Entanglement
In this chapter, we will find out how we can apply quantum mechanics to more than one system. In doing so, we encounter what is truly strange in quantum mechanics, namely entanglement. We also explore some of the more shocking applications of quantum mech
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Entanglement
In this chapter, we will find out how we can apply quantum mechanics to more than one system. In doing so, we encounter what is truly strange in quantum mechanics, namely entanglement. We also explore some of the more shocking applications of quantum mechanics, including teleportation and quantum computing.
6.1 The State of Two Electrons We have looked at thought experiments with a single photon in a Mach–Zehnder interferometer, a single electron in a Stern–Gerlach apparatus, and the interaction of a two-level atom with an optical pulse. We found that their states are all described by a two-dimensional complex vector, and the observables and evolution operators are described by 2 × 2 matrices. But what if we have two photons, or two electrons, or two atoms? Or three? Let’s consider two electrons, and assume that they are spatially well-separated so we can label “electron 1” and “electron 2” without any ambiguity. It is easy to write the state of electron 1 as |ψ1 = a|↑1 + b|↓1 with |a|2 + |b|2 = 1 ,
(6.1)
where we add a subscript “1” to emphasise that we refer to electron 1. Similarly, we can write down the state of electron 2:
Electronic supplementary material The online version of this chapter (https://doi.org/10.1007/978-3-319-92207-2_6) contains supplementary material, which is available to authorized users. © Springer International Publishing AG, part of Springer Nature 2018 P. Kok, A First Introduction to Quantum Physics, Undergraduate Lecture Notes in Physics, https://doi.org/10.1007/978-3-319-92207-2_6
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6 Entanglement
|φ2 = c|↑2 + d|↓2 with |c|2 + |d|2 = 1 .
(6.2)
The question is: what is the state of the composite system consisting of these two electrons? We can construct the states of the composite system from the states of the individual systems. Remember that the symbols and writing inside the ket are nothing more than convenient labels, indicating the measurement outcomes. We therefore have four possible measurement outcomes if we measure the spin of each electron in the z-direction: |electron 1 = ↑, electron 2 = ↑ , |electron 1 = ↑, electron 2 = ↓ , |electron 1 = ↓, electron 2 = ↑ , |electron 1 = ↓, electron 2 = ↓ .
(6.3)
This is hardly convenient, so instead we may write |↑1 , ↑2 , |↑1 , ↓2 , |↓1 , ↑2 , |↓1 , ↓2 .
(6.4)
By convention, the ordering of the ↑ and ↓ arrow is fixed, and do we really need the comma? Clearly, we can reduce this further to |↑↑ , |↑↓ , |↓↑ and |↓↓ .
(6.5)
These are four quantum states that make perfect sense in the light of measuring Sz on each electron separately. And because this is quantum mechanics, we can take superpositions of these states, such as |ψ =
1 1 1 1 |↑↑ + |↑↓ + |↓↑ + |↓↓ . 2 2 2 2
(6.6)
Next, suppose that the electrons are in the states given in Eqs. (6.1) and (6.2). How do we write this in terms of the states of Eq. (6.5)? We can look first at the probabilities of the measurement outcomes of Sz on the two electrons. Since the electrons are completely independent, the probabilities multiply: P
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