Studentische Forschung an der Fakultät für Mathematik

Beyond Time-Reversal

Masterarbeit von Alexander Müller-Hermes SS 2012
Betreuer: Prof. Dr. Michael M. Wolf

The goal of information theory is to describe information processing tasks in a precise mathematical framework. An important concept to do so is that of a channel, i.e. a physical device that can be used to transmit information. In the mathematical framework this corresponds to a map from some input space the sender can control to some output space the receiver can control. The simplest case of a channel is the identity channel, i.e. the channel given by the identity map on bits $${\rm id}:\{0,1\}\to\{0,1\}\,,~~{\rm id}(x):=x~.$$ This channel describes the situation where information is transmitted without any disturbance. Most physical systems do not behave that nice and may introduce errors, which mathematically corresponds to a stochastic map, that gives certain outputs depending on the input and some probabilities. To transmit information via those noisy channels one has to encode the data in order to protect it from the noise. A natural question is, at what rate information data can be sent through such a noisy channel using some optimal encoding and decoding. This leads to the important concept of a capacity, which measures this optimal rate.

As information theory is about the description of physical systems, the underlying physical theory has to be taken into account. We know, that the physical laws of our universe are fundamentally given by quantum mechanics.

In the mathematical framework of quantum mechanics the state of a finite-dimensional physical system is described by a density matrix, i.e. a linear, positive operator \( \rho\in{\mathfrak{L}}(H) \) with trace 1 acting on a Hilbert space \(H\) corresponding to the physical system.

A very important difference compared to classical systems arises, when we join two quantum systems together two form a bigger quantum system. Assume two physical systems with corresponding Hilbert spaces \(H_A\) and \(H_B\). The laws of quantum mechanics state, that the Hilbert space corresponding to the joint system is given by the tensor product $$H_{AB} = H_A\otimes H_B$$ of the two Hilbert spaces. This innocent looking law leads immediately to one of the fundamental features of quantum theory, "entanglement".

Let us first consider so called pure states. These states are represented by density matrices, which are projectors onto normalized vectors in the Hilbert space. They can be interpreted as states, in which we know everything there is to know about the system at hand.

For pure states we can distinguish the cases where the state can be written as a tensor product of two pure states $$\rho_{AB}=\left|\psi_A\right.\rangle\langle\left.\psi_A\right|\otimes\left|\psi_B\right.\rangle\langle\left.\psi_B\right|$$ and the cases, where this is not possible. The latter ones are called "pure entangled states" and have quite remarkable properties. In those states the two systems are correlated in a way that has no counterpart in classical theory as the total system is in a definite state while the subsystems are not.

For general density matrices \( \rho_{AB}\in{\mathfrak{L}}(H_A\otimes H_B)\) one has to distinguish between classical correlations and quantum correlations stemming from entanglement. It is in general a hard problem to detect entanglement in arbitrary quantum states.

As the name already suggests, quantum information theory is the synthesis of the two parts introduced above. The concept of a classical channel is generalized to that of a quantum channel, which is a map $${\mathcal{T}}:{\mathfrak{L}}(H)\to{\mathfrak{L}}(H)$$ that maps quantum states to quantum states and is compatible with the tensor product structure of joint quantum systems. One fundamental task of quantum information theory is to find capacities for transmitting quantum states over such quantum channels. The difference to the classical case is, that the entanglement with other systems has to be preserved, which makes it much harder to handle quantum capacities compared to the classical case.

'reversed clock
The transposition map in quantum theory can be interpreted as time-reversal.
An interesting observation about quantum channels, which is not hard to prove, is the connection between the quantum capacity and time-reversal. Time-reversal is a non-physical map, i.e. it cannot be implemented in a real physical system. It is modeled via the ordinary transposition map \(\vartheta(\rho) := \rho^T\). Reversing the time of the quantum system and then applying the quantum channel \({\mathcal{T}}\) means to concatenate it with the transposition map yielding the map $${\mathcal{T}}\circ\vartheta~.$$ In general this map is not a quantum channel, as the transposition is not a quantum channel itself. But in the cases, where the above construction gives a quantum channel, the original channel \({\mathcal{T}}\) is known to have vanishing quantum capacity, i.e. a quantum channel can transmit quantum states reliably while preserving the entanglement with other systems only if its time-reversal is not a physical operation.

One goal of my thesis was to generalize this and find maps different than the transposition, which yield similar criteria. This task is connected to many other problems of quantum information theory and the transposition map plays an important role in all of them. Many natural ways of doing this generalization fail because there are some, quite innocent-looking, properties used in the proof of the above statement, that already fix the corresponding map to be the transposition.

To close this summary I want to state my own motivation for working in quantum information theory. It is of course a very young field and as such has a lot of potential. There are still many interesting theorems to prove and one can start with quite simple tools and arrive at new questions that are far from trivial. The field lies at the intersection of mathematics, physics and computer science and has already caused many advances in those areas. In numerical mathematics there are tensor-network techniques leading to efficient algorithms for high-dimensional problems. They originate from many-body quantum physics and can be understood using ideas from quantum information theory. In computer science there is the idea of quantum cryptography for securing communication and also the idea of quantum computing, which is connected to quantum information theory. Finally in physics there is the possibility of studying physical processes as information processing tasks and apply quantum information theory to them, which may lead to new insights and a better understanding of the known physical laws. This were just a few applications and there are many more to come.

Zur vollständigen Master-Arbeit von Alexander Müller-Hermes hier (PDF, 1MB)

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Steckbrief des Autors


  • Alexander Müller-Hermes
  • Student an der TU seit WS 2007/08
  • B.Sc. in Mathematik, SS 2010
  • M.Sc. in Mathematik, SS 2012
  • Mathematische Interessen: Quantum Information Theory, Mathematical Physics
  • nach dem Master: PhD in Mathematischer Physik an der TUM
  • Webseite (M5, Mathematische Physik)

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