If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state's ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state psi relative to the symmetry group G are completely specified by the characteristic function of the state, defined as chi_psi(g)= where g\in G and U is the unitary representation of interest. For a symmetry described by a compact Lie group G, we show that two pure states can be reversibly interconverted one to the other by symmetric operations if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. Characteristic functions also allow us to easily identify the conditions for one pure state to be converted to another by symmetric operations (in general irreversibly) for the various paradigms of single-copy transformations: deterministic, state-to-ensemble, stochastic and catalyzed.
If a system undergoes symmetric dynamics, then the final state of the system can only break the symmetry in ways in which it was broken by the initial state, and its measure of asymmetry can be no greater than that of the initial state. It follows that for the purpose of understanding the consequences of symmetries of dynamics, in particular, complicated and open-system dynamics, it is useful to introduce the notion of a state's asymmetry properties, which includes the type and measure of its asymmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can also be understood in terms of information-theoretic concepts, for instance in terms of the state's ability to encode information about an element of the symmetry group. We show that the asymmetry properties of a pure state psi relative to the symmetry group G are completely specified by the characteristic function of the state, defined as chi_psi(g)= where g\in G and U is the unitary representation of interest. For a symmetry described by a compact Lie group G, we show that two pure states can be reversibly interconverted one to the other by symmetric operations if and only if their characteristic functions are equal up to a 1-dimensional representation of the group. Characteristic functions also allow us to easily identify the conditions for one pure state to be converted to another by symmetric operations (in general irreversibly) for the various paradigms of single-copy transformations: deterministic, state-to-ensemble, stochastic and catalyzed.
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. Characterizing these is found to be surprisingly useful for addressing a very common problem: to determine what follows from a system's dynamics (possibly open) having that symmetry. We demonstrate and exploit the fact that the asymmetry properties of a state can be understood in terms of information-theoretic concepts. We show that for a pure state $\ensuremath{\psi}$ and a symmetry group G, they are completely specified by the characteristic function of the state, defined as ${\ensuremath{\chi}}_{\ensuremath{\psi}}(g)\ensuremath{\equiv}\ensuremath{\langle}\ensuremath{\psi}|U(g)|\ensuremath{\psi}\ensuremath{\rangle}$, where $g\ensuremath{\in}G$ and U is the unitary representation of interest. Based on this observation, we study several important problems about the interconversion of pure states under symmetric dynamics, such as determining the conditions for reversible transformations, deterministic irreversible transformations, and asymptotic transformations.
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.
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.
Symmetry in quantum mechanicsRoughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is.Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map u : H → H satisfying one and hence all of the following equivalent conditions:• u is invertible and preserves the inner product, i.e., uϕ, uψ = ϕ, ψ (ϕ, ψ ∈ H).The discussion of symmetries in quantum physics is based on the above idea, but the mathematically obvious choices need not be the physically relevant ones.Even in elementary quantum mechanics, where A = B(H), i.e., the C*-algebra of all bounded operators on some Hilbert space H, the concept of a symmetry is already diverse.The main structures whose symmetries we shall study in this chapter are:1.The normal pure state space P 1 (H), i.e., the set of one-dimensional projections on H, with transition probability τ : P 1 (H) × P 1 (H) → [0, 1] defined by (2.44).2. The normal state space D(H), i.e. the convex set of density operators ρ on H. The self-adjoint operators B(H)sa on H, seen as a Jordan algebra (see below).4. The effects E (H) = [0, 1] B(H) , seen as a convex partially ordered set (poset). 5.The projections P(H) on H, seen as an orthocomplemented lattice.6.The unital commutative C*-subalgebras C (B(H)) of B(H), seen as a poset.Each of these structures comes with its own notion of a symmetry, but the main point of this chapter will be to show these notions are equivalent, corresponding in all cases to either unitary or-surprisingly-anti-unitary operators, both merely defined up to a phase.The latter subtlety will open the world of projective unitary group representation to quantum mechanics (without which the existence of spin-1 2 particles such as electrons, and therewith also of ourselves, would be impossible).
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.
A cornerstone of quantum mechanics is the characterization of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated with the dynamics of closed systems and the stochastic evolutions associated with the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time-reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time-symmetric formulation of quantum theory must either restrict the set of the allowed evolutions or modify the operational interpretation of quantum states and processes. Here we propose a time-symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time-symmetric version by introducing an operation of conditioning on the outcomes of past experiments.