#physcisblr

It's been silent on my blog for a while now - the past year has been a busy time for me since I was working on my Master's thesis as well as working as a research assistant at Fraunhofer in parallel. Both were great experiences but it became so time-consuming that I totally lost track of posting any content on this blog!

In April 24 I finished my Master's Thesis in Physics in which I was mainly working on tensor networks. We tried to make DMRG (Density Matrix Renormalization Group) for Quantum Chemistry-like setups more efficient by lifting degeneracies in the symmetry sectors of the tensor network structure. Besides from what I learned on the physics-side of this project, it helped me to foster my fascination for method-development.

My work at Fraunhofer was very rewarding as well - I did not only learn a lot, I also had the chance to contribute in two publications in the area of quantum circuit cutting. In case you are interested, I could write a blog entry for one or both of the papers(:

Cryptography methods that use classical physics can be cracked - as long as you have access to sufficient computing power. Of course “sufficient computing power” can indeed be a difficult obstacle. But the crucial point is, that physics does not fundamentally forbid to crack classical cryptography methods - but as you can imagine, we can make use of quantum physics in order to create safe encryption methods. Let’s have a look at a specific example:

Perfectly secure communication - is there such a thing?

One-time-pad is a perfectly secure classical encryption method, provided that you have a key of the length of the message you want to send. This key is not allowed to be used more than once and in addition the key itself needs to be completely random. Most of this might be considered as achievable tasks, except the transmission of the key. You can imagine to transport a perfectly sealed suitcase that contains a piece of paper with your key to the person you want to communicate with. So far so good, but if you do so, you could have transmitted the message itself this way. Even if one-time-pad might be a secure method, it does not help you if you cannot transmit the key with acceptable effort. But the transmission of the key always leaves the possibility that it can somehow be copied by a third party without you noticing. At least classical physics leaves this possibility open - but quantum mechanics does not. You can even prove that the laws of quantum mechanics forbid copying of states for the general case!

Impossibility of creating copies: the No-Cloning-Theorem

Let’s dive into some quantum mechanics - the main ingredient we are going to use is that changes in quantum mechanical systems are described by unitary transformations. And unitary transformations have an important property: They preserve inner products.

The impossibility of cloning arbitrary quantum states can be shown with a proof by contradiction. Therefore we act as there was a mapping U_C that is capable of copying quantum states:

And now we can go our way straight to the contradiction by taking the inner product of both equations (<0|0>=1 because |0> is a orthonormal basis vector):

The last equation only holds if the inner product of | Ψ > and | Φ > is either 0 or 1. Thus both vectors are either the same or orthogonal to each other. Thus there is only a very small amount of states that can be copied according to the laws of quantum mechanics. Any non orthogonal states cannot be cloned! This important result gives us the capability to transmit keys that cannot be copied “on their way” by a third party. One way of making use of the No-Cloning-Theorem is the BB84 protocol - a quantum key distribution method that creates a key for one-time-pad.

As a closing remark - even if it might seem a bit off topic - I’d like to quote Julian Assange: “Cryptography is the ultimate form of non-violent direct action.”