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Quantum Information Theory Quantum Information Theory Joseph M. Renes With elements from the script of Renato Renner & Matthias Christandl and contributions by Christopher Portmann February 4, 2015 Lecture Notes ETH Zürich HS2014 Revision: f4e02a9be866 Branch: full Date: Wed, 04 Feb 2015 14:51:33 +0100 Contents Contents i 1 Introduction 1 1.1 Bits versus qubits............................................. 1 1.2 No cloning................................................. 2 1.3 Measurement and disturbance .................................... 3 1.4 Quantum key distribution....................................... 3 1.5 Quantum computation is not like classical computation ................... 4 1.6 Notes & Further reading........................................ 6 2 Probability Theory7 2.1 What is probability?........................................... 7 2.2 Probability spaces and random variables.............................. 8 2.3 Discrete random variables....................................... 11 2.4 Channels.................................................. 14 2.5 Vector representation of finite discrete spaces........................... 16 2.6 Notes & Further reading........................................ 19 2.7 Exercises .................................................. 20 3 Quantum Mechanics 23 3.1 The postulates of quantum mechanics ............................... 23 3.2 Qubits.................................................... 24 3.3 Comparison with classical probability theory .......................... 26 3.4 Bipartite states and entanglement .................................. 27 3.5 No cloning & no deleting ....................................... 28 3.6 Superdense coding and teleportation ................................ 29 3.7 Complementarity ............................................ 31 3.8 The EPR paradox............................................. 35 3.9 Bell inequalities.............................................. 37 3.10 Notes & Further reading........................................ 41 3.11 Exercises .................................................. 42 4 Quantum States, Measurements, and Channels 45 4.1 Quantum states.............................................. 45 4.2 Generalized measurements....................................... 50 4.3 Quantum operations .......................................... 55 4.4 Everything is a quantum operation ................................. 62 4.5 Notes & Further Reading ....................................... 63 4.6 Exercises .................................................. 64 5 Quantum Detection Theory 69 5.1 Distinguishing states & channels................................... 69 5.2 Fidelity ................................................... 73 5.3 Distinguishing between many states................................. 76 5.4 Binary hypothesis testing........................................ 78 i CONTENTS 5.5 Convex optimization .......................................... 79 5.6 Notes & Further reading........................................ 82 5.7 Exercises .................................................. 84 6 Divergence Measures and Entropies 87 6.1 f -Divergence ............................................... 87 6.2 Quantum divergences.......................................... 90 6.3 von Neumann & Shannon entropies ................................ 91 6.4 Entropic uncertainty relations .................................... 94 6.5 Min and max entropies......................................... 96 6.6 Entropic measures of entanglement................................. 97 6.7 Notes & Further reading........................................ 99 6.8 Exercises .................................................. 100 7 Information Processing Protocols 103 7.1 Background: The problem of reliable communication & storage . 103 7.2 The resource simulation approach.................................. 103 7.3 Optimality of superdense coding and teleportation....................... 106 7.4 Compression of classical data..................................... 109 7.5 Classical communication over noisy channels .......................... 115 7.6 Compression of quantum data .................................... 121 7.7 Entanglement purification....................................... 123 7.8 Exercises .................................................. 126 8 Quantum Key Distribution 129 8.1 Introduction................................................ 129 8.2 Classical message encryption ..................................... 130 8.3 Quantum cryptography ........................................ 132 8.4 QKD protocol .............................................. 136 8.5 Security proof of BB84 ......................................... 137 8.6 Exercises .................................................. 144 A Mathematical background 147 A.1 Hilbert spaces and operators on them ............................... 147 A.2 The bra-ket notation........................................... 147 A.3 Representations of operators by matrices ............................. 148 A.4 Tensor products.............................................. 149 A.5 Trace and partial trace.......................................... 150 A.6 Decompositions of operators and vectors ............................. 151 A.7 Operator norms and the Hilbert-Schmidt inner product ................... 152 A.8 The vector space of Hermitian operators ............................. 153 A.9 Norm inequalities ............................................ 154 A.10 A useful operator inequality...................................... 154 B Solutions to Exercises 155 Bibliography 179 ii Introduction 1 “Information is physical” claimed the late physicist Rolf Landauer,1 by which he meant that Computation is inevitably done with real physical degrees of freedom, obeying the laws of physics, and using parts available in our actual physical universe. How does that re- strict the process? The interface of physics and computation, viewed from a very fun- damental level, has given rise not only to this question but also to a number of other subjects...[1] The field of quantum information theory is among these “other subjects”. It is the result of asking what sorts of information processing tasks can and cannot be performed if the underlying informa- tion carriers are governed by the laws of quantum mechanics as opposed to classical mechanics. For example, we might use the spin of a single electron to store information, rather than the magneti- zation of a small region of magnetic material (to say nothing of ink marks on a piece of paper). As this is a pretty broad question, the field of quantum information theory overlaps with many fields: physics, of course, but also computer science, electrical engineering, chemistry and materials science, mathematics, as well as philosophy. Famously, it is possible for two separated parties to communicate securely using only insecure classical and quantum transmission channels (plus a short key for authentication), using a protocol for quantum key distribution (QKD). Importantly, the security of the protocol rests on the correctness of quantum mechanics, rather than any assumptions on the difficulty of particular computational tasks—such as factoring large integers—as is usual in today’s cryptosystems. This is also fortunate from a practical point of view because, just as famously, a quantum computer can find prime factors very efficiently. On the other hand, as opposed to classical information, quantum information cannot even be copied, nor can it be deleted! Nevertheless, quantum and classical information theory are closely related. Because any such classical system can in principle be described in the language of quantum mechanics, classical information theory is actually a (practically significant) special case of quantum information theory. The goal of this course is to provide a solid understanding of the mathematical foundations of quantum information theory, with which we can then examine some of the counterintuitive phe- nomena in more detail. In the next few lectures we will study the foundations more formally and completely, but right now let’s just dive in and get a feel for the subject. 1.1 Bits versus qubits Classical information, as you already know, usually comes in bits, random variables which can take on one of two possible values. We could also consider “dits”, random variables taking on one of d values, but this can always be thought of as some collection of bits. The point is that the random variable takes a definite value in some alphabet. 2 In contrast, quantum information comes in qubits, which are normalized vectors in C . Given some basis 0 and 1 , the qubit state, call it , can be written = a 0 +b 1 , with a, b C such that 2 2 j i j i j i j i j i 2 a + b = 1. The qubit is generally not definitely in either state 0 or 1 ; if we make a measurement whosej j j twoj outcomes correspond to the system being in 0 and j1 i, thenj i the probabilities are j i j i 2 2 2 2 prob(0) = 0 = a prob(1) = 1 = b (1.1) jh j ij j j jh j ij j j 1Rolf Wilhelm Landauer, 1927-1999, German-American physicist. 1 1. INTRODUCTION 2n The state of n qubits is a vector in C , a basis for which is given by states of the form 0,...,0 = 0 0 , 0,...,1 , 0,...,1,0 , etc. Then we write the quantum state of the entire collectionj i as j i ⊗ ··· ⊗ j i j i j i X = s s , (1.2) j i s 0,1 n j i 2f g P 2 where s are binary strings of length n and once again s C with = 1 = s s . Allowed transformations of a set of qubits come in2 the formh ofjunitaryi operators,j j which just 2n transform
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