According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose-Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note presents examples of infinite-particle states with Bose-Einstein symmetry that arise as limits of Gibbs ensembles on finite dimensional spaces, and displays their de Finetti representations. We consider Gibbs ensembles for systems of bosons in a finite dimensional setting and discover limits as the number of particles tends to infinity, provided the temperature is scaled in proportion to particle number.
The quantum de Finetti theorem asserts that the k-body density matrices of a N-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed. In this note we review a construction due to Christandl, Mitchison, König and Renner valid for finite dimensional Hilbert spaces, which gives a quantitative version of the theorem. We first propose a variant of their proof that leads to a slightly improved estimate. Next we provide an alternative proof of an explicit formula due to Chiribella, which gives the density matrices of the constructed state …
The quantum versions of de Finetti's theorem derived so far express the convergence of n-partite symmetric states, i.e., states that are invariant under permutations of their n parties, towards probabilistic mixtures of independent and identically distributed (i.i.d.) states. Unfortunately, these theorems only hold in finite-dimensional Hilbert spaces, and their direct generalization to infinite-dimensional Hilbert spaces is known to fail. Here, we address this problem by considering invariance under orthogonal transformations in phase space instead of permutations in state space, which leads …
We prove a de Finetti theorem for exchangeable sequences of states on test spaces, where a test space is a generalization of the sample space of classical probability theory and the Hilbert space of quantum theory. The standard classical and quantum de Finetti theorems are obtained as special cases. By working in a test space framework, the common features that are responsible for the existence of these theorems are elucidated. In addition, the test space framework is general enough to imply a de Finetti theorem for classical processes. We conclude by discussing the ways in which our assumptio…
We prove the security of Gaussian continuous-variable quantum key distribution against arbitrary attacks in the finite-size regime. The novelty of our proof is to consider symmetries of quantum key distribution in phase space in order to show that, to good approximation, the Hilbert space of interest can be considered to be finite-dimensional, thereby allowing for the use of the postselection technique introduced by Christandl, Koenig and Renner (Phys. Rev. Lett. 102, 020504 (2009)). Our result greatly improves on previous work based on the de Finetti theorem which could not provide security f…
A universal entangler (UE) is a unitary operation which maps all pure product states to entangled states. It is known that for a bipartite system of particles $1,2$ with a Hilbert space $\mathbb{C}^{d_1}\otimes\mathbb{C}^{d_2}$, a UE exists when $\min{(d_1,d_2)}\geq 3$ and $(d_1,d_2)\neq (3,3)$. It is also known that whenever a UE exists, almost all unitaries are UEs; however to verify whether a given unitary is a UE is very difficult since solving a quadratic system of equations is NP-hard in general. This work examines the existence and construction of UEs of bipartite bosonic/fermionic syst…
We develop a framework to systematically investigate the influence of many-particle interference on the dynamics of generic $-$ possibly interacting $-$ bosonic systems. We consider mixtures of bosons which belong to several distinguishable species, allowing us to tune the level of many-particle interference, and identify the corresponding signatures in the time-dependent expectation values of observables. Interference contributions to these expectation values can be classified based on the number of interfering particles. Interactions are shown to generate a series of additional, higher-order…
