Imperial College London Department of Physics Is quantum steering spooky? Matthew Fairbairn Pusey September 2013 Supervised by Terry Rudolph & Jonathan Barrett Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Physics of Imperial College London Declaration The following thesis was written solely by me. It presents my own work, and work by others is appropriately referenced. Matthew Fairbairn Pusey The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 2 Abstract In quantum mechanics, if one party performs a measurement on one system, different outcomes can lead to different states for another system. This phenomenon is known as quantum steering. The thesis begins with some general results about steering: a classifi- cation of which states permit the most powerful type of steering, and the implications of this for quantum correlations. The first main topic is steering in two-qubit systems. It turns out that steering provides an excellent way to visualize two-qubit states, leading to a novel criteria for entanglement and a better understanding of separable states. Oddly, the structure of steering can be more subtle in separable states than it can be in entangled ones. Returning to general quantum systems, I then turn to the EPR paradox, and its generalisation using Local Hidden State models. I show that the lack of such models can be used to quantify the amount of entanglement shared by two parties, even when one of them does not trust their measuring devices. Finally, the desire to understand steering without invoking \spooky action- at-a-distance" leads to the idea that quantum states are states of knowledge. I explore some de Finetti theorems that help to make sense of this idea, but then show a significant roadblock the most natural formalisation of it. The main results in chapters 3 and 4 appear in [JPJR13] and [Pus13] respectively. The result in chapter 6 was improved in collaboration with my supervisors and published in [PBR12]. 3 Acknowledgements My research would not have been possible without my good fortune in hav- ing not one, but two, excellent supervisors. Since one was also the lecturer and irritating-problem-setter for my first and only course in quantum in- formation, it is fair to say that almost everything I have learnt about my field I have learnt from them. Terry's impatience with technicalities and Jon's impatience with handwaving complement each other brilliantly when discussing ideas and clash horribly when writing papers. Terry has also dealt with countless bureaucratic necessities with reckless speed, whilst Jon has demonstrated the enticing possibility of getting by in academia without promptly replying to any emails at all (except of course those that offer free items of furniture). The atmosphere in the Whiteley suite, and then the IMS, and then Level 12, has been consistently fantastic, both for learning physics and avoiding doing so. I need to thank the countless people I have discussed and argued about quantum mechanics with, both at Imperial and during conferences and workshops spanning three continents. Thanks also to my brother Joe for his help with proofreading a draft of this document, and to my examiners for finding many other typos. This thesis is dedicated to Sabrina, without whom I probably would have finished it much earlier, and where's the fun in that? 4 Contents 1. Introduction 9 1.1. Operational quantum theory . 9 1.2. Composite systems . 11 1.3. Steered states . 13 1.4. Digression: comparison with other formulations . 14 1.5. The road ahead . 15 2. Steering warm-up: extending two theorems 17 2.1. Complete steering from purifications . 17 2.2. Quantum correlations from local quantum theory . 22 2.3. Summary . 24 3. Two qubits: the quantum steering ellipsoid 25 3.1. One qubit: the Pauli basis and Bloch sphere . 25 3.2. Introducing the steering ellipsoid . 26 3.3. Entangled ellipsoids . 29 3.4. Minimal product states . 36 3.5. Incomplete steering . 37 3.6. Summary . 40 4. Linking steering with negativity 41 4.1. Local hidden state models . 42 4.2. Enter Einstein, Podolsky and Rosen . 43 4.3. LHS models and entanglement . 44 4.4. Quantifying lack of LHS models: steering inequalities . 45 4.5. Quantifying lack of separability: entanglement measures . 46 4.6. Semidefinite programming for LHS models . 47 4.7. The NPA hierarchy for probabilities . 48 4.8. The NPA hierarchy for assemblages . 51 5 Contents Contents 4.9. Making the link to negativity . 52 4.10. Results: stronger Peres conjecture? . 