Optical transition in 40Ca+: |S1/2 (m = 1/2)i ⌘ | #i |D5/2 (m = 1/2)i ⌘ | "i 2-photon transitions in bichromatic laser field: [2] Phys. Rev. Lett. 106, 130506 (2011) Creation of Cat States - Wineland’s Group [1] (I) 9Be+ ions, with hyperfine-splitting: |F = 2, mF = 1i ⌘ | #i |F = 1, mF = [1] Nature 438, 639 (2005) 1i ⌘ | "i Creation of Cat States - Wineland’s Group [1] (I) 9Be+ ions, with hyperfine-splitting: |F = 2, mF = 1i ⌘ | #i |F = 1, mF = Create | #, N i and apply unitary operator: UN = J~ = ✓ N X 1i ⌘ | "i ◆⇣ h ⇡ i h ⇡ i⌘ ⇣ h ⇡ i⌘ ⇠⇡ 2 exp i Jx exp i Jz exp i Jz exp i Jx 2 2 2 2 ~j S Dicke oprators j=1 ⇠ = 0/1 N even/odd [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (I) 9Be+ ions, with hyperfine-splitting: |F = 2, mF = 1i ⌘ | #i |F = 1, mF = Create | #, N i and apply unitary operator: UN = J~ = ✓ N X 1i ⌘ | "i ◆⇣ h ⇡ i h ⇡ i⌘ ⇣ h ⇡ i⌘ ⇠⇡ 2 exp i Jx exp i Jz exp i Jz exp i Jx 2 2 2 2 ~j S Dicke oprators j=1 ⇠ = 0/1 N even/odd [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (II) Implementation of phase gate: [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (II) Implementation of phase gate: ● Illumination of the ions with two laser beams with detuning !COM + . [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (II) Implementation of phase gate: ● Illumination of the ions with two laser beams with detuning !COM + . ● Adjustment of detuning and pulse length lead to a state dependent phase shift. | "i| "i| i ! | "i| "i| i | "i| #i| i ! ei⇡/2 | "i| #i| i | #i| "i| i ! ei⇡/2 | #i| "i| i | #i| #i| i ! | #i| #i| i [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (III) [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (III) First ⇡/2 -pulse: | ##i = | #i ⌦ | #i 1 1 ! p (| #i + | "i) ⌦ p (| #i + | "i) 2 2 1 = (| #i| #i + | "i| "i + | "i| #i + | #i| "i) 2 [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (III) First ⇡/2 -pulse: | ##i = | #i ⌦ | #i 1 1 ! p (| #i + | "i) ⌦ p (| #i + | "i) 2 2 1 = (| #i| #i + | "i| "i + | "i| #i + | #i| "i) 2 After phase pulse: 1 (| #i| #i + | "i| "i) |0i 2 + ei (t) (| #i| "i|↵(t)i + | "i| #i| ↵(t)i) [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (III) First ⇡/2 -pulse: | ##i = | #i ⌦ | #i 1 1 ! p (| #i + | "i) ⌦ p (| #i + | "i) 2 2 1 = (| #i| #i + | "i| "i + | "i| #i + | #i| "i) 2 After phase pulse: 1 (| #i| #i + | "i| "i) |0i 2 + ei (t) (| #i| "i|↵(t)i + | "i| #i| ↵(t)i) State after ⇡ and ⇡/2 pulse: 1 1 [(| #i| #i + | "i| "i)|0i + ei (t) (| #i| "i | "i| "i) 2 2 1 + | #i| #i | "i| #i)|↵(t)i + ei (t) (| "i| #i | "i| "i 2 + | #i| #i | #i| "i)| ↵(t)i] [1] Nature 438, 639 (2005) Creation of Cat States - Wineland’s Group [1] (III) First ⇡/2 -pulse: | ##i = | #i ⌦ | #i 1 1 ! p (| #i + | "i) ⌦ p (| #i + | "i) 2 2 1 = (| #i| #i + | "i| "i + | "i| #i + | #i| "i) 2 After phase pulse: 1 (| #i| #i + | "i| "i) |0i 2 + ei (t) (| #i| "i|↵(t)i + | "i| #i| ↵(t)i) State after ⇡ and ⇡/2 pulse: 1 1 [(| #i| #i + | "i| "i)|0i + ei (t) (| #i| "i | "i| "i) 2 2 1 + | #i| #i | "i| #i)|↵(t)i + ei (t) (| "i| #i | "i| "i 2 + | #i| #i | #i| "i)| ↵(t)i] An ideal ⇡ displacement pulse leaves only | ""i and | ##i components! [1] Nature 438, 639 (2005) Outline Qubits in Ion Traps Multiparticle Entanglement - ‘Cat States’ Creation of ‘Cat States’ Witnessing Multiparticle Entanglement Superdecoherence Conclusions Witnessing Multiparticle Entanglement (I) Quantum state tomography? Entanglement measures? Witnessing Multiparticle Entanglement (I) Quantum state tomography? Entanglement measures? Genuine N-particle