Detection of quantum entanglement in physical systems Carolina Moura Alves Merton College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2005 Abstract Quantum entanglement is a fundamental concept both in quantum mechanics and in quantum information science. It encapsulates the shift in paradigm, for the descrip- tion of the physical reality, brought by quantum physics. It has therefore been a key element in the debates surrounding the foundations of quantum theory. Entangle- ment is also a physical resource of great practical importance, instrumental in the computational advantages o®ered by quantum information processors. However, the properties of entanglement are still to be completely understood. In particular, the development of methods to e±ciently identify entangled states, both theoretically and experimentally, has proved to be very challenging. This dissertation addresses this topic by investigating the detection of entanglement in physical systems. Multipartite interferometry is used as a tool to directly estimate nonlinear properties of quantum states. A quantum network where a qubit undergoes single-particle interferometry and acts as a control on a swap operation between k copies of the quantum state ½ is presented. This network is then extended to a more general quantum information scenario, known as LOCC. This scenario considers two distant parties A and B that share several copies of a given bipartite quantum state. The construction of entanglement criteria based on nonlinear properties of quantum states is investigated. A method to implement these criteria in a simple, experimen- tally feasible way is presented. The method is based of particle statistics' e®ects and its extension to the detection of multipartite entanglement is analyzed. Finally, the experimental realization of the nonlinear entanglement test in photonic systems is investigated. The realistic experimental scenario where the source of entangled photons is imperfect is analyzed. Acknowledgements Contents Abstract i Acknowledgements ii 1 Introduction 1 1.1 Entanglement as a property of quantum systems . 1 1.2 Entanglement as a physical resource . 2 1.3 Detection and characterization of entanglement . 2 1.4 Outline of thesis . 3 1.5 Chapter outline . 4 2 Basic concepts 5 2.1 State Vectors . 5 2.1.1 Subsystems . 6 2.2 Density Operators . 6 2.2.1 Mathematical properties of density operators . 8 2.2.2 Ensemble interpretation of density operators . 9 2.3 Entanglement . 9 2.4 Superoperators . 10 2.4.1 Mathematical properties of superoperators . 10 2.4.2 Jamiolkowski isomorphism . 11 2.5 Mathematical characterization of bipartite entanglement . 11 2.5.1 Mixed states . 12 2.6 Experimental detection of entanglement . 13 2.6.1 Bell's inequalities . 13 2.6.2 Entanglement witnesses . 14 2.7 Multipartite entanglement . 15 2.7.1 Maximally entangled state . 16 2.7.2 W State . 16 2.7.3 Cluster state . 16 2.8 Quantum networks . 17 2.8.1 Universal set of gates . 17 2.8.2 Interferometry . 18 2.9 Summary . 19 iii CONTENTS iv 3 Direct estimation of density operators 21 3.1 Modi¯ed interferometry . 21 3.2 Multiple target states . 23 3.2.1 Spectrum estimation . 23 3.2.2 Quantum communication . 24 3.2.3 Extremal eigenvalues . 24 3.2.4 State estimation . 25 3.2.5 Arbitrary observables . 25 3.3 Quantum channel estimation . 25 3.4 Summary . 27 4 Direct estimation of density operators using LOCC 28 4.1 LOCC estimation of nonlinear functionals . 28 4.2 Structural Physical Approximations . 30 4.2.1 SPA using only LOCC . 31 4.3 Entanglement detection . 31 4.4 Channel capacities . 32 4.5 Summary . 32 5 Entanglement Detection in Bosons 33 5.1 Nonlinear entanglement inequalities . 33 5.2 Estimation of the purities . 34 5.2.1 Bipartite case . 34 5.2.2 Multipartite case . 35 5.3 Realization of the entanglement detection network . 36 5.4 Detection of entanglement . 37 5.5 Degree of macroscopicity . 39 5.5.1 Determination of ² ............................... 39 5.6 Summary . 40 6 Entropic inequalities 41 6.1 Entropic inequalities . 42 6.1.1 Graphical comparison between Bell-CHSH and entropic inequalities . 42 6.2 Experimental proposal . 45 6.2.1 Realistic sources of entangled photons . 46 7 Conclusion 49 Bibliography 51 List of Figures 2.1 The controlled-U gate. The top line represents the control qubit and the bottom line represents the target qubit. U acts on the target qubit i® the control qubit is in the logical state j1i................................. 18 2.2 The Mach-Zender interferometer. 19 2.3 The quantum network corresponding to the Mach-Zender interferometer (' = θ1 ¡ θ0). The visibility of the interference pattern associated with p0 varies as a function of ' according to Eq.