Visual Quantum Mechanics





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Chapter Summary:

6. Quantum Information Theory

The question of whether quantum mechanics violates the principle of locality and Einstein causality has been the subject of ongoing discussions. This is relevant for the theory of quantum communication, which makes use of the "nonlocal" correlations between the parts of a bipartite system in an entangled state. In Section 6.1, we describe the Bell states of two-qubit systems and discuss their measurement by means of exchanging locally available information.

In Section 6.2, we discuss the question of whether a manipulation of one qubit of an entangled pair can in any way influence the other qubit (which may be spatially separated from the first qubit). Closely related is the famous Einstein-Podolsky-Rosen (EPR) paradox, which is presented in Section 6.3. In Section 6.4, we describe the mathematical and statistical properties of the correlations arising from entanglement. In Section 6.5, we give an elementary derivation of Bell's inequality that is based on classical ideas about the correlation of measurement results ("local realism"). It has been confirmed experimentally that quantum mechanics violates Bell's inequality. While quantum mechanics is local in the sense required by Bell's theorem, it appears to violate the assumption of realism (that is, the assumption that observables do have values even if they are not measured).

The strange features of the correlations between the parts of an entangled system can be used for quantum information exchange. Alice and Bob can communicate rather efficiently if they share entangled pairs of qubits. The dense coding protocol describes the transmission of classical 2-bit information using a single qubit. The teleportation protocol sends one qubit with the help of two classical bits (see Section 6.6).

A quantum computer (Section 6.7) applies a sequence of unitary transformations ("reversible logic gates") to a quantum register (a system of n qubits). This transforms an initial state (the input) into a final state (the output) that is measured. In that way, any task that can be performed by a classical computer can also be performed by a quantum computer.

The ability of quantum logic gates to operate on superpositions of input states can be used to do computations with all possible classical inputs in parallel. For certain problems, one succeeds in using interference in a clever way so that the final measurement gives the desired result in a few steps, where a classical computer would need many steps. In Section 6.9, we give a detailed description of a simple algorithm, where a quantum computer is more efficient than a classical computer. An overview of important quantum algorithms concludes this chapter.





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    • 6.1. Entangled States of Two-Qubit Systems 272
      • 6.1.1. Bases of entangled states 272
      • 6.1.2. Global aspects of local measurements 273
      • 6.1.3. Determining a Bell state 275
      • 6.1.4. Preparing a Bell state 276
    • 6.2. Local and Nonlocal 278
      • 6.2.1. Nonlocal interaction? 278
      • 6.2.2. Local manipulations 280
    • 6.3. The Einstein-Podolsky-Rosen Paradox 281
    • 6.4. Correlations Arising from Entangled States 285
      • 6.4.1. Joint probabilities in two-qubit systems 285
      • 6.4.2. A protocol for verifying joint probabilities 288
    • 6.5. Bell Inequalities and Local Hidden Variables 290
      • 6.5.1. General setup for measuring correlations 291
      • 6.5.2. Realism and hidden variables 292
      • 6.5.3. Statistical independence 293
      • 6.5.4. Bell's theorem 295
      • 6.5.5. Violation of the Bell inequality 297
    • 6.6. Entanglement-Assisted Communication 300
      • 6.6.1. Dense coding 301
      • 6.6.2. Quantum state teleportation 302
    • 6.7. Quantum Computers 305
    • 6.8. Logic Gates 307
      • 6.8.1. Single-qubit gates: The Hadamard transformation 308
      • 6.8.2. The controlled-not gate 310
      • 6.8.3. Sequences of quantum logic gates 313
      • 6.8.4. An impossible gate 315
    • 6.9. Quantum Algorithms 316
      • 6.9.1. Function evaluation 316
      • 6.9.2. Simple quantum algorithms 317
      • 6.9.3. Further applications of quantum computers 320
        • 6.9.3.1. Grover's algorithm 320
        • 6.9.3.2. Simon's algorithm 321
        • 6.9.3.3. Period finding and the quantum Fourier transform 321
        • 6.9.3.4. Shor's algorithm 322
        • 6.9.3.5. Error correction 322