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arXiv:1712.09365 · Physical review. D/Physical review. D.

Quantum information measures for restricted sets of observables

Sudip Ghosh, Suvrat Raju

Published Aug 8, 2018• 29 citations
arXivHTMLPDF

Abstract

We study measures of quantum information when the space spanned by the set of accessible observables is not closed under products, i.e., we consider systems where an observer may be able to measure the expectation values of two operators, $⟨{O}_{1}⟩$ and $⟨{O}_{2}⟩$, but may not have access to $⟨{O}_{1}{O}_{2}⟩$. This problem is relevant for the study of localized quantum information in gravity since the set of approximately local operators in a region may not be closed under arbitrary products. While we cannot naturally associate a density matrix with a state in this setting, it is still possible to define a modular operator for a state, and distinguish between two states using a relative modular operator. These operators are defined on a ``little Hilbert space,'' which parametrizes small deformations of the system away from its original state, and they do not depend on the structure of the full Hilbert space of the theory. We extract a class of relative-entropy-like quantities from the spectrum of these operators that measure the distance between states, are monotonic under contractions of the set of available observables, and vanish only when the states are equal. Consequently, these distance measures can be used to define measures of bipartite and multipartite entanglement. We describe applications of our measures to ``coarse-grained'' and ``fine-grained'' subregion dualities in $\mathrm{AdS}/\mathrm{CFT}$ and provide a few sample calculations to illustrate our formalism.

DOI
10.1103/physrevd.98.046005

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We study measures of quantum information when the space spanned by the set of accessible observables is not closed under products, i.e., we consider systems where an observer may be able to measure the expectation values of two operators, ⟨O1⟩⟨{O}_{1}⟩⟨O1​⟩ and ⟨O2⟩⟨{O}_{2}⟩⟨O2​⟩, but may not have access to ⟨O1O2⟩⟨{O}_{1}{O}_{2}⟩⟨O1​O2​⟩. This problem is relevant for the study of localized quantum information in gravity since the set of approximately local operators in a region may not be closed under arbitrary products. While we cannot naturally associate a density matrix with a state in this setting, it is still possible to define a modular operator for a state, and distinguish between two states using a relative modular operator. These operators are defined on a little Hilbert space,'' which parametrizes small deformations of the system away from its original state, and they do not depend on the structure of the full Hilbert space of the theory. We extract a class of relative-entropy-like quantities from the spectrum of these operators that measure the distance between states, are monotonic under contractions of the set of available observables, and vanish only when the states are equal. Consequently, these distance measures can be used to define measures of bipartite and multipartite entanglement. We describe applications of our measures to coarse-grained'' and ``fine-grained'' subregion dualities in AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT and provide a few sample calculations to illustrate our formalism.
We present a measure of quantum correlation of an ensemble of multiparty states. It is based on the idea of minimal entropy production in a measurement whose elements are locally distinguishable. It is shown, for pure ensembles, to be a relative entropy distance from a set of ensembles. For pure bipartite ensembles, which span the whole bipartite Hilbert space, the measure is bounded below by the average entanglement. We naturally obtain a monotonicity axiom for any measure of quantum correlation of ensembles for single systems. We evaluate the measure for certain cases. Subsequently we use this measure to propose a complementarity relation between our measure and the accessible information obtainable about the ensemble under local operations. The measure is well defined even for the case of a single system, where the complementarity relation is seen to be yet another face of the ``Heisenberg uncertainty relation.''
Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity.
We introduce a formal notion of observer aperture, representing the measurement interfaceavailable to an observer interacting with a quantum system. Given a restricted family of POVMsM, we define the aperture-restricted distinguishability between two quantum states as thesupremum of classical relative entropy between the induced outcome distributions. This quantitycan be viewed as a measurement-restricted variant of measured quantum relative entropy. Weprove monotonicity under aperture expansion and establish a data-processing inequality underquantum channels via Heisenberg pullback of measurement families. We further show thatthe quantity is subadditive across independent systems. The resulting framework provides anoperational interpretation of observer bandwidth connecting measurement constraints, coarse-graining, and accessible information within standard quantum information theory.
We study quantum coarse-grained entropy and demonstrate that the gap in entropy between local and global coarse-grainings is a natural generalization of entanglement entropy to mixed states and multipartite systems. This ``quantum correlation entropy'' Sqc{S}^{q}cSqc is additive over independent systems, is invariant under local unitary operations, measures total nonclassical correlations (vanishing on states with strictly classical correlation), and reduces to the entanglement entropy for bipartite pure states. It quantifies how well a quantum system can be understood via local measurements and ties directly to nonequilibrium thermodynamics, including representing a lower bound on the quantum part of thermodynamic entropy production. We discuss two other measures of nonclassical correlation to which this entropy is equivalent and argue that together they provide a unique thermodynamically distinguished measure.
In this thesis we apply techniques from quantum information theory to study quantum gravity within the framework of the anti-de Sitter / conformal field theory correspondence (AdS/CFT). Through AdS/CFT, progress has been made in understanding the structure of entanglement in quantum field theories, and in how gravitational physics can emerge from these structures. However, this understanding is far from complete and will require the development of new tools to quantify correlations in CFT. This thesis presents refinements of a duality between operator product expansion (OPE) blocks in the CFT, giving the contribution of a conformal family to the OPE, and geodesic integrated fields in AdS which are diffeomorphism invariant quantities. This duality was originally discovered in the maximally symmetric setting of pure AdS dual to the CFT ground state. In less symmetric states the duality must be modified. Working with excited states within AdS3_33​/CFT2_22​, this thesis shows how the OPE block decomposes into more fine-grained CFT observables that are dual to AdS fields integrated over non-minimal geodesics. Additionally, this thesis contains results on the dynamics of entanglement measures for general quantum systems. Results are presented for the family of quantum Rényi entropies and entanglement negativity. Rényi entropies are studied for general dynamics by imposing special initial conditions. Around pure, separable initial states, all Rényi entropies grow with the same timescale at leading, and next-to-leading order. Mathematical tools are developed for the differentiation of non-analytic matrix functions with respect to constrained arguments and are used to construct analytic expressions for derivatives of negativity. We establish bounds on the rate of change of state distinguishability and the rate of entanglement growth for closed systems. Note: Abstract shortened.
The idea that information can be localized is pervasive in physics. In thermodynamics, entropy flows from one reservoir to another, and in quantum gravity, the entropy content of some spatial regions is bounded. In quantum information theory, information is transmitted from one place to another, and entanglement can provide an advantage when there are constraints on such transmission. Shannon entropy quantifies localizable information in the sense that marginalization defines an outer measure on the set of parts of a classical multipartite system. In contrast, von Neumann entropy does not quantify localizable information in this sense. However, a variant of von Neumann entropy, which originates in noncommutative geometry, does. We prove that this adjusted von Neumann entropy is the minimum quantum entropy that quantifies localizable information. We also prove that this is the unique quantum entropy that characterizes Bell states in terms of redundant information, directly generalizing the classical case. We work with finite-dimensional C∗C^*C∗-algebras throughout, modeling finite quantum systems that may have superselection sectors.