Holevo Information Research Articles

Quantum inception score

Motivated by the great success of classical generative models in machine learning, enthusiastic exploration of their quantum version has recently started. To depart on this journey, it is important to develop a relevant metric to evaluate the quality of quantum generative models; in the classical case, one such example is the (classical) inception score (cIS). In this paper, as a natural extension of cIS, we propose the quantum inception score (qIS) for quantum generators. Importantly, qIS relates the quality to the Holevo information of the quantum channel that classifies a given dataset. In this context, we show several properties of qIS. First, qIS is greater than or equal to the corresponding cIS, which is defined through projection measurements on the system output. Second, the difference between qIS and cIS arises from the presence of quantum coherence, as characterized by the resource theory of asymmetry. Third, when a set of entangled generators is prepared, there exists a classifying process leading to the further enhancement of qIS. Fourth, we harness the quantum fluctuation theorem to characterize the physical limitation of qIS. Finally, we apply qIS to assess the quality of the one-dimensional spin chain model as a quantum generative model, with the quantum convolutional neural network as a quantum classifier, for the phase classification problem in the quantum many-body physics. Published by the American Physical Society 2024

Relative Entropy of Coherence Quantifies Performance in Bayesian Metrology

The ability of quantum states to be in superposition is one of the key features that sets them apart from the classical world. This “coherence” is rigorously quantified by resource theories, which aim to understand how such properties may be exploited in quantum technologies. There has been much research on what the resource theory of coherence can reveal about quantum metrology, almost all of which has been from the viewpoint of Fisher information. We prove, however, that the relative entropy of coherence, and its recent generalization to positive operator-valued measures (POVMs), naturally quantify the performance of Bayesian metrology. In particular, we show how a coherence measure can be applied to an ensemble of states. We then prove that during parameter estimation, the ensemble relative entropy of coherence (C) is equal to the difference between the optimal Holevo information (X), and the mutual information attained by a measurement (I). We call this relation the CXI equality. The ensemble coherence lets us visualize how much information is locked away in superposition and hence is inaccessible with a given measurement scheme and quantifies the advantage that would be gained by using a joint measurement on multiple states. Our results hold regardless of how the parameter is encoded in the state, encompassing unitary, dissipative, and discrete settings. We consider both projective measurements and general POVMs. This work suggests new directions for research in coherence, provides a novel operation interpretation for the relative entropy of coherence and its POVM generalization, and introduces a new tool to study the role of quantum features in metrology. Published by the American Physical Society 2024

Quantum-optimal information encoding using noisy passive linear optics

The amount of information that a noisy channel can transmit has been one of the primary subjects of interest in information theory. In this work we consider a practically-motivated family of optical quantum channels that can be implemented without an external energy source. We optimize the Holevo information over procedures that encode information in attenuations and phase-shifts applied by these channels on a resource state of finite energy. It is shown that for any given input state and environment temperature, the maximum Holevo information can be achieved by an encoding procedure that uniformly distributes the channel's phase-shift parameter. Moreover for large families of input states, any maximizing encoding scheme has a finite number of channel attenuation values, simplifying the codewords to a finite number of rings around the origin in the output phase space. The above results and numerical evidence suggests that this property holds for all resource states. Our results are directly applicable to the quantum reading of an optical memory in the presence of environmental thermal noise.

Correlation measures of a quantum state and information characteristics of a quantum channel

We discuss the interconnections between basic correlation measures of a bipartite quantum state and basic information characteristics of a quantum channel, focusing on the benefits of these interconnections for solving specific problems concerning the characteristics of both types. We describe properties of the (unoptimized and optimized) quantum discord in infinite-dimensional bipartite systems. In particular, using the generalized Koashi–Winter relation, a simple condition is obtained that guarantees that a state with zero quantum discord is quantum-classical. Two possible definitions of the quantum discord for states with infinite one-way classical correlation are proposed and analysed. The generalized versions of Koashi–Winter and Xi–Lu–Wang–Li relations are used to obtain advanced continuity bounds for the Holevo information at the outputs of a channel and its complementary channel (as functions of a channel for a given ensemble of input states), for the Holevo capacity and the unregularized private capacity of a quantum channel depending either on the input dimension or on the input energy bound. We also discuss the properties of quantum channels which are “doppelgangers” of the monotonicity of the quantum discord and the entropy reduction of a local measurement under quantum channels acting on an unmeasured subsystem.