entanglement. Witnessing Multiparticle Entanglement (I) Quantum state tomography? Entanglement measures? Genuine N-particle entanglement. ‘Local operations and classical communication’ (LOCC) entanglement classes: cat state class. Wikipedia Witnessing Multiparticle Entanglement (I) Quantum state tomography? Entanglement measures? Genuine N-particle entanglement. ‘Local operations and classical communication’ (LOCC) entanglement classes: cat state class. Wikipedia Entanglement witness operator: expectation value negative entanglement. Witnessing Multiparticle Entanglement (I) Quantum state tomography? Entanglement measures? Genuine N-particle entanglement. ‘Local operations and classical communication’ (LOCC) entanglement classes: cat state class. Wikipedia Entanglement witness operator: expectation value negative entanglement. Method used in [1,2]. [1] Nature 438, 639 (2005) [2] Phys. Rev. Lett. 106, 130506 (2011) Witnessing Multiparticle Entanglement (II) ⇢N State Witnessing Multiparticle Entanglement (II) ⇢N W =1 State 2|N, CatihN, Cat| Entanglement witness Witnessing Multiparticle Entanglement (II) ⇢N W =1 State 2|N, CatihN, Cat| Entanglement witness hW i⇢N = 1 2FN,Cat Negative? FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 Witnessing Multiparticle Entanglement (II) ⇢N W =1 State 2|N, CatihN, Cat| Entanglement witness hW i⇢N = 1 2FN,Cat Negative? FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 ? Witnessing Multiparticle Entanglement (II) ⇢N W =1 State 2|N, CatihN, Cat| Entanglement witness hW i⇢N = 1 Negative? 2FN,Cat FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 ⇣ ⇡ exp i 4 j=1 N O (j) = (j) x (j) ⌘ cos + ? Collectively rotate ⇢N (j) y sin Vary phase Witnessing Multiparticle Entanglement (II) ⇢N W =1 State 2|N, CatihN, Cat| Entanglement witness hW i⇢N = 1 Negative? 2FN,Cat FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 ⇣ ⇡ exp i 4 j=1 N O (j) = (j) x (j) ⌘ Collectively rotate ⇢N cos + P( ) = Peven ( ) ? (j) y sin Podd ( ) A (P( )) = ⇢0...0,1...1 N Vary phase Probability of finding even / odd number of excitations Amplitude of oscillations Witnessing Multiparticle Entanglement (II) PRL 106, 130506 (2011) PHYSICAL REVIEW LETT Number of qubits 1 0 ⇢N hW i⇢N = 1 ⇣ ⇡ exp i 4 j=1 (j) = (j) x 3 0 -1 1 Negative? 2FN,Cat 4 0 FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 N O -1 1 (j) ⌘ ? -1 1 0 5 -1 1 Parity W =1 State 2|N, CatihN, Cat| Entanglement witness 2 Collectively rotate ⇢N 6 0 -1 1 8 0 cos + (j) y sin Vary phase -1 1 10 0 P( ) = Peven ( ) Podd ( ) A (P( )) = ⇢0...0,1...1 N Probability of finding even / odd number of excitations Amplitude of oscillations -1 1 12 0 -1 1 0 -1 14 0 0.2 0.4 0.6 0.8 Phase φ of analyzing pulse (π) 1 1 (color observed on [2]FIG. Phys. Rev.online). Lett.Parity 106,oscillations 130506 (2011) f2; 3; 4; 5; 6; 8; 10; 12; 14g-qubit GHZ states. The coherence of GHZ states as a function of time is FIG. 2 (color o probability !ðNÞ function of time 2 (green), 3 (red observed relative behavior proport coherence of an faster than the co magnetic field d ing coils. By de ence time impro coherence time Witnessing Multiparticle Entanglement (II) PRL 106, 130506 (2011) PHYSICAL REVIEW LETT Number of qubits 1 0 ⇢N hW i⇢N = 1 ⇣ ⇡ exp i 4 j=1 (j) = (j) x 3 0 -1 1 Negative? 2FN,Cat 4 0 FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 N O -1 1 (j) ⌘ ? -1 1 0 5 -1 1 Parity W =1 State 2|N, CatihN, Cat| Entanglement witness 2 Collectively rotate ⇢N 6 0 -1 1 8 0 cos + (j) y sin Vary phase -1 1 10 0 P( ) = Peven ( ) Podd ( ) A (P( )) = ⇢0...0,1...1 N Probability of finding even / odd number of excitations Amplitude of oscillations -1 1 -1 1 0 -1 Attention: rotations and measurements reliable? 