We prove an entanglement area law for a class of 1D quantum systems involving infinite-dimensional local Hilbert spaces. This class of quantum systems include bosonic models such as the Hubbard-Holstein model, and both U(1) and SU(2) lattice gauge theories in one spatial dimension. Our proof relies on new results concerning the robustness of the ground state and spectral gap to the truncation of Hilbert space, applied within the approximate ground state projector (AGSP) framework from previous work. In establishing this area law, we develop a system-size independent bound on the expectation va…
We introduce a novel approach to solve optimization problems on a boson sampling device assisted by classical machine-learning techniques. By virtue of the parity function, we map all measurement patterns, which label the basis spanning an $M$-mode bosonic Hilbert space, to the Hilbert space of $M$ qubits. As a result, the sampled probability function can be interpreted as a result of sampling a multiqubit circuit. The method is presented on several instances of a QUBO/Ising problem as well as portfolio optimization problems. Among many demonstrated properties of the parity function is the abi…
We study Hilbert-space localization of the many-body dynamics due to ergodicity breaking and analyze this effect in terms of the entanglement entropy and the entanglement spectrum. We find a transition from a regime driven by quantum tunneling to a regime that is dominated by boson-boson interaction, where the latter exhibits ergodicity breaking. Properties of this transition are captured by observation time averaging, which effectively suppresses the large dynamical entanglement fluctuations near the critical point. We employ this approach to the experimentally available Bosonic Josephson Jun…
Controllability -- the possibility of performing any target dynamics by applying a set of available operations -- is a fundamental requirement for the practical use of any physical system. For finite-dimensional systems, as for instance spin systems, precise criterions to establish controllability, such as the so called rank criterion, are well known. However most physical systems require a description in terms of an infinite-dimensional Hilbert space whose controllability properties are poorly understood. Here, we investigate infinite-dimensional bosonic quantum systems -- encompassing quantu…
ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology（ICFO-光子科学研究所，巴塞罗那科学与技术研究院）；Universidad de Buenos Aires, Facultad de Ciencias Exactas y Naturales（布宜诺斯艾利斯大学，理学院）；CONICET - Universidad de Buenos Aires, Instituto de Física de Buenos Aires (IFIBA)（阿根廷国家科学研究委员会-布宜诺斯艾利斯大学，布宜诺斯艾利斯物理研究所 (IFIBA)）
We discuss quantum correlations in systems of indistinguishable particles in relation to entanglement in composite quantum systems consisting of well separated subsystems. Our studies are motivated by recent experiments and theoretical investigations on quantum dots and neutral atoms in microtraps as tools for quantum information processing. We present analogies between distinguishable particles, bosons and fermions in low-dimensional Hilbert spaces. We introduce the notion of Slater rank for pure states of pairs of fermions and bosons in analogy to the Schmidt rank for pairs of distinguishabl…
The gauge invariant observables of the closed bosonic string are quantized without anomalies in four space-time dimensions by constructing their quantum algebra in a manifestly covariant approach. The quantum algebra is the kernel of a derivation on the universal envelopping algebra of an infinite-dimensional Lie algebra. The search for Hilbert space representations of this algebra is separated from its construction, and postponed.
The so-called equation of motion method is useful to obtain the explicit form of the eigenvectors and eigenvalues of certain non self-adjoint bosonic Hamiltonians with real eigenvalues. These operators can be diagonalized when they are expressed in terms of pseudo-bosons, which do not behave as ordinary bosons under the adjoint transformation, but obey the Weil-Heisenberg commutation relations.
We discuss the partitioning of the Hilbert space of a quantum system induced by the interaction with another system at thermal equilibrium, showing that the higher the temperature the more effective is the formation of Zeno subspaces. We show that our analysis keeps its validity even in the case of interaction with a bosonic reservoir, provided appropriate limitations of the relevant bandwidth.
