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Chapman & Hall/CRC eBooks

Quantum measurement 2: general pure states, Bell states

Helmut Bez, Tony Croft

Published Jan 6, 2023• 0 citations
Publisher

Abstract

The relatively simple examples of quantum computation (emulating digital computation) discussed in Chapters 15 , 17 and 18 involved only the simplest quantum inputs and produced, deterministically, outputs identical to their digital counterparts. The simplified statement of the von Neumann-Lüders measurement postulate, discussed in Chapter 14 , was applicable to these computations. In this chapter we introduce the processing of more general, i.e., super-imposed, quantum states. This is divergent from the digital case and, as will be clear later, imparts quantum computation with extended power.

DOI
10.1201/9781003264569-19

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The relatively simple examples of quantum computation (emulating digital computation) discussed in Chapters 15 , 17 and 18 involved only the simplest quantum inputs and produced, deterministically, outputs identical to their digital counterparts. The simplified statement of the von Neumann-Lüders measurement postulate, discussed in Chapter 14 , was applicable to these computations. In this chapter we introduce the processing of more general, i.e., super-imposed, quantum states. This is divergent from the digital case and, as will be clear later, imparts quantum computation with extended power.
This chapter aims to define quantum equivalents of a number of familiar digital gates. It argues that the quantum gates, with appropriately restricted state vector inputs, produce exactly the same numerical outputs as their digital counterparts. Hence, although current digital technology is not based on invertible Boolean functions, in order to emulate digital computation quantum mechanically, it is necessary to first construct invertible representations of Boolean functions. The chapter considers only some of the more familiar examples – up to and including the Peres half-adder. In the graphical representation of quantum gates and circuits, the horizontal lines indicate the time-ordered application of the unitary gates to the input qubits.
We have seen how information can be encoded onto a quantum system by selecting the state in which it is prepared. Retrieving the information is achieved by performing a measurement, and the optimal measurement in any given situation is usually a generalized measurement. In between preparation and measurement, the information resides in the quantum state of the system, which evolves in a manner determined by the Hamiltonian. The associated unitary transformation may usefully be viewed as quantum information processing; if we can engineer an appropriate Hamiltonian then we can use the quantum evolution to assist in performing computational tasks. Our objective in quantum information processing is to implement a desired unitary transformation. Typically this will mean coupling together a number, perhaps a large number, of qubits and thereby generating highly entangled states. It is fortunate, although by no means obvious, that we can realize any desired multiqubit unitary transformation as a product of a small selection of simple transformations and, moreover, that each of these need only act on a single qubit or on a pair of qubits. The situation is reminiscent of digital electronics, in which logic operations are decomposed into actions on a small number of bits. If we can realize and control a very large number of such operations in a single device then we have a computer. Similar control of a large number of qubits will constitute a quantum computer. It is the revolutionary potential of quantum computers, more than any other single factor, that has fuelled the recent explosion of interest in our subject. We shall examine the remarkable properties of quantum computers in the next chapter. In digital electronics, we represent bit values by voltages: the logical value 1 is a high voltage (typically +5 V) and 0 is the ground voltage (0 V). The voltage bits are coupled and manipulated by transistor-based devices, or gates. The simplest gates act on only one bit or combine two bits to generate a single new bit, the value of which is determined by the two input bits. For a single bit, with value 0 or 1, the only possible operations are the identity (which does not require a gate) and the bit flip.
In Chapters 2 and 3 we assumed a model according to which an experiment is split into two parts – preparation and measurement. This led us to the associated concepts of states and observables, respectively. On the basis of this picture we can think about two types of apparatus, which can be characterized abstractly as follows. Preparators are devices producing quantum states. No input is required, but a quantum output is produced. Measurements are input–output devices that accept a quantum system as their input and produce a classical output in the form of measurement outcome distributions. This setting is summarized in Figure 2.2. Furthermore, in subsection 3.5.2 we discussed coarse-graining relations; coarse-graining can be depicted as a box with a classical input and a classical output. This line of thought indicates that at least one type of input–output box is so far missing from our discussion – a device taking a quantum input and producing a quantum output. Hence, we say that • quantum channels are input–output devices transforming quantum states into quantum states. Each quantum channel can be placed between arbitrary preparation and measurement devices (Figure 4.1). Channels, and a slightly more general concept, that of operations , are the topics of this chapter. Transforming quantum systems In this section we discuss the mathematical definition of operations and channels. The notion of an operation was introduced by Haag and Kastler [70] within the framework of algebraic quantum field theory. Later Kraus [88], Lindblad [93] and Davies [51] discussed the requirement for operations to be completely positive.
We have seen in earlier chapters how digital computations can be emulated quantum mechanically. In this chapter, we cover further aspects of processing, and highlight some significant differences that occur between the digital and quantum cases. We first introduce the concepts of quantum parallelism and quantum swapping both of which have analogs in the digital case. We demonstrate that copying, which is a trivial operation in digital computing, is not generally achievable in the quantum case. The quantum concept of teleportation, which has no analog in digital computing, is also introduced.
Abstract The postulates presented at this point are generally agreed upon as being the primary set. But in the course of these postulates, there is no mention of the consequences of measurement. This chapter discusses this problem and the solution as provided by the Von-Neumann postulate. The concept of the projection operator is introduced, and this leads naturally to the study of the quantum entangled state. The results show in part the origin of the struggle that Einstein and others had with quantum, and the Einstein, Podolsky, and Rosen (EPR) paradox. Quantum entanglement is the key to advanced ideas in quantum encryption, teleportation, and quantum computing.
Although the basic ideas of quantum computation are fairly straightforward, the detailed manipulations required to extract a useful result from an entangled superposition of answers can be rather complicated. There are, however, a small number of algorithms which are simple enough to explain using only elementary methods. The best example is Deutsch's algorithm: as this only requires two qubits its properties are amenable to brute-force matrix calculations, and these calculations can be simplified using a variety of short cuts, which also give some insight into how and why the algorithm works. We will also explore the simplest case of Grover's quantum search algorithm, which is once again simple enough to be tacked by brute force. Finally we will look briefly at two methods used to stabilize quantum computers against errors. Deutsch's algorithm The invention of Deutsch's algorithm can be taken as defining the start of modern quantum computation, and it remains a key example, exhibiting many of the key properties of quantum algorithms in a particularly simple form. (Note, however, that the version of Deutsch's algorithm described here is not in fact the original but a later modification which is both more powerful and easier to understand.) Consider a binary function f from one bit to one bit, that is a function which takes in either 0 or 1 as its input and returns either 0 or 1 as its output. There are four such functions which may be conveniently labeled by their outputs as shown in Table 8.1.