Finding the consequences of symmetry for open-system quantum dynamics is a problem with broad applications, including describing thermal relaxation, deriving quantum limits on the performance of amplifiers, and exploring quantum metrology in the presence of noise. The symmetry of the dynamics may reflect a symmetry of the fundamental laws of nature or a symmetry of a low-energy effective theory, or it may describe a practical restriction such as the lack of a reference frame. In this paper, we apply some tools of harmonic analysis together with ideas from quantum information theory to this problem. The central idea is to study the decomposition of quantum operations---in particular, states, measurements, and channels---into different modes, which we call modes of asymmetry. Under symmetric processing, a given mode of the input is mapped to the corresponding mode of the output, implying that one can only generate a given output if the input contains all of the necessary modes. By defining monotones that quantify the asymmetry in a particular mode, we also derive quantitative constraints on the resources of asymmetry that are required to simulate a given asymmetric operation. We present applications of our results for deriving bounds on the probability of success in nondeterministic state transitions, such as quantum amplification, and a simplified formalism for studying the degradation of quantum reference frames.
Finding the consequences of symmetry for open-system quantum dynamics is a problem with broad applications, including describing thermal relaxation, deriving quantum limits on the performance of amplifiers, and exploring quantum metrology in the presence of noise. The symmetry of the dynamics may reflect a symmetry of the fundamental laws of nature or a symmetry of a low-energy effective theory, or it may describe a practical restriction such as the lack of a reference frame. In this paper, we apply some tools of harmonic analysis together with ideas from quantum information theory to this problem. The central idea is to study the decomposition of quantum operations---in particular, states, measurements, and channels---into different modes, which we call modes of asymmetry. Under symmetric processing, a given mode of the input is mapped to the corresponding mode of the output, implying that one can only generate a given output if the input contains all of the necessary modes. By defining monotones that quantify the asymmetry in a particular mode, we also derive quantitative constraints on the resources of asymmetry that are required to simulate a given asymmetric operation. We present applications of our results for deriving bounds on the probability of success in nondeterministic state transitions, such as quantum amplification, and a simplified formalism for studying the degradation of quantum reference frames.
It is impossible to overstate the importance of symmetry in physics and mathematics. Symmetry arguments play a central role in a broad range of problems from simplifying a system of linear equations to a deep role in organizing the fundamental principles of physics. They are used, for instance, in Noether’s theorem to find the consequences of symmetry of a dynamics. For many systems of interest, the dynamics are sufficiently complicated that one cannot hope to characterize their evolution completely, whereas by appealing to the symmetries of the dynamical laws one can easily infer many useful results. \n \nIn part I of this thesis we study the problem of finding the consequences of symmetry of a (possibly open) dynamics from an information-theoretic perspective. The study of this problem naturally leads us to the notion of asymmetry of quantum states. The asymmetry of a state relative to some symmetry group specifies how and to what extent the given symmetry is broken by the state. Characterizing these is found to be surprisingly useful to constrain which final states of the system can be reached from a given initial state. Another motivation for the study of asymmetry comes from the field of quantum metrology and relatedly the field of quantum reference frames. It turns out that the degree of success one can achieve in many metrological tasks depends only on the asymmetry properties of the state used for metrology. We show that some ideas and tools developed in the field of quantum information theory are extremely useful to study the notion of asymmetry of states and therefore to find the consequences of symmetry of an open or closed system dynamics. \n \nIn part II of this thesis we present a novel application of symmetry arguments in the field of quantum estimation theory. We consider a family of multi-copy estimation problems wherein one is given n copies of an unknown quantum state according to some prior distribution and the goal is to estimate certain parameters of the given state. In particular, we are interested to know whether collective measurements are useful and if so to find an upper bound on the amount of entanglement which is required to achieve the optimal estimation. We introduce a new approach to this problem by considering the symmetries of the prior and the symmetries of the parameters to be estimated. We show that based on these symmetries one can find strong constraints on the amount of entanglement required to implement the optimal measurement. In order to infer properties of the optimal estimation procedure from the symmetries of the parameters and the prior we come up with a generalization of Schur-Weyl duality. Just as Schur-Weyl duality has many applications to quantum information theory and quantum algorithms so too does this generalization. In this thesis we explore some of these applications.