12 0 14 0 0.2 0.4 0.6 0.8 Phase φ of analyzing pulse (π) 1 1 (color observed on [2]FIG. Phys. Rev.online). Lett.Parity 106,oscillations 130506 (2011) f2; 3; 4; 5; 6; 8; 10; 12; 14g-qubit GHZ states. The coherence of GHZ states as a function of time is FIG. 2 (color o probability !ðNÞ function of time 2 (green), 3 (red observed relative behavior proport coherence of an faster than the co magnetic field d ing coils. By de ence time impro coherence time all ions. The electronic and vibrational states of the ion string are manipulated by setting the frequency, duration, intensity, and phase of the pulses. Finally, the state of the ion qubits is measured by scattering light at 397 nm on the S1=2 $ P1=2 transition and detecting the fluorescence with a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual qubit in the fj0i; j1ig basis, while the PMT only detects the number of ions being in j0i or j1i. Sufficient statistics is achieved by repeating each experiment 100 times for each setting. f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed parity oscillations are shown in Fig. 1. Although N-particle distillability can be inferred from the criterion in Ref. [14] by many standard deviations, according to the criteria in Ref. [15] the obtained data support genuine N-particle entanglement for 14 qubits with a confidence of 76%. The 12-qubit state is likely not fully entangled. The Poissonian statistics of the PMT fluorescence data is accounted for by a data analysis based on Bayesian inference [16]. Witnessing Multiparticle Entanglement (III) Experimental results from [2]: TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria supported by & standard deviations. All errors in parenthesis, 1 standard deviation. Number of ions Populations, % Coherence, % Fidelity, % Distillability criterion [14], & Entanglement criterion [15], & 2 3 4 5 6 8 10 12 14 99.50(7) 97.8(3) 98.6(2) 283 265 97.6(2) 96.5(6) 97.0(3) 151 143 97.5(2) 93.9(5) 95.7(3) 181 167 96.0(4) 92.9(8) 94.4(5) 100 101 91.6(4) 86.8(8) 89.2(4) 95 96 84.7(4) 78.7(7) 81.7(4) 96 92 67.0(8) 58.2(9) 62.6(6) 40 25 53.3(9) 41.6(10) 47.4(7) 18 $6 56.2(11) 45.4(13) 50.8(9) 17 0.7 130506-2 [2] Phys. Rev. Lett. 106, 130506 (2011) all ions. The electronic and vibrational states of the ion string are manipulated by setting the frequency, duration, intensity, and phase of the pulses. Finally, the state of the ion qubits is measured by scattering light at 397 nm on the S1=2 $ P1=2 transition and detecting the fluorescence with a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual qubit in the fj0i; j1ig basis, while the PMT only detects the number of ions being in j0i or j1i. Sufficient statistics is achieved by repeating each experiment 100 times for each setting. f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed parity oscillations are shown in Fig. 1. Although N-particle distillability can be inferred from the criterion in Ref. [14] by many standard deviations, according to the criteria in Ref. [15] the obtained data support genuine N-particle entanglement for 14 qubits with a confidence of 76%. The 12-qubit state is likely not fully entangled. The Poissonian statistics of the PMT fluorescence data is accounted for by a data analysis based on Bayesian inference [16]. Witnessing Multiparticle Entanglement (III) Experimental