Phase Covariant Channel: Quantum Speed Limit of Evolution

AbstractThe quantum speed of evolution for the phase covariant map is investigated. This involves absorption, emission, and dephasing processes. The maps under various combinations of the above processes are considered to investigate the effect of phase covariant maps on quantum speed limit time. For absorption‐free phase covariant maps, combinations of dissipative and CP‐(in)divisible (non)‐Markovian dephasing noises are considered. The role of coherence‐mixedness balance on the speed limit time is checked in the presence of both vacuum and finite temperature effects. The rate at which Holevo's information changes and the action quantum speed of evolution for specific cases of the phase covariant map are also investigated.

Phase-transition-like behavior in information retrieval of a quantum scrambled random circuit system

Information in a chaotic quantum system will scramble across the system, preventing any local measurement from reconstructing it. The scrambling dynamics is key to understanding a wide range of quantum many-body systems. Here we use Holevo information to quantify the scrambling dynamics, which shows a phase-transition-like behavior. When applying long random Clifford circuits to a large system, no information can be recovered from a subsystem of less than half the system size. When exceeding half the system size, the amount of stored information grows by two bits of classical information per qubit until saturation through another sharp unanalytical change. We also study critical behavior near the transition points. Finally, we use coherent information to quantify the scrambling of quantum information in the system, which shows similar phase-transition-like behavior.

Irreducibly SU(2)-covariant quantum channels of low rank

We investigate information theoretic properties of low rank (less than or equal to 3) quantum channels with [Formula: see text]-symmetry, where we have a complete description. We prove that PPT property coincides with entanglement-breaking property and that degradability seldomly holds in this class. In connection with these results, we will demonstrate how we can compute Holevo and coherent information of those channels. In particular, we exhibit a strong form of additivity violation of coherent information, which resembles the superactivation of coherent information of depolarizing channels.

Quantum information scrambling in quantum many-body scarred systems

Quantum many-body scarred systems host special nonthermal eigenstates that support periodic revival dynamics and weakly break the ergodicity. Here, we study the quantum information scrambling dynamics in quantum many-body scarred systems, with a focus on the ``PXP'' model. We use the out-of-time-ordered correlator (OTOC) and Holevo information as measures of the information scrambling and apply an efficient numerical method based on matrix product operators to compute them up to 41 spins. We find that both the OTOC and Holevo information exhibit a linear light cone and periodic oscillations inside the light cone for initial states within the scarred subspace, which is in sharp contrast to thermal or many-body localized systems. The periodic revivals of OTOCs and Holevo information signify unusual breakdown of quantum chaos and are not equivalent to the revival dynamics of state fidelity or local observables studied in the previous literature. To explain the formation of the linear light-cone structure, we provide a perturbation-type calculation based on a phenomenological model. In addition, we demonstrate that the OTOC and Holevo information dynamics of the PXP model can be measured using the Rydberg-atom quantum simulators with current experimental technologies and numerically identify the measurable signatures using experimental parameters.

Holevo information and ensemble theory of gravity

Holevo information is an upper bound for the accessible classical information of an ensemble of quantum states. In this work, we use Holevo information to investigate the ensemble theory interpretation of quantum gravity. We study the Holevo information in random tensor network states, where the random parameters are the random tensors at each vertex. Based on the results in random tensor network models, we propose a conjecture on the holographic bulk formula of the Holevo information in the gravity case. As concrete examples of holographic systems, we compute the Holevo information in the ensemble of thermal states and thermo-field double states in the Sachdev-Ye-Kitaev model. The results are consistent with our conjecture.

