We review the framework of operator Hilbert space and introduce the one-and two-particle super reduced density matrices (1-SRDMs and 2-SRDMs), as well as the super mutual information (SMI). The eigenvectors of the 1-SRDMs define what we term natural single particle operator bases, and provide a way to compress vibrational and vibronic Hamiltonians with controlled error. The SMI is defined from the operator entanglement entropy of the 1-SRDMs and 2-SRDMs, and captures the correlation between operators acting on different one-mode subspaces, which may be used to reveal and quantify both direct and indirect couplings that might otherwise be difficult to extract. Efficient numerical approaches for the calculation of SRDMs and the SMI are developed and applied to a set of prototypical vibrational and vibronic Hamiltonians, as well as approximations to the corresponding time-evolution operators. Through this, we demonstrate that: (i) commonly used vibronic Hamiltonians are amenable to extremely high levels of compression without compromising accuracy, and; (ii) SMI analysis can be used to systematically and quantitatively reveal both direct and indirect couplings that might otherwise be difficult to extract, including indirect couplings of vibrational modes via intermediary electronic-vibrational interactions.
We review the framework of operator Hilbert space and introduce the one-and two-particle super reduced density matrices (1-SRDMs and 2-SRDMs), as well as the super mutual information (SMI). The eigenvectors of the 1-SRDMs define what we term natural single particle operator bases, and provide a way to compress vibrational and vibronic Hamiltonians with controlled error. The SMI is defined from the operator entanglement entropy of the 1-SRDMs and 2-SRDMs, and captures the correlation between operators acting on different one-mode subspaces, which may be used to reveal and quantify both direct and indirect couplings that might otherwise be difficult to extract. Efficient numerical approaches for the calculation of SRDMs and the SMI are developed and applied to a set of prototypical vibrational and vibronic Hamiltonians, as well as approximations to the corresponding time-evolution operators. Through this, we demonstrate that: (i) commonly used vibronic Hamiltonians are amenable to extremely high levels of compression without compromising accuracy, and; (ii) SMI analysis can be used to systematically and quantitatively reveal both direct and indirect couplings that might otherwise be difficult to extract, including indirect couplings of vibrational modes via intermediary electronic-vibrational interactions.
We review the framework of operator Hilbert space and introduce the one-and two-particle super reduced density matrices (1-SRDMs and 2-SRDMs), as well as the super mutual information (SMI). The eigenvectors of the 1-SRDMs define what we term natural single particle operator bases, and provide a way to compress vibrational and vibronic Hamiltonians with controlled error. The SMI is defined from the operator entanglement entropy of the 1-SRDMs and 2-SRDMs, and captures the correlation between operators acting on different one-mode subspaces, which may be used to reveal and quantify both direct and indirect couplings that might otherwise be difficult to extract. Efficient numerical approaches for the calculation of SRDMs and the SMI are developed and applied to a set of prototypical vibrational and vibronic Hamiltonians, as well as approximations to the corresponding time-evolution operators. Through this, we demonstrate that: (i) commonly used vibronic Hamiltonians are amenable to extremely high levels of compression without compromising accuracy, and; (ii) SMI analysis can be used to systematically and quantitatively reveal both direct and indirect couplings that might otherwise be difficult to extract, including indirect couplings of vibrational modes via intermediary electronic-vibrational interactions.
We review the framework of operator Hilbert space and introduce the one- and two-particle super reduced density matrices (1-SRDMs and 2-SRDMs), as well as the super mutual information (SMI). The eigenvectors of the 1-SRDMs define what we term natural single particle operator bases, and provide a way to compress vibrational and vibronic Hamiltonians with controlled error. The SMI is defined from the operator entanglement entropy of the 1-SRDMs and 2-SRDMs, and captures the correlation between operators acting on different one-mode subspaces, which may be used to reveal and quantify both direct and indirect couplings that might otherwise be difficult to extract. Efficient numerical approaches for the calculation of SRDMs and the SMI are developed and applied to a set of prototypical vibrational and vibronic Hamiltonians, as well as approximations to the corresponding time-evolution operators. Through this, we demonstrate that: (i) commonly used vibronic Hamiltonians are amenable to extremely high levels of compression without compromising accuracy, and; (ii) SMI analysis can be used to systematically and quantitatively reveal both direct and indirect couplings that might otherwise be difficult to extract, including indirect couplings of vibrational modes via intermediary electronic-vibrational interactions.
