I'd need a whole book to go into quantum entanglement (:-)), but the summary version is that quantum physics predicts that, for instance, you can get a pair of quantum particles into an entangled state where making a measurement of a property of one (its spin, for instance) will instantly influence the other particle, however far apart they are. And this has been experimentally verified many times since the 1980s. Entanglement can't be used for what seems the obvious application - sending an instantaneous message - as the 'information' transmitted is random. However it can act as a linking mechanism to do things that would otherwise be impossible.
In the case of the impressive-sounding teleportation, the apparent impossibility that entanglement can help with is the so-called 'no cloning theorem.' It was proved reasonably early that it is impossible to make a duplicate of a quantum particle while preserving the original. However, with a mix of entanglement and conventional information transfer, it is possible to transfer a property of a quantum particle to a similar particle elsewhere, in effect making a remote copy of at least one aspect of the particle. In the process, the original particle's properties are altered (so you don't end up with an identical pair) and you never find out what the property's value was.
Despite these provisos, if you could do this for all the significant properties of a particle - or a collection of particles - it would be as if the original particle had been teleported from its original position to the location of the modified particle. In effect, to use the inevitable simile, it would be like putting the particle through a Star Trek transporter. This mechanism in its simplest form is already valuable for applications like quantum computing, but inevitably there was interest in doing it for real - all the significant properties - and with something bigger than a single particle.
It ought to be stressed this is never going to produce a Star Trek transporter, whether as Amazon's latest way to deliver goods or to avoid the rigours of air travel. This is because of the sheer number of particles in an sizeable object, which would take thousands of years to scan and reassemble. If we're talking a product, you don't need an atomic level duplicate - you can just send the instructions for a factory to make it. If we're talking a person, even if you got over the fact that the original 'you' would be disintegrated and only a perfect copy was produced, that timescale is simply impractical.
Over the years we've seen various properties of particles and simple molecules teleported. And it would be fascinating if it were possible to teleport a virus or bacterium. However, it should be stressed that this is not what has happened here. Firstly, nothing has actually happened. It's a proposed mechanism, not an experiment that has been carried out. And secondly we have to be clear what's meant by that 'teleport the memory' headline. In more detail, Tongcang Li at Purdue University and Zhang-qi Yin at Tsinghua University have suggested a way to use electromechanical oscillators and superconducting circuits to teleport the internal quantum state and center-of-mass motion state of a microorganism.
What it essentially means is that they may be able to transfer as a package some of the states of molecules in the bacterium to another organism. As these states are a form of information, they are described as teleporting memories. There are a few provisos, however. To make the system work, the organisms would have to be cryogenically frozen. They would not be alive. And what isn't made clear is how the setup would deal with the reality that any two bacteria are not identical in their molecular makeup. But the theoretical experiment is interesting in the way it accesses internal properties of the organism for teleportation, rather than expecting it to be stripped down, particle by particle.
You can, in principle, see more in the original paper, but unfortunately it is a pay access.
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