Holevo information of black hole mesostates

We define a bulk wormhole geometry interpolating between horizons of differing size and determine characteristics of the HRT surface in these geometries. This construction is dual to black hole mesostates, an intermediate coarse-graining of states between black hole microstates and the full black hole state. We analyze the distinguishability of these objects using the recently derived holographic Holevo information techniques, demonstrating novel phase transition behavior for such systems.

Multifold behavior of the information transmission by the quantum 3-switch

We uncover new behaviors of the transmission of information by three quantum channels in superposition of causal orders subject to some level of noise. We find that the transmission can exhibit three different behaviors as the level of noise is varied. This multifold behavior can be explained by the different equivalence classes of quantum switch matrices related to specific combinations of causal orders. We classify these matrices using their characteristic polynomials and matrix invariants, and we calculate analytical expressions for the Holevo information in three representative cases. Our results are a step forward to understand and harness quantum control of causal orders with different levels of noise. We also study the Holevo information as function of a continuous order parameter and analyze transitions at integer values.

Entanglement spectrum of geometric states

The reduced density matrix of a given subsystem, denoted by ρA, contains the information on subregion duality in a holographic theory. We may extract the information by using the spectrum (eigenvalue) of the matrix, called entanglement spectrum in this paper. We evaluate the density of eigenstates, one-point and two-point correlation functions in the microcanonical ensemble state ρA,m associated with an eigenvalue λ for some examples, including a single interval and two intervals in vacuum state of 2D CFTs. We find there exists a microcanonical ensemble state with λ0 which can be seen as an approximate state of ρA. The parameter λ0 is obtained in the two examples. For a general geometric state, the approximate microcanonical ensemble state also exists. The parameter λ0 is associated with the entanglement entropy of A and Rényi entropy in the limit n → ∞. As an application of the above conclusion we reform the equality case of the Araki-Lieb inequality of the entanglement entropies of two intervals in vacuum state of 2D CFTs as conditions of Holevo information. We show the constraints on the eigenstates. Finally, we point out some unsolved problems and their significance on understanding the geometric states.

Shannon unicity distance, ε-secrecy criterion and Holevo information in quantum cryptography

In this paper, we consider the relationship between C Shannon’s unicity distance, which is determined by the number of symbols of the encrypted message required to uniquely find the encryption key of the cryptosystem in information-theoretical sense, ε-secrecy criterion in quantum cryptography and Holevo information. A lower bound for the unicity distance for ε-secret keys in quantum cryptography is obtained. The relationship between unicity distance, ε-secrecy criterion in quantum cryptography and Holevo information is demonstrated.

Correlations in geometric states

In this paper we explore the correlations in the geometric states. Here the geometric state means the state in CFTs that can be effectively described by classical geometry in the bulk in the semi-classical limit G → 0. By using the upper bound of Holevo information we show the convex combination of geometric states cannot be a geometric state. To understand the duality between thermofield double state and eternal black hle, we construct several correlated states of two CFTs. In all the examples we show their correlations are too weak to produce the a connected spacetime. Then we review the measure named quantum discord and use it to characterize the classical and quantum correlations in quantum field theories. Finally, we discuss the correlations between two intervals A and B with distance d in the vacuum state of 2D CFTs with large central charge c. The feature is the phase transition of the mutual information I (ρAB). We analyse the quasi-product state of ρAB for large d. By using the Koashi-Winter relation of tripartite states the quantum and classical correlations between A and B can expressed as Holevo information, which provides a new understanding of the correlations as accessible information.

Discrete Weyl Channels With Markovian Memory

We discuss discrete Weyl channels (DWCs) with correlated noise in consecutive uses. Some special cases of DWCs have been shown to have the so-called transition behavior, where the optimal signal states, which maximize the Holevo information between the transmitter and the receiver, sharply transition from product states to maximally entangled states as the degree of memory between the consecutive channel uses increases from a threshold value. We first show the general existence of this transition behavior for a class of DWCs where the product states are optimal for the memoryless case. We also provide two methods to estimate the degree of memory exhibited by the correlated noise. These estimators rely on generating classical data by measuring the channel output which shows the same memory characteristics as the consecutive uses of the channel. We specifically consider the Markovian noise but one of our proposed estimators can be straightforwardly generalized to an arbitrary type of noise as it provides the direct access to the sequence in which noise affects the consecutive input states.

