Entanglement and Quantum Measurements

Entangled states also show up, perhaps surprisingly, in some descriptions of quantum measurements. John von Neumann (1903–57), one of the pioneers of quantum theory, proposed that measurements in quantum mechanics should be understood as creating entangled states between the quantum system state (a qubit’s state in our language) and the measurement system (usually built out of a lot of qubits). At first sight, that formulation seems to contradict Bob’s mantra “measurement crushes entanglement.” Let’s explore these ideas.

A measurement device is engineered, based on our knowledge of physics, so that it has a state $|M_0⟩$ correlated with the Alice’s basis state $|A_0⟩$, for example, and another state $|M_1⟩$ correlated with Alice’s basis state $|A_1⟩$. That means that if we observe the measurement device in the state $|M_1⟩$ after the measurement interaction, then we know the incoming state had a state component along $|A_1⟩$. If you understand those last two sentences, you have come a long way in developing your understanding of QIS!

In von Neumann’s picture, measurement systems are set up so that the output of a measurement device is expressed as

$$\frac{1}{\sqrt{2}} \{{|A_0⟩ |M_0⟩ + |A_1⟩ |M_1⟩}\}.$$

Often, authors will say, pointing to an expression like the equation above, that the measurement interaction leaves the measurement device in a superposition of the two states $|M_0⟩$ and $|M_1⟩$. Schrödinger’s infamous cat, viewed as a measurement device for radioactive decay, is, in this view, both dead and alive. Based on our discussion of entangled states earlier, that kind of statement is just plain wrong. We have seen that if a system of qubits is described by an entangled state, we cannot say anything about the states of the individual qubits. In fact, in some sense, the individual qubits don’t “have” quantum states if the overall state is an entangled one.

The key question is then: How does the combined object-measurement device system “collapse” to a specific measurement outcome? Von Neumann’s model does not address that issue.

If we want a piece of equipment to serve as a measurement device, its state after the measurement interaction must be persistent so we can record the result without changing the measurement device state. For example, in the situation described in the equation above, if we obtain the result $|M_0⟩$ for the measurement apparatus, we know from the design of the apparatus that the qubit state must have had a component along $|A_0⟩$. So, the measurement problem is then the problem of figuring out under what conditions a measurement device state is persistent and not significantly disturbed by itself being measured. We know how to build such devices, and they work reliably. But how do we provide a theory that tells us, for example, how many qubits the measurement device must have to end up in a persistent state.