On the adiabatic limit of Hadamard states

We consider the adiabatic limit of Hadamard states for free quantum Klein–Gordon fields, when the background metric and the field mass are slowly varied from their initial to final values. If the Klein–Gordon field stays massive, we prove that the adiabatic limit of the initial vacuum state is the (final) vacuum state, by extending to the symplectic framework the adiabatic theorem of Avron–Seiler–Yaffe. In cases when only the field mass is varied, using an abstract version of the mode decomposition method we can also consider the case when the initial or final mass vanishes, and the initial state is either a thermal state or a more general Hadamard state.     

Applications of quantum Fourier transform in photon-added coherent state

Quantum Fourier transform is realized by the Hadamard gate in a quantum computer, which can also be considered as a Hadamard transform. We introduce the Hadamard transformed photon-added coherent state （HTPACS）, which is obtained by letting the photon-added coherent state （PACS） across the quantum Hadamard gate, from this result. It is found that the HTPACS can be considered as a coordinate-momentum mutual exchanging followed by a squeezing transform of the PACS. In addition, the non-classical statistical properties of HTPACS, such as squeezing coefficient, Mandel parameter, etc., are also discussed.     

Construction of Hadamard States by Pseudo-Differential Calculus

We give a new construction based on pseudo-differential calculus of quasi-free Hadamard states for Klein–Gordon equations on a class of space-times whose metric is well-behaved at spatial infinity. In particular on this class of space-times, we construct all pure Hadamard states whose two-point function (expressed in terms of Cauchy data on a Cauchy surface) is a matrix of pseudo-differential operators. We also study their covariance under symplectic transformations. As an aside, we give a new construction of Hadamard states on arbitrary globally hyperbolic space-times which is an alternative to the classical construction by Fulling, Narcowich and Wald.     

Micro-local approach to the Hadamard condition in quantum field theory on curved space-time

For the two-point distribution of a quasi-free Klein-Gordon neutral scalar quantum field on an arbitrary four dimensional globally hyperbolic curved space-time we prove the equivalence of (1) the global Hadamard condition, (2) the property that the Feynman propagator is a distinguished parametrix in the sense of Duistermaat and Hörmander, and (3) a new property referred to as the wave front set spectral condition (WFSSC), because it is reminiscent of the spectral condition in axiomatic quantum field theory on Minkowski space. Results in micro-local analysis such as the propagation of singularities theorem and the uniqueness up toC of distinguished parametrices are employed in the proof. We include a review of Kay and Wald's rigorous definition of the global Hadamard condition and the theory of distinguished parametrices, specializing to the case of the Klein-Gordon operator on a globally hyperbolic space-time. As an alternative to a recent computation of the wave front set of a globally Hadamard two-point distribution on a globally hyperbolic curved space-time, given elsewhere by Köhler (to correct an incomplete computation in [32]), we present a version of this computation that does not use a deformation argument such as that used in Fulling, Narcowich and Wald and is independent of the Cauchy evolution argument of Fulling, Sweeny and Wald (both of which are relied upon in Köhler's proof). This leads to a simple micro-local proof of the preservation of Hadamard form under Cauchy evolution (first shown by Fulling, Sweeny and Wald) relying only on the propagation of singularities theorem. In another paper [33], the equivalence theorem is used to prove a conjecture by Kay that a locally Hadamard quasi-free Klein-Gordon state on any globally hyperbolic curved space-time must be globally Hadamard.To my parents     

Nested Construction of Families of Complex Hadamard Matrices

We present a new parametrization of families of complex Hadamard matrices stemming from the Fourier matrices in every prime power dimension. We connect continuous Abelian groups with families of complex Hadamard matrices and conjecture that the constructed families are maximal. Also, we derive new relations for complex Hadamard matrices in every prime power dimension and prove that some real Hadamard matrices can be written as a product of an arbitrarily large number of real Hadamard matrices.     

Singularity structure of the two point function of the free Dirac field on a globally hyperbolic spacetime

We give an introduction to the techniques from microlocal analysis that have successfully been applied in the investigation of Hadamard states of free quantum field theories on curved spacetimes. The calculation of the wave front set of the two point function of the free Klein‐Gordon field in a Hadamard state is reviewed, and the polarization set of a Hadamard two point function of the free Dirac field on a curved spacetime is calculated.     