Upper Bounds for the Holevo Information Quantity and Their Use

We present a family of easily computable upper bounds for the Holevo (information) quantity of an ensemble of quantum states depending on a reference state (as a free parameter). These upper bounds are obtained by combining probabilistic and metric characteristics of the ensemble. We show that an appropriate choice of the reference state gives tight upper bounds for the Holevo quantity which in many cases improve the estimates existing in the literature. We also present an upper bound for the Holevo quantity of a generalized ensemble of quantum states with finite average energy depending on the metric divergence of an ensemble. In the case of a multi-mode quantum oscillator, this upper bound is tight for large energy. Upper bounds for the Holevo capacity of finite-dimensional quantum channels depending on metric characteristics of the channel output are obtained.

Distinguishing Black Hole Microstates using Holevo Information.

We use Holevo information in the two-dimensional conformal field theory (CFT) with a large central charge c to distinguish microstates from the underlying thermal state. Holographically, the CFT microstates of a thermal state are dual to black hole microstate geometries in anti-de Sitter space. It was found recently that holographic Holevo information shows plateau behaviors at both short and long interval regions. This indicates that the black hole microstates are indistinguishable from the thermal state by measuring over a small region, and perfectly distinguishable over a region with its size comparable to the whole system. In this Letter, we demonstrate that the plateaus are lifted by including the 1/c corrections from both the vacuum and nonvacuum conformal families of CFT in either the canonical ensemble or microcanonical ensemble thermal state. Our results imply that the aforementioned indistinguishability and distinguishability of black hole microstate geometries from the underlying black hole are spoiled by higher order Newton constant G_{N} corrections of quantum gravity.

Entropic Heisenberg limits and uncertainty relations from the Holevo information bound

Strong and general entropic and geometric Heisenberg limits are obtained, for estimates of multiparameter unitary displacements in quantum metrology, such as the estimation of a magnetic field from the induced rotation of a probe state in three dimensions. A key ingredient is the Holevo bound on the Shannon mutual information of a quantum communication channel. This leads to a Bayesian bound on performance, in terms of the prior distribution of the displacement and the asymmetry of the input probe state with respect to the displacement group. A geometric measure of performance related to entropy is proposed for general parameter estimation. It is also shown how strong entropic uncertainty relations for mutually unbiased observables, such as number and phase, position and momentum, energy and time, and orthogonal spin-1/2 directions, can be obtained from elementary applications of Holevo’s bound. A geometric interpretation of results is emphasised, in terms of the ‘volumes’ of quantum and classical statistical ensembles.

Complete elimination of information leakage in continuous-variable quantum communication channels

In all lossy communication channels realized to date, information is inevitably leaked to a potential eavesdropper. Here we present a communication protocol that does not allow for any information leakage to a potential eavesdropper in a purely lossy channel. By encoding information into a restricted Gaussian alphabet of squeezed states we show, both theoretically and experimentally, that the Holevo information between the eavesdropper and the intended recipient can be exactly zero in a purely lossy channel while minimized in a noisy channel. This result is of fundamental interest, but might also have practical implications in extending the distance of secure quantum key distribution.

Corrections in the relative entropy of black hole microstates

Inspired by the recent work of Bao and Ooguri (BO), we study the distinguishability of the black hole microstates from the thermal state as captured by the average of their relative entropies: the Holevo information. Under the assumption that the vacuum conformal block dominates the entropy calculation, BO find that the average relative entropy vanishes on spatial regions smaller than half the size of the CFT. However, vacuum block dominance fails for some microstates of the M = 0 BTZ black hole. We show that this renders the average relative entropy nonzero even on infinitesimal intervals at O(c0).