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Many good questions on this site have explored how entanglement lies at the boundary between the quantum world and the classical. For example in computational speedups, or teleportation or superdense coding, at least two qubits are entangled in some form or another, and entanglement seems to be at the heart of, or required for, the quantum improvement.

The no-cloning theorem, on the other hand, is a statement applicable to even a single qubit. Nonetheless, there is no classical analogue of the no-cloning theorem, and yet the no-cloning theorem can be the basis for interesting applications of quantum mechanics/quantum information theory.

Two "entanglement-free" applications of the no-cloning theorem that come to immediate mind are:

  1. Weisner's quantum money scheme, which begat
  2. The BB84 quantum key distribution scheme.

Although there are "entangled" versions of the above (e.g. the E91 scheme), the "entanglement-free" versions are just as valid applications of qubits.

Can qubits that are not entangled and instead in a product state be used in other interesting applications, in a manner that does not seem to have a classical analogue?

If so, are the applications merely also a version of the no-cloning theorem, or is there some other aspect of quantum information theory at play?

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One (obvious) application is the generation as True Random Number Generators, e.g. IDQ, or you can download some here Free True Random Numbers (please do not use these for security relevant application).

In order to build such a TRNG, from a quantum circuit perspective, all you need is a single qubit, a Hadamard gate and a measurement. Although there might be more efficient ways to do it.

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Certainly not exhaustive, but to get the ball rolling...

One possible application is blind quantum computation. In this, there is a user who wants to complete a computation, but only has the capability of producing single-qubit (non-entangled) states. These are sent to a server who can (locally) entangle them for the purposes of performing a measurement-based quantum computation. However, the trick is that the server never find out what computation is being performed.

In general, it seems likely that any security-related protocol will be using the indistinguishability of non-orthogonal states to achieve part of that security, and hence you can relate it to no-cloning.

Moving away from security-related applications, one might think about quantum metrology. Usually, quantum metrology is referring to the enhancement in measurement achieved by using entangled states. However, think about a qubit whose energy gap between $|0\rangle$ and $|1\rangle$ is affected by a magnetic field. If you prepare many qubits in the $|+\rangle$ state, leave them for some time in the magnetic field, and then measure how many are in the $|+\rangle$ or $|-\rangle$ state, you get a measurement of the magnetic field. This might almost be considered a classical technique, and is more or less the process that goes on inside e.g. MRI machines. This would not seem to have anything to do with no-cloning.

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  • $\begingroup$ Your metrology example reminds me that the emission spectrum of a lowly hydrogen atom acts a bit like a quantum computer to "solve" nontrivial eigenproblems associated with spherical harmonics, in much the same way that couscous on a metal sheet acts as a computer to "solve" nontrivial eigenproblems in (planar) harmonics. $\endgroup$ – Mark S Aug 1 '19 at 23:32
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In lecture 4 of O'Donnell's series on quantum computing, he introduces the Elitzur-Vaidman bomb tester, which is an interesting application of the quantum Zeno effect.

O'Donnell introduces the bomb tester prior to discussing multi-qubit entanglement in lecture 5.

In the Elitzur-Vaidman tester, a single qubit in a superposition can be used to probe and detect the presence of a bomb, importantly without triggering the bomb. The success probability is naively low, but can be amplified arbitrarily high by rotating the qubit by a small $\epsilon$ amount, and feeding the qubit back through the tester. The bomb only goes off with probability only $\epsilon^2$ for each test.

This computation of the presence/absence of a bomb certainly requires a qubit in superposition, along with a unitary gate to apply a rotation, along with the Born rule to determine the success probabilities; however, it does not seem to require any entanglement. Furthermore it's not clear to me how the quantum Zeno effect is related to the no-cloning Theorem, or if the no-cloning theorem and the quantum Zeno effect are two sides to the same coin.

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