13.6.3: Entangled qubits

We have previously introduced the notion of entanglement in terms of wave functions of a system that allows two measurements to be made. Here we see that it follows from the properties of the tensor product α a as a means to concatenate systems. Consider two qubits \(|\; \psi\rangle = \dbinom{\alpha}{\beta}\) and \(|\; \phi\rangle = \dbinom{a}{b}\), according to the definition of the tensor product,

\[|\;\psi\rangle \otimes|\;\varphi\rangle=\left(\begin{array}{l}
\alpha a \\
\alpha b \\
\beta a \\
\beta b
\end{array}\right) \tag{13.32} \]

If we operate in the ensemble system (i.e. ignoring that it is composed by two subsystems), it is not unthinkable to reach a state described by the following ket

\[\left|\;\psi_{12}\right\rangle=\frac{1}{\sqrt{2}\;}\left(\begin{array}{l}
0 \\
1 \\
1 \\
0
\end{array}\right) \tag{13.33} \]

It turns out that the composite system \(|\; \psi_{12}\rangle\) cannot be expressed as a tensor product of two independent qubits. That is, operating directly on the ensemble, it is possible to reach states that cannot be described by two isolated systems.

To see why it is so, try to equal equations 13.32 and 13.33: the first element of the ket requires \(\alpha a = 0\); this implies that either \(\alpha\) or \(a\) must be zero. However if \(\alpha = 0\) the second element cannot be 1, and similarly, if \(a\) = 0 the third element would have to be zero instead of one. So there is no combination of \(\alpha, \beta, a\), and \(b\) that allows us to write the system described in equation 13.33 as a tensor product like the one described in equation 13.32. We conclude that

\[\left|\;\psi_{12}\right\rangle \neq|\;\psi\rangle \otimes|\;\varphi\rangle . \tag{13.34} \]

We have already encountered similar situations in the context of “mixing” in chapter 11. There, we noted that intensive variables could no longer be well defined in terms of the subsystems, furtermore, if the process of mixing two subsystems was not reversible, the entropy in the final system was bigger than the sum of the entropies, and it no longer made sense to try to express the composite system in terms of its original constituents. The two situations are different but there is certainly room for analogy.

Whenever this situation arises at the quantum level, we say that the two qubits (or any two systems) are entangled. This word is a translation from the German word “Verschränkung”, that is often also translated as “interleaved”, Schrödinger coined this word to describe the situation in which:

“Maximal knowledge of a total system does not necessarily include total knowledge of all of its parts, not even when these are fully separated from each other and at the moment are not influencing each other at all.”.\(^4\)

The most salient difference with what we saw in the mixing process arises from the fact that, as Schrödinger points out in the paragraph above, this “interleaving” of the parts remains even after separating them, and the measurement of one of the parts will condition the result on the other. This is what Einstein called “spooky action at a distance”.

We can illustrate the effect of entanglement on measurements rewriting equation 13.33 by means of a superposition of two tensor products

\begin{align*}
\left|\;\psi_{12}\right\rangle &=\frac{1}{\sqrt{2}}\;\left(\begin{array}{l}
0 \\
1 \\
1 \\
0
\end{array}\right) \tag{13.35} \\
&=\frac{1}{\sqrt{2}}\;\left[\left(\begin{array}{l}
0 \\
0 \\
1 \\
0
\end{array}\right)+\left(\begin{array}{l}
0 \\
1 \\
0 \\
0
\end{array}\right)\right] \tag{13.36}\\
&=\frac{1}{\sqrt{2}}\left[|0\rangle_{1}|1\rangle_{2}+|1\rangle_{1}|0\rangle_{2}\right] \\
&=\frac{1}{\sqrt{2}}\left[\left|0_{1} 1_{2}\right\rangle+\left|1_{1} 0_{2}\right\rangle\right] \tag{13.37}
\end{align*}

where we have added subindices to distinguish between the two qubits. This is also a good example to start appreciating the power of bracket notation to simplfy expressions.

If we measure the first qubit, we will obtain either \(|\; 0\rangle_1\) or \(|\; 1\rangle_1\). To compute the probability that the outcome is \(|\; 0\rangle_1\), we take the dot product of the entangled state with \(| 0_1, ?_2\rangle =|\; 0\rangle_1\)

\begin{align*}
\left\langle 0_{1}, ?_{2} \mid \psi_{12}\right\rangle &=\langle 0_{1}, ?_{2}|\cdot \frac{1}{\sqrt{2}}[|\;0\;\rangle_{1}|\;1\rangle_{2}+|\;1\rangle_{1}|\;0\rangle_{2}] \tag{13.38}\\
&=\frac{1}{\sqrt{2}}\left(\langle 0 \mid 0\rangle_{1}\langle ? \mid 1\rangle_{2}+\langle 0 \mid 1\rangle_{1}\langle ? \mid 0\rangle_{2}\right) \tag{13.39}\\
&=\frac{1}{\sqrt{2}}\langle ? \mid 1\rangle_{2} \tag{13.40}
\end{align*}

This results says that the outcome will be \(|0\rangle_1\) with a probability of 1/2, if and only if the second system collapses at the same time to the state \(|\; 1\rangle_2\) (note that otherwise if the question mark represented a 0, the probability would be equal to zero). So the measurement on the first system conditions the value on the second system even if the systems are far apart.

To appreciate the spookyness of entanglement, it may be worth thinking of it in a more mundane setting. Imagine you have such a great connection with a colleague that every time he yawns you systematically yawn as well. Your common friends will pay no attention to it as it is a normal thing to happen, we know that when somebody yawns people in the surroundings tend to yawn. You would certainly scare them though if your colleague went to Europe and you remained in the US, and every once in a while you were driven to have a yawn, precisely when your friend had one. In fact, to be scared your friends would need the input from a referee at each side of the ocean to record the events and match time tags. The question would arise as to whether you and your colleague can use your the yawn-connection for the purpose of inmediate communication, since there appears to be a need for a referee. This cartoon example is certainly not quantum mechanical, however it illustrates what is it about entanglement that has fascinated and scared at the same time some of the greatest minds of our time. And at the same time, it introduces one caveat of quantum communication: the need for a classical exchange to verify that communication has existed (the referees). You will have a chance to appreciate this in a real quantum mechanical setting when we discuss teleportation.

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\(^4\)Extract from “Die gegenwartige Situation in der Quantenmechanik,” Erwin Schrödinger, Naturwissenschaftern. 23 : pp. 807-812; 823-823, 844-849. (1935). English translation: John D. Trimmer, Proceedings of the American Philosophical Society, 124, 323-38 (1980).