We use Bub's (2016) correlation arrays and Pitowksy's (1989b) correlation polytopes to analyze an experimental setup due to Mermin (1981) for measurements on the singlet state of a pair of spin-$\frac12$ particles. The class of correlations allowed by quantum mechanics in this setup is represented by an elliptope inscribed in a non-signaling cube. The class of correlations allowed by local hidden-variable theories is represented by a tetrahedron inscribed in this elliptope. We extend this analysis to pairs of particles of arbitrary spin. The class of correlations allowed by quantum mechanics i…
Efficiently Computable Strategies and Limits for Bosonic Channel Discrimination
School of Science, STEM College, RMIT University（斯威本科技大学理学院STEM学院）；Scuola Normale Superiore（比萨高等师范学院）；Korteweg–de Vries Institute for Mathematics, University of Amsterdam（阿姆斯特丹大学科特韦格-德弗里斯数学研究所）；Mathematical Quantum Information RIKEN Hakubi Research Team, RIKEN Pioneering Research Institute (PRI) and RIKEN Center for Quantum Computing (RQC)（理科院系量子信息RIKEN Hakubi研究团队，理化学研究所前沿研究院（PRI）和理化学研究所量子计算中心（RQC））；School of Electrical and Computer Engineering, Cornell University（康奈尔大学电气与计算机工程学院）
We present explicit evaluations of quantum speed limit times pertinent to the Markovian dynamics of an open continuous-variable system. Specifically, we consider the standard setting of a cavity mode of the quantum radiation field weakly coupled to a thermal bosonic reservoir. The evolution of the field state is ruled by the quantum optical master equation, which is known to have an exact analytic solution. Starting from a pure input state, we employ two indicators of how different the initial and evolved states are, namely, the fidelity of evolution and the Hilbert-Schmidt distance of evoluti…
Quantum simulation of quantum field theory is a flagship application of quantum computers that promises to deliver capabilities beyond classical computing. The realization of quantum advantage will require methods to accurately predict error scaling as a function of the resolution and parameters of the model that can be implemented efficiently on quantum hardware. In this paper, we address the representation of lattice bosonic fields in a discretized field amplitude basis, develop methods to predict error scaling, and present efficient qubit implementation strategies. A low-energy subspace of …
Quantum processors enable computational speedups for machine learning through parallel manipulation of high-dimensional vectors. Early demonstrations of quantum machine learning have focused on processing information with qubits. In such systems, a larger computational space is provided by the collective space of multiple physical qubits. Alternatively, we can encode and process information in the infinite-dimensional Hilbert space of bosonic systems such as quantum harmonic oscillators. This approach offers a hardware-efficient solution with potential quantum speedups to practical machine lea…
In this paper we analyze two different functional formulations of classical mechanics. In the first one the Jacobi fields are represented by bosonic variables and belong to the vector (or its dual) representation of the symplectic group. In the second formulation the Jacobi fields are given as condensates of Grassmannian variables belonging to the spinor representation of the metaplectic group. For both formulations we shall show that, differently from what happens in the case presented in paper no. (I), it is possible to endow the associated Hilbert space with a positive definite scalar produ…
We study a non-autonomous, non-linear evolution equation on the space of operators on a complex Hilbert space. We specify assumptions that ensure the global existence of its solutions and allow us to derive its asymptotics at temporal infinity. We demonstrate that these assumptions are optimal in a suitable sense and more general than those used before. The evolution equation derives from the Brocket-Wegner flow that was proposed to diagonalize matrices and operators by a strongly continuous unitary flow. In fact, the solution of the non-linear flow equation leads to a diagonalization of Hamil…
Quantum simulations of bosonic field theories require a truncation in field space to map the theory onto finite quantum registers. Ideally, the truncated theory preserves the symmetries of the original model and has a critical point in the same universality class. In this paper, we explore two different truncations that preserve the symmetries of the 1+1-dimensional $O(3)$ non-linear $σ$-model - one that truncates the Hilbert space for the unit sphere by setting an angular momentum cutoff and a fuzzy sphere truncation inspired by non-commutative geometry. We compare the spectrum of the truncat…