results from [2]: TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria supported by & standard deviations. All errors in parenthesis, 1 standard deviation. Number of ions Populations, % Coherence, % Fidelity, % Distillability criterion [14], & Entanglement criterion [15], & 2 3 4 5 6 8 10 12 14 99.50(7) 97.8(3) 98.6(2) 283 265 97.6(2) 96.5(6) 97.0(3) 151 143 97.5(2) 93.9(5) 95.7(3) 181 167 96.0(4) 92.9(8) 94.4(5) 100 101 91.6(4) 86.8(8) 89.2(4) 95 96 84.7(4) 78.7(7) 81.7(4) 96 92 67.0(8) 58.2(9) 62.6(6) 40 25 53.3(9) 41.6(10) 47.4(7) 18 $6 56.2(11) 45.4(13) 50.8(9) 17 0.7 130506-2 [2] Phys. Rev. Lett. 106, 130506 (2011) all ions. The electronic and vibrational states of the ion string are manipulated by setting the frequency, duration, intensity, and phase of the pulses. Finally, the state of the ion qubits is measured by scattering light at 397 nm on the S1=2 $ P1=2 transition and detecting the fluorescence with a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual qubit in the fj0i; j1ig basis, while the PMT only detects the number of ions being in j0i or j1i. Sufficient statistics is achieved by repeating each experiment 100 times for each setting. f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed parity oscillations are shown in Fig. 1. Although N-particle distillability can be inferred from the criterion in Ref. [14] by many standard deviations, according to the criteria in Ref. [15] the obtained data support genuine N-particle entanglement for 14 qubits with a confidence of 76%. The 12-qubit state is likely not fully entangled. The Poissonian statistics of the PMT fluorescence data is accounted for by a data analysis based on Bayesian inference [16]. Witnessing Multiparticle Entanglement (III) Experimental results from [2]: TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria supported by & standard deviations. All errors in parenthesis, 1 standard deviation. Number of ions Populations, % Coherence, % Fidelity, % Distillability criterion [14], & Entanglement criterion [15], & 2 3 4 5 6 8 10 12 14 99.50(7) 97.8(3) 98.6(2) 283 265 97.6(2) 96.5(6) 97.0(3) 151 143 97.5(2) 93.9(5) 95.7(3) 181 167 96.0(4) 92.9(8) 94.4(5) 100 101 91.6(4) 86.8(8) 89.2(4) 95 96 84.7(4) 78.7(7) 81.7(4) 96 92 67.0(8) 58.2(9) 62.6(6) 40 25 53.3(9) 41.6(10) 47.4(7) 18 $6 56.2(11) 45.4(13) 50.8(9) 17 0.7 130506-2 FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 hW i⇢N = 1 [2] Phys. Rev. Lett. 106, 130506 (2011) 2FN,Cat all ions. The electronic and vibrational states of the ion string are manipulated by setting the frequency, duration, intensity, and phase of the pulses. Finally, the state of the ion qubits is measured by scattering light at 397 nm on the S1=2 $ P1=2 transition and detecting the fluorescence with a photomultiplier tube (PMT). The camera detection effectively corresponds to a measurement of each individual qubit in the fj0i; j1ig basis, while the PMT only detects the number of ions being in j0i or j1i. Sufficient statistics is achieved by repeating each experiment 100 times for each setting. f2–6; 8; 10; 12; 14g ions and achieved the populations, coherences, and fidelities shown in Table I. The observed parity oscillations are shown in Fig. 1. Although N-particle distillability can be inferred from the criterion in Ref. [14] by many standard deviations, according to the criteria in Ref. [15] the obtained data support genuine N-particle entanglement for 14 qubits with a confidence of 76%. The 12-qubit state is likely not fully