A useful and unifying concept in quantum information theory is the idea of a consumable "resource". The basic idea behind quantum resource theories is that whenever we face restrictions on the set of operations we can perform, some states (resource states) enable us to perform particular tasks that otherwise would have been impossible. In this thesis we first review a resource theory that recently has gained a lot of attention, namely the resource theory of asymmetry in which one is restricted to symmetric operations and therefore asymmetric states are the resource states. The asymmetry properties of a state relative to some symmetry group specify how and to what extent the given symmetry is broken by the state. In the remainder of the thesis we explain how the ideas developed in this resource theory are related to three different areas in quantum theory: theory of quantum reference frames, the Wigner-Araki-Yanase(WAY) theorem and the conditional probability interpretation of time in quantum mechanics. Theory of quantum reference frames treats reference frames like any other physical system within the formalism of quantum theory, which causes the measurements to be an approximation of the measurements against their classical counterparts. We consider the dynamics of a quantum directional reference frame undergoing repeated interactions. These interactions induce a back-action on the reference which is the central focus of our study. The effect of a precise sequence of measurement outcomes on the reference frame is studied by looking at both the case that the measurement record is averaged over and the case wherein it is retained. We find, in particular, that there is interesting dynamics in the latter situation, which cannot be revealed by considering the average case. We then consider in detail how a sequence of rotationally invariant unitary interactions affects the reference frame, a situation, which leads to quite different dynamics than the case of repeated measurements. Different strategies for correcting reference frame drift is considered given that we have access to a set of particles with polarization opposite to the direction of drift. In particular, we find that, by implementing a suitably chosen unitary interaction after every two measurements, we can eliminate the rotational drift of the reference frame. The WAY theorem establishes an important constraint that conservation laws impose on quantum mechanical measurements. We formulate the WAY theorem in the broader context of resource theories, where one is constrained to a subset of quantum mechanical operations described by a symmetry group. Establishing connections with the theory of quantum state discrimination we obtain optimal unitaries describing the measurement of arbitrary observables, explain how prior information can permit perfect measurements that circumvent the WAY constraint, and provide a framework that establishes a natural ordering on measurement apparatuses through a decomposition into asymmetry and charge subsystems. Finally we review two di erent schemes in the conditional probability interpretation (CPI) of time in quantum mechanics. In these schemes parameter time "t" as it appears in Schrodinder equation is considered to be unobservable, yet one can use an extension of conditional probabilities in order to study dynamics of a system relative to another system, namely the quantum clock. We phrase CPI as a quantum communication protocol which enables us to use the machinery developed in the resource theory of asymmetry. This provides us with a deeper understanding of the decoherence effect caused due to the inaccessibility of the parameter time "t" and that how this effect can be suppressed by choosing an optimal initial state for the quantum clock.
Abstract Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently, a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However, these conditions are difficult to use in practice and their physical implications unclear. In the present theoretical work, we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed to make a given quantum state accessible under channels covariant with any given symmetry group.
