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arXiv:quant-ph/0506223 · International Journal of Quantum Information

QUANTUM ESTIMATION OF RELATIVE INFORMATION

Netanel H. Lindner, Petra F. Scudo, Dagmar Bruß

Published Feb 1, 2006• 11 citations
arXivHTMLPDF

Abstract

We derive optimal schemes for preparation and estimation of relational degrees of freedom between two quantum systems. We specifically analyze the case of rotation parameters representing relative angles between elements of the SU(2) symmetry group. Our estimation procedure does not assume prior knowledge of the absolute spatial orientation of the systems and as such does not require information on the underlying classical reference frame in which the states are prepared.

DOI
10.1142/s0219749906001657

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We derive optimal schemes for preparation and estimation of relational degrees of freedom between two quantum systems. We specifically analyze the case of rotation parameters representing relative angles between elements of the SU(2) symmetry group. Our estimation procedure does not assume prior knowledge of the absolute spatial orientation of the systems and as such does not require information on the underlying classical reference frame in which the states are prepared.
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.
The maximum likelihood strategy for the estimation of group parameters allows one to derive in a general fashion optimal measurements, optimal signal states, and their relations with other information theoretical quantities. These results provide deep insight into the general structure underlying optimal quantum estimation strategies. The entanglement between representation spaces and multiplicity spaces of the group action appears to be the unique kind of entanglement which is really useful for the optimal estimation of group parameters.
Quantum limits to parameter estimation Introduction Many experiments can be thought of as comprising two steps: (i) a preparation procedure in which the system to be measured is isolated and prepared, and the apparatus is initialized; and (ii) a measurement step in which the system is coupled to an apparatus and the measurement result recorded. The preparation procedure can be specified by a set of classical parameters, or settings of a physical device. The measurement results are random classical variables that will be correlated with the preparation procedure. In this chapter we are concerned with the case in which the classical parameters specifying the preparation of the state are imperfectly known. Then, assuming that the physical system is well understood, these correlations allow the unknown parameters to be estimated from the measurement results. As we saw in the last chapter, in quantum mechanics the results of measurements are generally statistical, even when one has complete knowledge of the preparation procedure. A single preparation step and measurement step might not be sufficient to estimate a parameter well. Thus it is common to repeat the two steps of preparation and measurement on a large number of systems, either all at one time or sequentially. Whether measuring one quantum system or many, one is faced with a number of questions. How should one prepare the system state? What sort of measurement should one make on the system?
In the literature on quantum reference frames, the internal (relative) properties of a system are defined as those which are preserved under an arbitrary change of reference frame. For a system of quantum spins, these are all properties preserved by proper spatial rotations of the laboratory. However, this approach does not account for the hypothetical possibility of the laboratory becoming entangled to the system, as described by a second laboratory (the `Wigner's friend' scenario), in which case the relationship between the two laboratories is not a rotation, but is fundamentally quantum. To overcome this limitation, we re-define the reference frame transformations to be those that preserve the fidelities between subsystems. This enables us to derive U(2) as the correct symmetry group for transformations of a system of N spin-half particles. Next, we propose that systems having the same internal properties should be regarded as physically equivalent in the absence of an external frame. Remarkably, this implies that a single spin in a superposition relative to a spin magnet is equivalent to a macroscopic superposition of the magnet relative to the spin. We discuss the implications of this result for the Wigner's friend paradox.
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox “absolute” quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.