entangled. The Poissonian statistics of the PMT fluorescence data is accounted for by a data analysis based on Bayesian inference [16]. Witnessing Multiparticle Entanglement (III) Experimental results from [2]: TABLE I. Populations, coherence, and fidelity with a N-qubit GHZ state of experimentally prepared states. Entanglement criteria supported by & standard deviations. All errors in parenthesis, 1 standard deviation. Number of ions Populations, % Coherence, % Fidelity, % Distillability criterion [14], & Entanglement criterion [15], & 2 3 4 5 6 8 10 12 14 99.50(7) 97.8(3) 98.6(2) 283 265 97.6(2) 96.5(6) 97.0(3) 151 143 97.5(2) 93.9(5) 95.7(3) 181 167 96.0(4) 92.9(8) 94.4(5) 100 101 91.6(4) 86.8(8) 89.2(4) 95 96 84.7(4) 78.7(7) 81.7(4) 96 92 67.0(8) 58.2(9) 62.6(6) 40 25 53.3(9) 41.6(10) 47.4(7) 18 $6 56.2(11) 45.4(13) 50.8(9) 17 0.7 130506-2 FN,Cat = F (⇢N , |N, CatihN, Cat|) ⌘ 1 ⇣ 0...0,0...0 1...1,1...1 = ⇢N + ⇢N + ⇢0...0,1...1 N 2 hW i⇢N = 1 Other methods. [2] Phys. Rev. Lett. 106, 130506 (2011) 2FN,Cat Outline Qubits in Ion Traps Multiparticle Entanglement - ‘Cat States’ Creation of ‘Cat States’ Witnessing Multiparticle Entanglement Superdecoherence Conclusions Superdecoherence (I) Decoherence (fidelity)? PRL 106, 130506 (2011) Superdecoherence (I) PHYSICAL REVIEW LETTERS week ending 1 APRIL 2011 Number of qubits 1 0 -1 1 0 2 Decoherence (fidelity)? 3 -1 1 0 4 -1 1 0 5 Parity -1 1 0 6 -1 1 0 8 -1 1 0 -1 1 0 10 [2] Phys. Rev. Lett. 106, 130506 (2011) 12 FIG. 2 (color online). Coherence decay and relative error PRL 106, 130506 (2011) Superdecoherence (I) PHYSICAL REVIEW LETTERS week ending 1 APRIL 2011 Number of qubits 1 0 -1 1 0 -1 1 0 2 Decoherence (fidelity)? Gate time: ⇠ 100µs. 3 4 -1 1 0 5 Parity -1 1 0 6 -1 1 0 8 -1 1 0 -1 1 0 10 [2] Phys. Rev. Lett. 106, 130506 (2011) 12 FIG. 2 (color online). Coherence decay and relative error PRL 106, 130506 (2011) Superdecoherence (I) PHYSICAL REVIEW LETTERS week ending 1 APRIL 2011 Number of qubits 1 0 -1 1 0 -1 1 0 -1 1 0 Parity -1 1 0 2 Decoherence (fidelity)? Gate time: ⇠ 100µs. 3 Sources of noise: magnetic field noise, laser beam fluctuations, spontaneous emission, ... 4 5 6 -1 1 0 8 -1 1 0 -1 1 0 10 [2] Phys. Rev. Lett. 106, 130506 (2011) 12 FIG. 2 (color online). Coherence decay and relative error PRL PRL106, 106,130506 130506(2011) (2011) Superdecoherence (I) PPHHYYSSI ICCAALL RREEVVIIEEW W LLEETTTTEERRSS week week ending 11 APRIL APRIL 2011 Number Number qubits ofofqubits 11 00 -1 -1 11 00 -1 -1 11 00 -1 -1 11 00 Parity Parity -1 -1 11 00 -1 -1 11 00 22 Decoherence (fidelity)? Gate time: ⇠ 100µs. 33 Sources of noise: magnetic field noise, laser beam fluctuations, spontaneous emission, ... Decay proportional N 2.0(1), superdecoherence [2]! 44 55 66 88 -1 -1 11 00 -1 -1 1 1 0 0 10 10 [2] Phys. Rev. Lett. 106, 130506 (2011) 12 12 FIG.22 (color (color online). online). Coherence Coherence decay decay and and relative relative error error FIG. PRL PRL106, 106,130506 130506(2011) (2011) Superdecoherence (I) PPHHYYSSI ICCAALL RREEVVIIEEW W LLEETTTTEERRSS week week ending 11 APRIL APRIL 2011 Number Number qubits ofofqubits 11 00 -1 -1 11 00 -1 -1 11 00 -1 -1 11 00 Parity Parity -1 -1 11 00 -1 -1 11 00 -1 -1 11 00 -1 -1 1 1 0 0 22 Decoherence (fidelity)? Gate time: ⇠ 100µs. 33 Sources of noise: magnetic field noise, laser beam fluctuations, spontaneous emission, ... Decay proportional N 2.0(1), superdecoherence [2]! 