# Quantum Correlations, Data Hiding, and Quantum Many-body Systems

Lead Research Organisation:
Imperial College London

Department Name: Dept of Physics

### Abstract

Two or more quantum systems can be correlated in a way that defies any classical explanation. In the last twenty years, it has emerged that this kind of quantum correlations, termed entanglement, has a distinguished role in information processing. It turns out that entangled quantum systems can be harnessed to transmit, store, and manipulate information in a more efficient and secure manner than possible in the realm of classical physics. From a different perspective, the existence of entanglement in quantum theory also has dramatic consequences to our ability to simulate quantum many-body systems: It is widely believed that it is impossible to simulate efficiently in a classical computer the dynamics of quantum many-body systems. While this is a major problem when studying such systems, e.g. in the condensed matter context, it naturally leads to the idea of a quantum computer, in which controlled quantum systems are employed to perform computation in a more efficient way than possible by classical means. The field of quantum information and computation is concerned with the usefulness and limitations of quantum-mechanical systems to computation and information processing. The objective of the research is to make progress on several outstanding theoretical questions of quantum information science.

In the first theme one will address questions that would represent key progress to our understanding of quantum entanglement and its use in quantum information transmission. The first topic is focused on understanding the inherent irreversibility in the manipulation of entanglement. The second topic, in turn, seeks to achieve a better understanding of the non-additivity of quantum information in quantum communication channels.

The second theme will focus on quantum data hiding, correlations that are not accessible by restricted measurements (e.g. local ones), and in particular how one can address several current challenges in quantum information science by pursuing an in-depth understanding of data hiding in quantum systems. The research will focus on three topics related to quantum data hiding. The first two are related to the difficulties brought by data hiding states to two outstanding open problems in quantum information theory - the task of deciding if a state is entangled and the establishment of area laws for gapped local Hamiltonians -, together with proposals for overcoming them. The third addresses the question of generating quantum data hiding states by very simple procedures, such as constant depth quantum circuits, and its impact to the problem of understanding equilibration of quantum systems from first principles

The final theme is concerned with quantum hamiltonian complexity, an exciting new area linking condensed matter physics and quantum many-body theory to computational complexity theory and quantum computation. The research will address two topics in this direction. The first concerns the possibility of performing quantum computation by cooling down physical systems. The second is concerned with determining the computational complexity of estimating properties of thermal states of local models.

Together these 3 themes will enable us to widen our understanding of quantum correlations, quantum many-body systems, and the use of quantum-mechanical systems for information processing.

In the first theme one will address questions that would represent key progress to our understanding of quantum entanglement and its use in quantum information transmission. The first topic is focused on understanding the inherent irreversibility in the manipulation of entanglement. The second topic, in turn, seeks to achieve a better understanding of the non-additivity of quantum information in quantum communication channels.

The second theme will focus on quantum data hiding, correlations that are not accessible by restricted measurements (e.g. local ones), and in particular how one can address several current challenges in quantum information science by pursuing an in-depth understanding of data hiding in quantum systems. The research will focus on three topics related to quantum data hiding. The first two are related to the difficulties brought by data hiding states to two outstanding open problems in quantum information theory - the task of deciding if a state is entangled and the establishment of area laws for gapped local Hamiltonians -, together with proposals for overcoming them. The third addresses the question of generating quantum data hiding states by very simple procedures, such as constant depth quantum circuits, and its impact to the problem of understanding equilibration of quantum systems from first principles

The final theme is concerned with quantum hamiltonian complexity, an exciting new area linking condensed matter physics and quantum many-body theory to computational complexity theory and quantum computation. The research will address two topics in this direction. The first concerns the possibility of performing quantum computation by cooling down physical systems. The second is concerned with determining the computational complexity of estimating properties of thermal states of local models.

Together these 3 themes will enable us to widen our understanding of quantum correlations, quantum many-body systems, and the use of quantum-mechanical systems for information processing.

### Planned Impact

Although the scope of the research proposed is theoretical, it is underpinned by the knowledge that certain aspects of it could contribute to a breakthrough in modern technology. The field of quantum information science is driven by the goal of revolutionizing our most fundamental possibilities of processing information in the physical world.

One specific long-term objective of the field is to build a quantum computer. This is a special type of computer that would outperform any of today's supercomputers. One of the main applications of a quantum computer would to simulate quantum systems efficiently, something that is a major challenge in today's computers. The simulation quantum of quantum systems is a key problem in a myriad of areas including material sciences, quantum chemistry, microbiology, and nanosciences. Thus a full working quantum computer would likely lead to technological breakthroughs in all of this areas, with their evident consequences for the development of society.