(2.70). 20 3.1 A modi¯ed Mach-Zender interferometer with coupling to an ancilla by a controlled- U gate. The interference pattern is modi¯ed by the factor vei® = Tr [U½]. 22 3.2 Quantum network for direct estimations of both linear and non-linear functions of a quantum state. 23 3.3 A quantum channel ¤ acting on one of the subsystems of a bipartite maxi- P p mally entangled state of the form jÃ+i = k jkijki= d. The output state 1 P %¤ = d kl jkihlj ­ ¤ (jkihlj), contains a complete information about the channel. 26 4.1 Network for remote estimation of non-linear functionals of bipartite density op- erators. Since Tr[V (k)%­k] is real, Alice and Bob can omit their respective phase shifters. 29 5.1 Network of BS acting on pairs of identical bosons. The two rows of N atoms, labelled I and II respectively, are identical, and the state of each of the rows is ½123:::N . The total state of the system is ½123:::N ­ ½123:::N . 34 2 5.2 In Fig. 4.2(a), we plot the violation V of the inequalities Eq. (5.2), V1 = Tr(½123)¡ 2 2 2 2 2 Tr(½12) (dashed), V2 = Tr(½12) ¡ Tr(½1) (grey) and V3 = Tr(½12) ¡ Tr(½2) (solid), as a function of the phase Á, for N = 3 atoms. Whenever V > 0, entanglement is detected by our network. In Fig. 4.2(b) we plot di®erent purities associated with a cluster state of size N, as a function of Á. B is any one atom not at an end (dotted), any two atoms not at ends and with at least two others between them (dashed), any two or more consecutive atoms not including an end (dash-dotted), any one or more consecutive atoms including one end (solid). The plotted purities are independent of N. ................................. 38 v LIST OF FIGURES vi 5.3 Plot of the purity ¦N¡m for m = 1 (solid black), m = 7 (dashed black), m = 14 (solid grey) and m = 20 (dashed grey), as a function of ², for N = 300 atoms. 40 6.1 A graphical comparison of the Bell-CHSH inequalities with the entropic inequali- ties (6.2). All points inside the ball satisfy the entropic inequalities and all points within the Steinmetz solid satisfy all possible Bell-CHSH inequalities. NB not all the points in the outlining cube represent quantum states. 43 6.2 In a special case of locally depolarized states, represented by points within the tetrahedron, the set of separable states can be characterized exactly as an octahe- dron. All states in the ball but not in the octahedron are entangled states which are not detectable by the entropic inequalities. 44 6.3 An outline of our experimental set-up which allows to test for the violation of the entropic inequalities. 45 6.4 Possible emissions leading to four-photons coincidences. The central diagram shows the desired emission of two independent entangled pairs { one by source S1 and one by source S2. The top and the bottom diagrams show unwelcome emissions of four photons by one of the two sources. 47 CHAPTER 1 Introduction The subject of this dissertation is the detection of quantum entanglement in physical systems. Quantum entanglement was singled out by Erwin SchrÄodingeras \...the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought." [1]. Indeed, after playing a signi¯cant role in the development of the foundations of quantum me- chanics [1, 2, 3], quantum entanglement has been recently rediscovered as a physical resource in the context of quantum information science [4, 5, 6, 7]. This set of correlations, to which a clas- sical counterpart does not exist, arises from the interaction between distinct quantum systems. Entanglement is instrumental in the improvements of classical computation and classical com- munication results, of which two particularly important examples are the exponential speedup of certain classes of algorithms [8, 9] and physically secure cryptographic protocols [4]. 1.1 Entanglement as a property of quantum systems Entanglement was ¯rst used by Einstein, Podolski and Rosen (EPR) [2] to illustrate the con- ceptual di®erences between quantum and classical physics. In their seminal paper published in 1935, EPR argued that quantum mechanics is not a complete theory of Nature, i.e. it does not include a full description of the physical reality, by presenting an example of an entangled quantum state to which it was not possible to ascribe de¯nite elements of reality. EPR de¯ned an element of reality as a physical property, the value of which can be predicted with certainty, before the actual property measurement. This condition is straightforwardly obeyed in the con- text of classical physics, but not in the context of quantum mechanics. The predictive power of quantum mechanics is limited to, given a quantum state and an observable, the probabilities of the di®erent measurement outcomes.