Interview with Peter Johnson
Picture. Peter Johnson and Lorenza Viola
Who are you?
My name is Peter Johnson. I am a PhD student in the Quantum Information Science group of Professor Lorenza Viola at Dartmouth College in Hanover, NH. With Professor Viola I research topics that involve the quantum part-whole relationship, such as the quantum marginal problem, asymmetric quantum cloning, and the characterization and engineering of quasi-local dissipative dynamics. These topics draw heavily on the mathematical tools of linear algebra, representation theory, combinatorics, dynamical systems theory and functional analysis.
What prompted you to pursue this field of research?
This work stemmed from a simple question: "Why is entanglement monogamous?" Correlations described by classical probability distributions are not subject to such a constraint. Why, then, should quantum correlations be? I'll sketch the thinking that led from this question to our paper.
My favorite example of entanglement is the two-qubit singlet state. This state may arise as the spin state of the two electrons in the Helium atom ground state. Making the same measurement on the two qubits is certain to return either ��up-down�� or ��down-up��, demonstrating perfect anti-correlation. The singlet state exhibits monogamy: never can three qubit-carrying observers Alice, Bob, and Charlie, expect Alice and Bob's qubits as well as Alice and Charlie's qubits to be in the singlet state. An explanation for this limitation is that, were such a scenario to arise, then Bob and Charlie's qubits would be endowed with perfect correlation; but, no two-qubit state can express perfect correlation.
This simple proof by contradiction leads to a new question: "Why is there no perfectly-correlated two-qubit state?" The catch here is that there is! Although there is no such state per se, such correlations can arise between two measurements, and in a trivial way: measure the same system twice. The trivial evolution (identity channel) gives rise, in an appropriate sense, to the "anti-singlet" state. This idea, introduced to us by Sandu Popescu, marked a shift in perspective. From this platform, Professor Viola and I were able to phrase and answer questions about the link between monogamy of entanglement and analogous dynamical limitations, such as the no cloning theorem.
What is this latest paper all about?
This paper explores the connection between two familiar features of quantum mechanics: "no cloning" and "monogamy of entanglement". Remarkably, both concepts were developed in papers by Bill Wootters. These two "principles of limitation" are now ubiquitous in the quantum information community. Depending on the context, ranging from quantum cryptography to area-laws in quantum many-body systems, each feature might be understood as an ally or an adversary. Thus, a goal of quantum information science is to gain a handle on how these principles manifest in quantum-related tasks. We aim to illuminate the mathematical structures and more-basic principles which unite these two basic rules of quantum mechanics.
In order to compare these seemingly disparate scenarios, we develop a framework that treats dynamical relationships (quantum channels) and kinematic relationships (quantum states) on equal footing. This allows us to consider the Alice-Bob-Charlie qubit scenario (of above) to be completely analogous to one in which a channel transmits quantum information from Alice to Bob-Charlie. The question of how the Alice-Bob correlation limits the Alice-Charlie correlation is then analogous to the question of how the flow of quantum information from Alice to Bob limits flow from Alice to Charlie.
The general question of determining when a set of sub-relationships (be they states or channels) admit a consistent global description is called the general quantum joinability problem. Our paper explores some simple examples of this problem and draws the following conclusions: 1) quantum states and quantum channels are subject to complementary limitations on their correlations (disagree vs agree) and 2) these complementary limits on their correlations inform their respective joinability properties.
We hope that our framework and perspective might plant the seed for future insights into the workings of the microscopic world. Why are quantum states limited in the agreement they express? Why are quantum channels limited in the disagreement they may express? Might these opposing statistical properties shed light on connections between quantum spatial/temporal correlations and the structure of spacetime?
What do you plan to do next?
One short-coming of the framework we introduce is that it only describes temporal correlations between systems at two instances. We wish to understand either how to incorporate multi-time correlations or why such an incorporation is not possible. This is another intriguing contrast between quantum channels and states in that the former are manifestly bipartite though the latter need not be.
Another avenue we are considering comes from the work of Professor Viola and others in the early 2000s. They developed a framework to analyze the quantum part-whole relationship in the case where a tensor product structure is not relevant/available. For example, in a system of bosons, there is no operation which "singles out" Alice's boson vs Bob's boson. Given the framework we developed in this paper, one can ask the following question: since a collection of identical particles (bosons, say) at a given instant of time doesn't admit a natural tensor product structure, then what of the channel operators expressing their temporal correlations? Should they manifest a tensor product structure? If so, then, why the asymmetry between spatial and temporal identical particle statistics?