Another long-term goal of quantum information science is to establish ways of performing secure communication, for example between an online buyer and her credit card provider. Secure communication is a central task is modern society. However, the existing protocols only work under assumptions that are not known to be true and are contingent on bounds on the computational power of an adversary party who seeks to eavesdrop the communication. The area of quantum communication and quantum cryptography provide a fundamentally new solution to this problem by proposing the use of quantum mechanical systems to the establishment of secure communication, with security guaranteed not by a priori assumptions, but by the very fundamental laws of nature. Although quantum cryptography over short distances is already a reality, it is a long-term goal of the field to create paths for the establishment of quantum secure communication on a global scale. Here too the consequences to the development of society would be substancial.

The present research contributes to these two grand goals by advancing in the theoretical foundations of quantum information science. It will also advance in the relation of the field with several other important areas of knowledge (e.g. computational complexity, statistical mechanics, condensed matter physics) in the hope that this will bring closer to reality all the potential of using quantum mechanical systems to computation and information processing, and at the same time improve our understanding of quantum mechanics by considering such informational and computational perspective of the theory.

One specific long-term objective of the field is to build a quantum computer. This is a special type of computer that would outperform any of today's supercomputers. One of the main applications of a quantum computer would to simulate quantum systems efficiently, something that is a major challenge in today's computers. The simulation quantum of quantum systems is a key problem in a myriad of areas including material sciences, quantum chemistry, microbiology, and nanosciences. Thus a full working quantum computer would likely lead to technological breakthroughs in all of this areas, with their evident consequences for the development of society.

Another long-term goal of quantum information science is to establish ways of performing secure communication, for example between an online buyer and her credit card provider. Secure communication is a central task is modern society. However, the existing protocols only work under assumptions that are not known to be true and are contingent on bounds on the computational power of an adversary party who seeks to eavesdrop the communication. The area of quantum communication and quantum cryptography provide a fundamentally new solution to this problem by proposing the use of quantum mechanical systems to the establishment of secure communication, with security guaranteed not by a priori assumptions, but by the very fundamental laws of nature. Although quantum cryptography over short distances is already a reality, it is a long-term goal of the field to create paths for the establishment of quantum secure communication on a global scale. Here too the consequences to the development of society would be substancial.

The present research contributes to these two grand goals by advancing in the theoretical foundations of quantum information science. It will also advance in the relation of the field with several other important areas of knowledge (e.g. computational complexity, statistical mechanics, condensed matter physics) in the hope that this will bring closer to reality all the potential of using quantum mechanical systems to computation and information processing, and at the same time improve our understanding of quantum mechanics by considering such informational and computational perspective of the theory.

### Publications

Brandão F
(2013)

*An area law for entanglement from exponential decay of correlations*in Nature Physics
Brandão F
(2015)

*The second laws of quantum thermodynamics.*in Proceedings of the National Academy of Sciences of the United States of America
Brandão F
(2015)

*Area law for fixed points of rapidly mixing dissipative quantum systems*in Journal of Mathematical Physics
Brandão F
(2016)

*Product-State Approximations to Quantum States*in Communications in Mathematical Physics
Brandão F
(2014)

*Exponential Decay of Correlations Implies Area Law*in Communications in Mathematical Physics
Brandão F
(2015)

*Entanglement area law from specific heat capacity*in Physical Review B
Brandão FG
(2015)

*Reversible Framework for Quantum Resource Theories.*in Physical review letters
Brandão FG
(2015)

*Quantum Conditional Mutual Information, Reconstructed States, and State Redistribution.*in Physical review letters
Brandão FG
(2015)

*Generic emergence of classical features in quantum Darwinism.*in Nature communications### Related Projects

Project Reference | Relationship | Related To | Start | End | Award Value |
---|---|---|---|---|---|

EP/J017280/1 | 01/02/2013 | 16/06/2013 | £975,171 | ||

EP/J017280/2 | Transfer | EP/J017280/1 | 17/06/2013 | 31/12/2016 | £916,190 |

Description | - I have discovered a fundamental relation between entanglement and correlation length, showing that the latter implies the amount of entanglement is limited in one-dimensional quantum systems. This was a surprising result as previous research put in doubt such relation. It has applications both to our ability to simulate quantum systems efficiently, with applications e.g. in material science, and in understanding limitations for building a quantum computer |

Exploitation Route | It might lead to better methods to simulate quantum systems. |

Sectors | Digital/Communication/Information Technologies (including Software) |

URL | https://www.sciencedaily.com/releases/2013/09/130915134224.htm |