Quantum physics has become the lens through which most physical phenomenas can be explained. It has had a vast impact on the study of a number of subfields of physics, including but not restricted to solid state and condensed matter physics, thermodynamics, electromagnetism, optics etc. Fundamentally, quantum physics describes the time-evolution of any physical system with a linear differential equation for its state. Solving this linear differential equation as a function of time (often called the Schroedinger’s equation), in principle, describes completely the behavour of the physical system.

However, while this description of a system in clean and elegant, it is only useful for understanding the dynamics of small ‘closed’ systems – systems which are completely isolated from and can’t exchange energy from their environment. The state of any closed systems can be well approximated by a finite set of states, and possible states of the closed system can be linear combinations of this finite set of states. In such cases, it is possible to numerically solve Schroedinger’s equations, and completely characterize the dynamics of such states. However, quantum systems often exchange energy with their environment and an accurate description of such an exchange of energy is crucial to gain an understanding of their dynamics. We call such systems open quantum systems. Most quantum systems that are currently being researched on for building quantum computing and quantum communication systems (e.g. superconducting qubits, single photons) are accurately described as open quantum systems.

While initially the theory of open quantum systems was studied and developed as a means of describing losses in the otherwise ideal and closed quantum systems, recent technological advances have enabled us to engineer the environment of these quantum systems to an unprecedented degree. It is now experimentally possible to ensure that a open quantum system interacts with a well defined environment mode – for example, coupling a superconducting qubit (almost completely) to an optical fiber would be possible in the near future. In such cases, it is possible to get these quantum systems to emit a well defined state into its environment, and such states can be used for a variety of applications. Perhaps the most important application of such engineered open quantum systems is the design of non-classical sources of light, which are useful for quantum communication.

For such states to be useful in quantum information processing applications, it is important for them to be entangled. Entanglement is a property of states that are accessible in quantum physics, and doesn’t have a classical counterpart. To intuitively understand the concept of entanglement, consider the example of the possible states of two interacting quantum systems. Suppose that each quantum system can be described by two states – an up state and a down state. Possible states of the two quantum systems taken together are: both quantum systems being in up state, both quantum systems being in the down state, and one of them being in the up state and the other being in the down state. However, quantum mechanically, one can also equivalently have a state which is a superposition of these four states. For example, we could have a state which is a superposition of both the systems being in up state and both the systems being in the down state. This is an example of the system being in an entangled state, for if through measurement of some sort on one of these systems we arrive at the conclusion that it is in the up state, then the second system is definitely in the up state. As might be evident, there are a lot more entangled states than there are unentangled states. A popular method for quantifying entanglement is entanglement entropy, which we describe intuitively. Consider a quantum system which comprises of multiple interacting quantum systems. As a result of their interaction, the state of the full quantum system can be entangled (as described above) – to characterize this entanglement, we focus on one of the interacting sub systems. Since this system is, in general, entangled with the other quantum systems, its state cannot be written out independently. However, we can associate probabilities of this system being in multiple states, and by a process called schmidth decomposition, we can ensure that the mutliple states being considered are orthogonal to each other (as an example, taking the state described above – an equal superposition of both states being up or both states being down, one of the systems will be in an up state with probability 0.5 and in down state with probability 0.5). Then the entanglement entropy is defined as the entropy of this probability distribution. If the local system has dimensionality D (i.e. has D linearly independent states), then the entanglement entropy of this system is at most log(D) (logarithm is to the base 2).

The above description of entanglement entropy would suggest that the larger the quantum system is (i.e. larger the value of D), the more entangled it can be. In particular, if we focus on states of environments in open quantum system, which live in extremely large (in fact infinite dimensional) hilbert spaces, it might appear that there is the possibility of creating arbitrary entanglements in such systems. However, since we are generating these states through a much smaller system (which is the open quantum system interacting with the environment), there must exist some bound on how entangled the environment state can be. This is the question I attempted to, and successfully answered, in this project.

There were two components to answering this question. The first component was to develop a finite dimensional approximation to the hilbert space of the environment. This could be done by simply discretizing the hilbert space of the environment, and a rigorous convergence of such a process can be proved by noting the fact that the hilbert space of the environment is complete, and hence any fundamental sequence of approximations to the state would converge to within the hilbert space. Having done this, we could then derive an expression for the environment state within this finite dimensional approximation – a careful study of the resulting expression immediately revealed that the state was a type of state studied extensively in many-body physics called a matrix product state. By utilizing the existing bounds on the entanglement entropy of the matrix product states, it was straightforward to show that the entanglement entropy of the environment state would also be bounded by 2log(D), where D was the dimensionality of the (finite-dimensional) open quantum system which was responsible for setting up the environment state. This analysis therefore made rigorous the idea that the entanglement entropy of the states emitted by a small open quantum system would, in fact, be bounded by the size of the open quantum system as opposed to the size of the environment.

[Note: I will attempt to put up a mathematically detailed account of this work on https://www.overleaf.com/6568571343scjdvjdrshmj and on arXiv very soon]