Non-classical Effects of the Photon-Added Hadamard Transformed Vacuum State

Theoretical analysis are given for the non-classical effects of the photon-added Hadamard transformed vacuum state (PAHTVS) generated by repeatedly acting photon addition operation on a vacuum state which passing through a Hadamard gate. It is shown that the normalization constant of the PAHTVS is a Legendre polynomial. Furthermore, we study the analytical expressions of several quasi-probability distributions for the PAHTVS. We discuss the negative values of the Wigner function for the PAHTVS, which implies the non-classical properties of the PAHTVS.     

A generalized Hadamard transformation from new entangled state

A new entangled state | eta ;theta rangle is proposed by the technique of integral within an ordered product. A generalized Hadamard transformation is derived by virtue of | eta ;theta rangle , which plays a role of Hadamard transformation for (hat a_1 sin theta - hat a_2 cos theta ) and (hat a_1 cos theta + hat a_2 sin theta ).     

利用三粒子部分纠缠GHZ 态和二粒子纠缠态概率隐形传送任意二粒子态

我们提出了利用三粒子非最大纠缠GHZ态和二粒子非最大纠缠态从一个发送者概率隐形传送任意未知二粒子态至两个接收者中的一个的方案。发送者进行两次Bell-state 测量，接收者以另一个可能的接收者进行的Hadamard操作和投影测量的结果为条件引入两个适当的幺正变换就可以概率隐形传送任意未知二粒子态。     

“Tunnelling” Black-Hole Radiation with ϕ 3 Self-Interaction: One-Loop Computation for Rindler Killing Horizons

Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory discovering interesting interplay with the Hawking radiation. A more precise and general account of that phenomenon has been subsequently given within the framework of QFT in curved spacetime by two of the authors of the present paper. In particular, it has been shown that, in the limit of sharp localization on opposite sides of a Killing horizon, the quantum correlation functions of a scalar field appear to have thermal nature, and the tunnelling probability is proportional to exp{?β _Hawking E}. This local result is valid in every spacetime including a local Killing horizon, no field equation is necessary, while a suitable choice for the quantum state is relevant. Indeed, the two-point function has to verify a short-distance condition weaker than the Hadamard one. In this paper we consider a massive scalar quantum field with a ? ³ self-interaction and we investigate the issue whether or not the black hole radiation can be handled at perturbative level, including the renormalisation contributions. We prove that, for the simplest model of the Killing horizon generated by the boost in Minkowski spacetime, and referring to Minkowski vacuum, the tunnelling probability in the limit of sharp localization on opposite sides of the horizon preserves the thermal form proportional to exp{?β _H E} even taking the one-loop renormalisation corrections into account. A similar result is expected to hold for the Unruh state in the Kruskal manifold, since that state is Hadamard and looks like Minkowski vacuum close to the horizon.     

未知三粒子GHZ纠缠态的网络多人控制传送

提出一种多人控制的三粒子GHZ纠缠态的量子隐形传送方案,为了实现传送,Alice需要对自己的三对粒子实施Bell测量并将结果通知Bob,异地的众多监控者对各自的控制位粒子实施Hadamard变换和投影测量.接受者Bob在Alice和所有监控的者发送的经典信息的协助下只需要施行简单的幺正变换就能成功实现量子态的隐形传送,传送过程中任意一个参与者的缺席都将导致传送的失败.     

Teleportation of an unknown bipartite state via non-maximally entangled two-particle state

In this paper a new scheme for teleporting an unknown entangled state of two particles is proposed. To weaken the requirement for the quantum channel, without loss of generality, two communicators only share a non-maximally entangled two-particle state. Teleportation can be probabilistically realized if sender performs Bell-state measurements and Hadamard transformation and receiver introduces two auxiliary particles, operates C-not operation, single-qubit measurements and appropriate unitary transformations. The probability of successful teleportation is determined by the smaller one among the coefficients' absolute values of the quantum channel.     