Finding pure states in an enlarged Hilbert space that encode the mixed state of a quantum field theory as a partial trace is necessarily a challenging task. Nevertheless, such purifications play the key role in characterizing quantum information-theoretic properties of mixed states via entanglement and complexity of purifications. In this article, we analyze these quantities for two intervals in the vacuum of free bosonic and Ising conformal field theories using, for the first time, the~most general Gaussian purifications. We provide a comprehensive comparison with existing results and identif…
Spacetime quantum mechanics for bosonic and fermionic systems
Center for Nonlinear Studies (CNLS), Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究中心）；Information Sciences, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室信息科学部）；Departamento de Física-IFLP/CONICET, Universidad Nacional de La Plata（拉普拉塔国立大学物理系/IFLP/阿根廷国家科学研究委员会）；Comisión de Investigaciones Científicas (CIC)（科学调查委员会）
Quantum particle statistics fundamentally controls the way particles interact, and plays an essential role in determining the properties of the system at low temperature. Here we study how the quantum statistics affects the computational power of quantum annealing. We propose an annealing Hamiltonian describing quantum particles moving on a square lattice and compare the computational performance of the atomic quantum annealers between two statistically-different components: spinless fermions and hard-core bosons. In addition, we take an Ising quantum annealer driven by traditional transverse-…
Learning and simulating bosonic systems via finite-energy locality
Technical University of Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Univ. Grenoble Alpes, CNRS, Grenoble INP, LIG（格勒诺布尔阿尔卑斯大学、法国国家科学研究中心、格勒诺布尔理工学院、信息学实验室）；University of Warwick（华威大学）；University of Copenhagen（哥本哈根大学）
Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences（奥地利科学院量子光学与量子信息研究所）；Institute for Experimental Physics, University of Innsbruck（因斯布鲁克大学实验物理研究所）；Atominstitut, TU Wien（维也纳技术大学原子研究所）；PSI Center for Photon Science（保罗·谢勒研究所光子科学中心）；Qruise GmbH（Qruise有限公司）；Dipartimento di Fisica e Astronomia “G. Galilei”, Università degli Studi di Padova（帕多瓦大学物理与天文系（G.伽利略））；INFN Istituto Nazionale di Fisica Nucleare, Sezione di Padova（意大利国家核物理研究所帕多瓦分所）；University of Innsbruck（因斯布鲁克大学）
Non-Hermiticity greatly expands existing physical laws beyond the Hermitian framework, revealing various novel phenomena with unique properties. Up to now, most exotic nonHermitian effects, such as exceptional points and non-Hermitian skin effects, are discovered in single-particle systems. The interplay between non-Hermitian and manybody correlation is expected to be a more fascinating but much less explored area. Due to the complexity of the problem, current researches in this field mainly stay at the theoretical level. The experimental observation of predicted non-Hermitian manybody phases …
Bosonic quantum field theories, even when regularized using a finite lattice, possess an infinite dimensional Hilbert space and, therefore, cannot be simulated in quantum computers with a finite number of qubits. A truncation of the Hilbert space is then needed and the physical results are obtained after a double limit: one to remove the truncation and another to remove the regulator (the continuum limit). A simpler alternative is to find a model with a finite dimensional Hilbert space belonging to the same universality class as the continuum model (a "qubitization"), so only the space continu…
An infinite dimensional system such as a quantum harmonic oscillator offers a potentially unbounded Hilbert space for computation, but accessing and manipulating the entire state space requires a physically unrealistic amount of energy. When such a quantum harmonic oscillator is coupled to a qubit, for example via a Jaynes-Cummings interaction, it is well known that the total Hilbert space can be separated into independently accessible subspaces of constant energy, but the number of subspaces is still infinite. Nevertheless, a closed four-dimensional Hilbert space can be analytically construct…
Estimating truncation effects of quantum bosonic systems using sampling algorithms
School of Mathematical Sciences, Queen Mary University of London（伦敦玛丽女王大学数学科学学院）；qBraid Co.（qBraid公司）；Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普利兹克分子工程学院）；Chicago Quantum Exchange（芝加哥量子交易所）；Kadanoff Center for Theoretical Physics, The University of Chicago（芝加哥大学卡达诺夫理论物理中心）；Quantinuum K.K.（Quantinuum株式会社）；Physics Department, University of Michigan（密歇根大学物理系）；Theoretical Quantum Physics Laboratory, RIKEN（理化学研究所理论量子物理实验室）；Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) Program, RIKEN（理化学研究所跨学科理论与数学科学项目）；Department of Physics, Kyoto University（京都大学物理系）
Topological phases of bosons with local parity coupling on a dimerized lattice
Univ. Toulouse, CNRS, Laboratoire de Physique Théorique（图卢兹大学，法国国家科学研究中心，理论物理实验室）；International Centre for Theoretical Sciences, Tata Institute of Fundamental Research（国际理论科学中心，塔塔基础研究所）