Entanglement is the fundamental difference between classical and quantum systems and has become one of the guiding principles in the exploration of high-and low-energy physics.The calculation of entanglement entropies in interacting quantum field theories, however, remains challenging.Here, we present the first method for the explicit computation of reduced density matrices of interacting quantum field theories using truncated Hamiltonian methods.The method is based on constructing an isomorphism between the Hilbert space of the full system and the tensor product of Hilbert spaces of subintervals.This naturally enables the computation of the von Neumann and arbitrary Rényi entanglement entropies as well as mutual information, logarithmic negativity, and other measures of entanglement.Our method is applicable to equilibrium states and nonequilibrium evolution in real time.It is model independent and can be applied to any Hamiltonian truncation method that uses a free basis expansion.We benchmark the method on the free Klein-Gordon theory finding excellent agreement with the analytic results.We further demonstrate its potential on the interacting sine-Gordon model, studying the scaling of von Neumann entropy in ground states and real-time dynamics following quenches of the model.
The generation and exploitation of entangled states has become one of the most important technological challenges of the twenty-first century. In order to reliably generate multi-particle entangled states, one needs controllable two (or more) qubit interactions, and there have been many proposals for suitable systems. We have recently begun investigating adiabatic transfer protocols (Greentree et al. 2004), which are distinguished from conventional, nonadiabatic approaches by their robustness and relative insensitivity to noise. Focussing on the phosphorus in silicon paradigm we have shown how such transport leads to new quasi-2D architectures (Hollenberg et al. 2006), and surprising particle coupling schemes based on operator measurements (Devitt et al. 2005; Greentree et al. 2006), which in turn may lead to efficient mechanisms for syndrome identification in quantum error correction. Here we describe the realisation of these operator measurements more fully based on adiabatic passage of a control particle, highlighting the applicability of our protocols by showing how they can improve the scalability of other implementations. Introduction The pursuit of devices which can exploit quantum coherence is widely seen as one of the most important and challenging tasks facing twenty-first Century Technology. Defined by the use of coherent quantum systems (at their simplest: qubits, two-state systems) such devices enable a range of tasks, from quantum key distribution via single qubits (Bennett and Brassard 1984) or entangled states (Ekert 1991), to protocols for quantum teleportation (Bennett et al. 1993), efficient quantum simulation (Feynmann 1982; Abrams and Lloyd 1997) and factorisation (Shor 1994). At the heart of most such protocols is the exploitation of entanglement, where qubits that have been entangled, in some sense feel an interaction even if space-like separated. In fact, entanglement is a ubiquitous feature of quantum systems: the task for quantum information devices is to generate controlled and isolated entanglement, so that only the desired qubits are entangled with each other, and deleterious entanglement to the environment avoided. Broadly speaking, entanglement can be generated in one of two ways, either by direct interactions between qubits, or by collective measurement of two (or more) particles in particular bases that are, by their very nature, entangled. Such projective measurements are called operator measurements, and a flexible mechanism to realise them forms the basis of this paper. Operator measurements are intimately linked to the concept of quantum error correction (QEC) (Nielsen and Chuang 2000), and as QEC is an extremely important task for quantum computing, it is possible that our schemes may prove extremely useful in future quantum computers (QC). Operator measurements are also compatible with the concept of ‘repeat-until-success’ quantum computing (Lim et al. 2005; Barrett and Kok 2005; Lim et al. 2006) which has potential advantages for QC in realistic settings. Our method for generating operator measurements is based on protocols to transfer particles adiabatically around a quantum network. There are now several different protocols for such spatial adiabatic passage, which are all extensions of the original STIRAP (STImulated Raman Adiabatic Passage) (Vitanov et al. 2001) protocol. Spatial adiabatic passage protocols can be mediated via electromagnetic pulses (Brandes et al. 2001; Siewert et al. 2006), or direct control of wavefunction overlap (Greentree et al. 2004; Eckert et al. 2004; Petrosyan 2006). All such schemes share a common theme, namely that a particular eigenstate, which is in the null space of the Hamiltonian, can be smoothly evolved as a function of time. Such protocols are therefore extremely robust, requiring little regard for timing provided that the adiabaticity criterion (Vitanov et al. 2001) is respected. For concreteness, we will review the MRAP protocol (Multiple Recipient Adiabatic Passage) introduced by us earlier (Devitt et al. 2005; Greentree et al. 2006), and show explicitly how it can be used to mediate operator measurements. We will then comment on some of the implications of this protocol for QC designs, especially when concepts such as defect-tolerance are introduced. Multiple-Recipient Adiabatic Passage Consider a network, defined by four quantum dots that share a single electron, and two attached qubits, as depicted in Fig. 1(a). Employing quantum communication terminology, we suppose that Alice wishes to send a particle (with information encoded in the electronic spin states) to either Bob 1 or Bob 2, or to some arbitrary superposition of the Bobs, Australian Institute of Physics 17th National Congress 2006 – Brisbane, 3-8 December 2006 RiverPhys Paper No. 474 using the network. For clarity we refer to the particle transferred through the network as the bus particle. We label the quantum dots, A (Alice), 1 B (Bob 1), 2 B (Bob 2), C (Central). Also, the Bobs have qubits 1 Q and 2 Q which will be used later. We further assume that each participant has control of the coupling of their quantum dot to the central dot, and we label this coupling Ωα for α = A, B1, B2. With all dots maintained at constant energy, the Hamiltonian for this four dot system (ignoring the spin degree of freedom) is ( ) ( ) ( ) . . 2 2 1 1 c h C B t C B t A C t H B B A + Ω + Ω + Ω = where the couplings are allowed to vary as a function of time (for example by local control of surface gates to modify the barrier potential). The null space of the Hamiltonian above is a two-dimensional space, spanned by the vectors
We investigate the entanglement between electronic and vibrational degrees of freedom produced by a vibronic coupling in a molecular system described in the Born-Oppenheimer approximation. Entanglement in a pure state of the Hilbert space H=Hel⨂Hvib is quantified using the von Neumann entropy of the reduced density matrix and the reduced linear entropy. Expressions for these entanglement measures are derived for the $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ and $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ cases of the bipartite entanglement, where 2 and 3 are the dimensions of the electronic Hilbert space Hel, and Nv is the dimension of Hvib. We study the entanglement dynamics for two electronic states coupled by a laser pulse (a $2\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case), taking as an example a coupling between the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$ and 1g(6s,6p3/2) states of the Cs2 molecule. The reduced linear entropy expression obtained for the $3\ifmmode\times\else\texttimes\fi{}{N}_{v}$ case is used to follow the entanglement evolution in a scheme proposed for the control of the vibronic dynamics in a Cs2 cold molecule, implying the ${a}^{3}{{\ensuremath{\Sigma}}_{u}}^{+}(6s,6s)$, ${0}_{g}{}^{\ensuremath{-}}(6s,6{p}_{3/2})$, and ${0}_{g}{}^{\ensuremath{-}}(6s,5d)$ electronic states, which are coupled by a nonadiabatic radial coupling and a sequence of chirped laser pulses.
Quantum information theory has become a central area in the two fields of quantum mechanics and information science. The focus of this work is mathematical modelling of quantum information with the use of operator algebras, in a way that accentuates the modelling, and acting or manipulation of observables and quantum states. The fundamental aim is to utilise C*-algebras, von Neumann algebras, and other frameworks in operator algebra to formally describe the behaviour of quantum systems, understand the nature of quantum entanglement, and support quantum error correction. Operator algebra usage provides a firm mathematical foundation that guarantees accuracy and contributes to the incorporation of topological mechanisms and algebraic rules necessary for explaining noncommutative concepts in quantum mechanics. In this paper, we will present some original notions of operator algebra and explain their usefulness when it comes to modeling the key dynamics, features of quantum states, measurement, and evolution. We will use a method that combines algebraic techniques to chart qubit interactions, entropic inequalities, and characterize transformations of level in unitary evolution. Through the use of Hilbert spaces and bounded linear operators, we build models in multi-qubit systems and see what the specifications have to say about quantum computing and quantum communication protocols. The paper provide a comprehensive discussion, including algebraic formulations, simulation outputs, and a critical comparison with traditional Hilbert space methods. It is quite notable that the operator algebras can say more about the non-local correlations and decoherence processes. There are extensive literature reviews, useful modelling tools, and implementation outcomes in the paper when symbolic computation tools have been utilised. This combination gives both theoretical strength and modelling instruments to develop quantum technologies in the future.