44 55 66 Why not N? 88 10 10 [2] Phys. Rev. Lett. 106, 130506 (2011) 12 12 FIG.22 (color (color online). online). Coherence Coherence decay decay and and relative relative error error FIG. Superdecoherence (II) Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms coherence time for single qubit [2]! [2] Phys. Rev. Lett. 106, 130506 (2011) Superdecoherence (II) Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms coherence time for single qubit [2]! Magnetic field noise accounts for collective phase errors: N Hnoise = E(t) X 2 j=1 (j) z [2] Phys. Rev. Lett. 106, 130506 (2011) Superdecoherence (II) Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms coherence time for single qubit [2]! Magnetic field noise accounts for collective phase errors: N Hnoise = E(t) X 2 j=1 (j) z For collective Gaussian phase noise it can be shown that: F (N ) / F (1) N2 [2] Phys. Rev. Lett. 106, 130506 (2011) Superdecoherence (II) Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms coherence time for single qubit [2]! Magnetic field noise accounts for collective phase errors: N Hnoise = E(t) X 2 j=1 (j) z For collective Gaussian phase noise it can be shown that: F (N ) / F (1) N2 Decoherence free subspaces: [2] Phys. Rev. Lett. 106, 130506 (2011) Superdecoherence (II) Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms coherence time for single qubit [2]! Magnetic field noise accounts for collective phase errors: N Hnoise = E(t) X 2 j=1 (j) z For collective Gaussian phase noise it can be shown that: F (N ) / F (1) N2 1 p (| ####""""i + | """"####i) 2 Decoherence free subspaces: has a decoherence time of 324(42) ms [2]! [2] Phys. Rev. Lett. 106, 130506 (2011) Superdecoherence (II) Reduce fluctuations of magnetic field: 8(1) ms vs. 95(7) ms coherence time for single qubit [2]! Magnetic field noise accounts for collective phase errors: N Hnoise = E(t) X 2 j=1 (j) z Collective Gaussian phase noise from magnetic field fluctuations For collective Gaussian phase noise it can be shown that: F (N ) / F (1) N2 1 p (| ####""""i + | """"####i) 2 Decoherence free subspaces: has a decoherence time of 324(42) ms [2]! [2] Phys. Rev. Lett. 106, 130506 (2011) Outline Qubits in Ion Traps Multiparticle Entanglement - ‘Cat States’ Creation of ‘Cat States’ Witnessing Multiparticle Entanglement Superdecoherence Conclusions Conclusions Create cat states of up to 14 trapped ions. Cat states are crucial for manifold applications in quantum information science. Collective phase noise, superdecoherence. Use decoherence free subspaces. A priori technical problem to reduce noise, but scalability problematic. Create Schrödinger’s cat? Check out: http://www.nobelprize.org/nobel_prizes/physics/ PHYSICS: Particle control in a quantum world 09:10–09:40 Controlling photons in a box and exploring the quantum to classical boundary SERGE HAROCHE, COLLÈGE DE FRANCE AND ECOLE NORMALE SUPÉRIEURE, PARIS, FRANCE 09:45–10:15 Superposition, entanglement, and raising Schroedinger’s cat DAVID J. WINELAND, NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY (NIST) AND UNIVERSITY OF COLORADO BOULDER, CO, USA CHEMISTRY: Check out: http://www.nobelprize.org/nobel_prizes/physics/ PHYSICS: Particle control in a quantum world 09:10–09:40 Controlling photons in a box and exploring the quantum to classical boundary SERGE HAROCHE, COLLÈGE DE FRANCE AND ECOLE NORMALE SUPÉRIEURE, PARIS, FRANCE 09:45–10:15 Superposition, entanglement, and raising Schroedinger’s cat DAVID J. WINELAND, NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY (NIST) AND UNIVERSITY OF COLORADO BOULDER, CO, USA CHEMISTRY: That’s it.