Symmetries of both closed- and open-system dynamics imply many significant constraints. These generally have instantiations in both classical and quantum dynamics (Noether's theorem, for instance, applies to both sorts of dynamics). We here provide an example of such a constraint which has no counterpart for a classical system, that is, a uniquely quantum consequence of symmetric dynamics. Specifically, we demonstrate the impossibility of broadcasting asymmetry (symmetry breaking) relative to a continuous symmetry group, for bounded-size quantum systems. The no-go theorem states that if two initially uncorrelated systems interact by symmetric dynamics and asymmetry is created at one subsystem, then the asymmetry of the other subsystem must be reduced. We also find a quantitative relation describing the trade-off between the subsystems. These results cannot be understood in terms of additivity of asymmetry, because, as we show here, any faithful measure of asymmetry violates both subadditivity and superadditivity. Rather, they must be understood as a consequence of an (intrinsically quantum) information-disturbance principle. Our result also implies that if a bounded-size quantum reference frame for the symmetry group, or equivalently, a bounded-size reservoir of coherence (e.g., a clock with coherence between energy eigenstates in quantum thermodynamics) is used to implement any operation that is not symmetric, then the quantum state of the frame or reservoir is necessarily disturbed in an irreversible fashion, i.e., degraded.
Dynamical symmetries in closed systems lead to conservation laws and to selection rules that are based on those conservation laws. However, conservation laws are not applica- ble to open systems that undergo irreversible transformations. More general selection rules are needed to determine whether, given two states, the transition from one state to the other is possible. Characterizing the asymmetry properties of quantum states using quantum information theoretical tools and methods is particularly fruitful for this purpose. This thesis is concerned with the problem of finding new general selection rules that hold for both open and closed systems. The usual approach to this problem relies heavily on group theory and involves a detailed study of the structure of the symmetry group. In this thesis, we approach the problem in a completely new way. Our approach is to use entanglement to investigate the asymmetry properties of quantum states. In order to do that, we embed the system’s Hilbert space in a larger tensor product Hilbert space, whereby all symmetric states are mapped to separable states, asymmetric states are mapped to entangled states, and the symmetric transformations between two states are replaced by local operations on their bipartite images. Our method is not restricted to only a specific group but applies, in general, to all symmetries that are associated with semi-simple compact Lie groups and their associated Lie algebras. The mapping of the original states to bipartite states that act on the larger Hilbert space enables us to use the well studied theory of entanglement to investigate the consequences of dynamic symmetries. For example, the monotonicity condition on measures of entanglement provide us with new selection rules. Under reversible transfor- mations, the entanglement of the bipartite image states becomes a conserved quantity. These entanglement-based conserved quantities are new and different from the conserved quantities based on expectation values of the Hamiltonian symmetry generators.
This thesis investigates the use of concepts from quantum information theory to analyze and quantify symmetry breaking in extended quantum systems, particularly focusing on the key measure known as Entanglement Asymmetry. The core of the work introduces and applies Entanglement Asymmetry. This quantity is defined based on the difference in information content between a quantum state and a version of that state where the symmetry has been enforced (or "symmetrized"). It represents an alternative approach to traditional, often problematic, local order parameters for measuring symmetry breaking: asymmetry is zero only if the state perfectly preserves the symmetry. The major contributions of the thesis are structured as follows: Symmetry Breaking in Bulk Critical Systems: The research analyzes Entanglement Asymmetry in two-dimensional Conformal Field Theories (CFTs) where the symmetry is explicitly broken. This work derives the asymptotic behavior of the asymmetry as a function of the subsystem size, extending earlier finite-size results to the critical continuum limit. Boundary Effects and Quenches: The analysis is extended to cases where the symmetry holds in the bulk of a CFT but is broken at the edge by a boundary condition. The thesis provides formulas describing this boundary-induced asymmetry, including power-law corrections whose exponents are linked to the properties of boundary-changing operators. It also explores the time evolution of asymmetry in quantum quench protocols. Generalization of Asymmetry to Modern Symmetries: The concept of Entanglement Asymmetry is generalized to include finite generalized symmetries, such as higher-form and non-invertible symmetries, which are a major topic in modern QFT. This involves establishing a generalized procedure for "symmetrizing" a state without relying on traditional group mathematics. Entanglement in Non-Hermitian Systems: The final chapter shifts focus to explore other entanglement properties, specifically symmetry-resolved entanglement and the entanglement Hamiltonian, within non-Hermitian models and their corresponding non-unitary CFTs.