Category: My Papers

Conservation Laws For Every Quantum Measurement Outcome

In Quantum Mechanics, a cylinder can simultaneously rotate clockwise and anti-clockwise at the same time. This phenomenon, called superposition, is crucial for paradoxes like Schrödinger’s cat, and applications such as Quantum Computing. If we measure which way the cylinder rotates, we will find it to be clockwise or anti-clockwise. This gives us a paradox, as there is a law in physics which states that angular momentum (rotational mass * rotational speed) should be conserved. e.g. if the cylinder rotates clockwise, it shouldn’t flip to rotating anti-clockwise unless something physically stops it and sends it spinning the other way.

This paradox, involving the randomness at the heart of quantum mechanics, has troubled physicists since the dawn of quantum mechanics. Several solutions have been presented. However we have written a new paper which, we believe, solves the paradox. The change in angular momentum of the cylinder when it is measured is perfectly offset by a change in the device which prepared the rotating superposition in the first place.

This solution, inspired by a paper by Aharonov, Popescu and Rohrlich, says that this preparation device is also the frame of reference for the angle of rotation of the cylinder. Without such a frame, we would not be able to make the superposition of clockwise and anti-clockwise rotation in the first place.

Not only that, but we also show that the preparation can be arranged so that the preparation device is a finite object. In principle we can perform a laboratory experiment to test this, i.e. to show that the total angular momentum of the system and preparation device is conserved.

We can, we believe, rewrite the conservation laws in quantum mechanics, which so far only talked about conservation of the distribution over many repeated experiments of e.g. angular momentum. Now we say conservation applies for each individual outcome. We do have some remaining work to prove this holds in all cases: so far we only dealt with angular momentum on a circle. Pursuing this solution to the end should be fun, and lead to a better understanding of randomness, frames of reference, and conservation laws in physics.

Angular Momentum Flows without anything carrying it

Yakir Aharonov, Sandu Popescu and I have written a new paper where we show how Angular Momentum, a conserved quantity, can travel from one place to another across a region of space which is empty except for a vanishingly small probability of containing a particle. Remarkably, this probability, which can be made as small as we like, allows the angular momentum to flow, apparently without any carrier. This is an advance in the interaction free research field, which began with Elitzur and Vaidman’s interaction free measurement.

Teleportation of Post-Selected States

I’ve written a new paper showing how to teleport a state which is post-selected. This combines three of my favourite topics: teleportation, post-selection, and the reality of an a non-local quantum state. My final result gives a simplified measurement for checking that a shared entangled state is really “in” that state at a given time, regardless of how far apart the parts of the state may be. I’ll explain what this means in a little more detail below.

First, Teleportation starts with the question: “How can I send a quantum state from one place to another?” This is tricky as quantum states tend to decohere, i.e. become irretrievably noisy in transmission, the further they are sent, so we prefer sending classical bits instead. Since the uncertainty principle prevents us from measuring the quantum state to convert it into classical bits, it seems we can’t avoid the decoherence. Bennett et. al. showed that we can in fact send it perfectly using classical communication, pre-shared entangled states, and local operations.

Second, Post-Selection is conditioning on a particular final state of a system. In classical deterministic physics, conditioning on a particular final state is equivalent to conditioning on some particular initial state. However in quantum mechanics post-selection tells us more. For example if a particle is initially prepared at a particular position, and later we condition on a final measurement of its momentum being a particular value, we can predict that any intermediate measurement of position/momentum would give the initial/final values, apparently letting us know the position and momentum of a particle at the same time.

My main result was to show that a post-selected state can be teleport ed from one place to another. That is, if we have two systems where a post-selection is applied to A but we would like instead it to be applied to B, we can make it happen with certainty using entanglement, classical communication and local operations. Since a post-selection is usually evolved backwards in time to see what it tells us about intermediate time measurements, this can be called teleportation of a state evolving backwards in time.

Finally, I addressed the question of how one can verify that an entangled state of two spatially separated systems really is what we think it is at a fixed time. If we cannot, it would make us query whether an entangled state is the right description for our system after all. Now the usual way you check a state of one system is by doing a Von Neumann measurement. However it is impossible to perform this instantaneously for two spatially separated systems as such procedure would allow faster than light communication. Instead Vaidman showed that one could perform an instantaneous verification measurement, which checks but destroys the state, using a large amount of shared entanglement and later classical communication and processing. My contribution is to show how to perform such a verification measurement on a set of pre and post-selected systems using exponentially less entanglement than the earlier protocols. This gives us a simplified demonstration that the systems really are described by that state at a given time.

Classical to Quantum Non-Signalling Boxes

CM Ferrera, R Simmons, J Purcell, S Popescu and I have written a new paper where we define and study C-Q boxes. A C-Q box is shared across many parties, each of which can enter a classical input and receive a quantum state as output. These boxes are defined to be non-signalling: they do not allow any party to send a message to any other party.

The motivation for looking at these boxes comes from the non-local correlations which Bell Inequalities show to be present in quantum mechanics. The simplest form of a Bell Inequality thought experiment has two parties who share an entangled quantum mechanical state. Each party chooses one of two measurements and get one of two classical outputs. Bell demonstrated that the correlations between these outputs cannot be reproduced by any local classical theory. The Bell experiment can be thought of as a no-signalling C-C box, which takes classical inputs and gives classical outputs, and which happens to have an entangled quantum state and measurements inside. Popescu and Rohrlich showed that no-signalling allows for C-C boxes which give even stronger non-local correlations than quantum mechanics, and several works have speculated what additional constraints may be required to explain why nature is quantum mechanical and does not have these super-strong correlations.

One of the main questions of our research is whether all C-Q boxes can be implemented using pre-shared entanglement and C-C Boxes. One of our main results is that we can implement any bi-partite C-Q box outputting pure quantum states in such a way. However we are still investigating whether there is in fact another C-Q box, perhaps outputting mixed quantum states, which cannot be reduced to C-C boxes in this way. Regardless of the answer, we believe that studying these super-strong non-local correlations will help us to understand the quantum mechanical non-local correlations themselves.