54 4.11. Summary . 58 5. Quantum states as states of knowledge: de Finetti theorems 59 5.1. The de Finetti theorem for probabilities . 60 5.2. The de Finetti theorem for quantum states . 62 5.3. The de Finetti theorem for channels . 63 5.4. The de Finetti theorem for POVMs . 65 5.5. The de Finetti theorem for quantum instruments . 69 5.6. Summary . 70 6. Quantum states as states of knowledge: an obstacle? 71 6.1. Ontological models . 71 6.2. -ontic versus -epistemic . 72 6.3. Basic limits on ontological models: PP-incompatibility . 73 6.4. Preparation independence . 76 6.5. Summary . 79 7. Epilogue: Bell's theorem 80 8. Conclusions and outlook 84 A. Ellipses inside tetrahedra fit inside triangles 86 Bibliography 88 Index 99 6 List of Tables 4.1. Form of χ(ρAB) when mA = nA = mB = nB = 2 . 50 4.2. Steering inequalities consistent with the Peres conjecture . 57 4.3. Parameters for randomly generated steering inequalities . 58 7.1. Summary of three scenarios for bipartite entanglement . 82 7 List of Figures 1.1. The simplest quantum experiment . 9 1.2. Preparation of a mixture . 11 1.3. Preparation of a product state . 12 1.4. Preparation of a reduced state . 13 1.5. Preparation of a steered state . 14 2.1. Summary of argument for condition 3 = condition 2 . 21 ) 3.1. Ellipsoid of a separable state . 29 3.2. The various quantities used in proving lemma 3.1 . 30 3.3. Effect of in 2-dimensional case . 33 A 4.1. Negativity bounds for a simple steering inequality . 55 4.2. Negativity bounds for another steering inequality . 55 5.1. A quantum channel . 63 6.1. n = 2 case in proof of theorem 6.2 . 78 7.1. Setup for Bell test . 83 A.1. Illustration of Poncelet's porism . 86 A.2. Illustrations for the proof of theorem A.2 . 87 8 1. Introduction To investigate quantum steering, we first need to review a little quantum theory. Familiarity with the basics of linear algebra is assumed, chapter 2 of [NC00] provides an good review of the relevant material. Once the key concepts have been established this chapter concludes with a guide to the remainder of this thesis. 1.1. Operational quantum theory Quantum mechanics is a vast subject, but for the purposes of this the- sis we will only require the following \bare-bones" version of the finite- dimensional non-relativistic theory. I will take the following notions to be primitive: quantum system, preparation, and measurement. Examples of quantum systems include the polarization of a photon and the orbital en- ergy of an electron. A preparation is some repeatable procedure which outputs a quantum system. A measurement is some repeatable procedure that receives a quantum system and then produces an \outcome" from some finite set, for example a particular light may switch on. Figure 1.1.: The simplest quantum experiment: preparation of ρ followed by measurement of E , here resulting outcome a = 2. f ag 9 1.1 Operational quantum theory The theory associates these primitives with the following mathematical objects, as described, for example in [NC00, Per95, SW10]. A quantum sys- tem is associated with a finite-dimensional Hilbert space . A preparation H is associated with a unit trace positive operator on , called the quantum H state ρ. A measurement is described by associating each outcome with a positive operator Ea on , with Ea = I, the identity operator. The set H a E is called a Positive Operator Valued Measure (POVM). Finally, the f ag P probability that a preparation associated with ρ followed by a measurement associated with E gives outcome a is given by the Born rule Tr(ρE ), f ag a where Tr is the trace. A word of warning: I will often (indeed, starting with the next paragraph) abuse terminology by referring to the above primitives and their associated mathematical objects interchangeably, but this should not be taken to suggest they are in any sense conceptually equivalent. As Peres put it [Per95]: \Quantum phenomena do not occur in a Hilbert space. They occur in a laboratory.". In this thesis it will be assumed that any quantum state can, at least in principle, be prepared, and likewise any POVM can be measured. For some justifications in the case of specific physical situations, see for example [RZBB94, SW80]. If a POVM E spans the Hermitian operators on , then it is called f ag H informationally-complete [Pru77]. This implies that for all ρ = σ there ex- 6 ists an a such that Tr(ρE ) = Tr(σE ), i.e. the probabilities associated with a 6 a an informationally-complete POVM suffice to specify the quantum state.