### 4. Qubits

A qubit (quantum bit) is a quantum-mechanical two-state system. Any quantum system that can have two different states can also assume an arbitrary superposition of these states. Compared to a classical bit, a qubit has a significantly higher complexity (a continuum of possible states versus two distinct states 0 and 1). But nevertheless, because the state space is just two-dimensional, the measurement of any observable can produce at most two different results.

A canonical example of a qubit is provided by the spin of a spin-1/2 particle. Many ideas in this chapter are formulated with this system in mind. But there are other realizations of qubits, for example, the polarization states of photons, which we are going to discuss in Section 4.5.

Because of the relative simplicity of a qubit system, it is worthwhile to review some peculiarities of quantum mechanics. In Section 4.2, we use a typical Stern-Gerlach experiment to illustrate the projection postulate, the state preparation by single-particle measurements, and the state verification by ensemble measurements (Section 4.3). We ask whether it is meaningful to talk about the state of a single qubit, and we describe how one can determine (or rather estimate) an unknown quantum state. Moreover, we discuss the impossibility of "classical" teleportation in quantum mechanics.

In Section 4.4, we associate a unique "spin-up direction" with every qubit state. We describe the implementation of rotations as unitary transformations in the qubit's Hilbert space and compute the transition probabilities between different qubit states.

The strange topic of single-particle interference is presented in Section 4.6. We introduce interferometers and discuss the problems of acquiring the "which-way" information. We describe a variant of the double-slit experiment and discuss what it means to rotate a qubit through an angle of 2 Pi. Interaction-free measurement (the detection of a bomb without actually looking at it) is presented in Section 4.6.4 as an example illustrating the meaning of the interference of probability amplitudes.

Section 4.7 deals with quantum cryptography. We present an example of a secure key distribution protocol that allows one to establish a secure communication via the classical one-time pad. The security of the method depends on the fact that quantum mechanics indeed gives a complete description of the state of a qubit. In a hidden variable theory, one assumes that the state of a qubit can be described by some additional parameters whose knowledge would enable us to make more accurate predictions. Section 4.8 presents an example of a hidden variable theory and discusses its implications.

We conclude this chapter with a section about the spin in a time-dependent magnetic field (Section 4.9). In particular, we discuss the time evolution in a periodically time-dependent magnetic field and the phenomenon of spin resonance, or magnetic resonance. The results are relevant for technological applications like nuclear spin tomography.

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• 4.1. States and Observables 158
• 4.1.1. The Hilbert space of a qubit 158
• 4.1.2. States of a qubit 159
• 4.1.3. Qubit observables 160
• 4.2. Measurement and Preparation 162
• 4.2.1. Stern-Gerlach experiment 162
• 4.2.2. Projection postulate 164
• 4.2.3. Stern-Gerlach filter and state preparation 165
• 4.3. Ensemble Measurements 167
• 4.3.1. State verification 167
• 4.3.2. Determining an unknown state 168
• 4.3.3. Classical teleportation is impossible 170
• 4.4. Qubit Manipulations 171
• 4.4.1. All states are "spin-up" in some direction 171
• 4.4.2. Rotations of a qubit 174
• 4.4.3. Time evolution of the spin in a magnetic field 176
• 4.4.4. Special topic: Spinor rotations 177
• 4.4.5. Transition probabilities between qubit states 179
• 4.5. Other Qubit Systems 181
• 4.5.1. Photon polarizations 181
• 4.5.2. Spatial states of photons 184
• 4.5.3. Two states of a harmonic oscillator 187
• 4.6. Single-Particle Interference 189
• 4.6.1. Interferometer 189
• 4.6.2. A double-slit experiment 192
• 4.6.3. A rotation through $2\pi$ 193
• 4.6.4. Interaction-free measurement 194
• 4.7. Quantum Cryptography 197
• 4.7.1. One-time pad 197
• 4.7.2. Quantum key distribution 198
• 4.8. Hidden Variables 200
• 4.8.1. Failure of classical picture 200
• 4.8.2. Hidden-variable interpretation 201
• 4.9. Special Topic: Qubit Dynamics 204
• 4.9.1. Time-dependent Hamiltonian 204
• 4.9.2. Time dependence generated by unitary operators 205
• 4.9.3. Magnetic resonance 206