Constructing Hadamard States via an Extended Møller Operator

We consider real scalar field theories, whose dynamics is ruled by normally hyperbolic operators differing only by a smooth potential V. By means of an extension of the standard definition of Møller operator, we construct an isomorphism between the associated spaces of smooth solutions and between the associated algebras of observables. On the one hand, such isomorphism is non-canonical, since it depends on the choice of a smooth time-dependant cut-off function \({\chi}\). On the other hand, given any quasi-free Hadamard state for a theory with a given V, such isomorphism allows for the construction of another quasi-free Hadamard state for a different potential. The resulting state preserves also the invariance under the action of any isometry, whose associated Killing field \({\xi}\) is complete and fulfilling both \({\mathcal{L}_\xi V=0 \,\, {\rm and} \,\, \mathcal{L}_\xi\chi=0}\). Eventually, we discuss a sufficient condition to remove on static spacetimes, the dependence on the cutoff via a suitable adiabatic limit.     

Meta-Entanglement

We give a meta-logical interpretation of the entanglement mechanism of quantum space-time in terms of the sequent calculus of a quantum sub-structural logic. This meta-logical picture is based mainly on the two meta-rules cut and EPR, and on the new meta-theorem “teleportation” (TEL), built by the use of the above meta-rules, both performed in parallel. The proof of (TEL)-theorem fairly reproduces the protocol of quantum teleportation. In the framework of space-time entanglement, the conclusion of the (TEL)-theorem is that the entangled space-time can convey the quantum teleportation of an unknown quantum state. We also introduce two new structural rules: the Hadamard (H)-rule and the CNOT-rule, the latter being used, together with the cut, in the proof of the new theorem “Entanglement” (ENT).

     

Joint Remote State Preparation Schemes for Two Different Quantum States Selectively

The scheme for joint remote state preparation of two different one-qubit states according to requirement is proposed by using one four-dimensional spatial-mode-entangled KLM state as quantum channel. The scheme for joint remote state preparation of two different two-qubit states according to requirement is also proposed by using one four-dimensional spatial-mode-entangled KLM state and one three-dimensional spatial-mode-entangled GHZ state as quantum channels. Quantum non-demolition measurement, Hadamard gate operation, projective measurement and unitary transformation are included in the schemes.     

Controlled Teleportation of an Arbitrary Two-Particle Pure or Mixed State by Using a Five-Qubit Cluster State

A new scheme for controlled teleportation of an arbitrary two-particle pure or mixed state with the help of a five-qubit cluster state is proposed in detail. In this scheme, a five-particle cluster state is shared by a sender, a controller and a receiver. At first, the sender performs a four-qubit von-Neumann measurement on the qubits at hand, and the controller performs a Hadamard measurement on his qubit. Then the receiver can reconstruct the arbitrary two-particle pure or mixed state by performing some appropriate unitary transformations on his particles after he knows the measure results of the sender and the controller. This controlled teleportation scheme is deterministic.     

Teleportation of an Arbitrary Two-Particle State by Two Partial Entangled Three-Particle GHZ States

We present a scheme for teleporting an unknown arbitrary two-particle state from a sender to either one of two receivers. The quantum channel is composed of two partial entangled three-particle GHZ states. An unknown arbitrary two-particle state can be perfectly teleported probabilistically if the sender performs two generalized Bell-state measurements and each receiver introduces an appropriate unitary transformation with the help of the other receiver's Hadamard operations and simple measurements.     

Stability of Quantum Systems at Three Scales: Passivity, Quantum Weak Energy Inequalities and the Microlocal Spectrum Condition

Quantum weak energy inequalities have recently been extensively discussed as a condition on the dynamical stability of quantum field states, particularly on curved spacetimes. We formulate the notion of a quantum weak energy inequality for general dynamical systems on static background spacetimes and establish a connection between quantum weak energy inequalities and thermodynamics. Namely, for such a dynamical system, we show that the existence of a class of states satisfying a quantum weak inequality implies that passive states (e.g., mixtures of ground- and thermal equilibrium states) exist for the time-evolution of the system and, therefore, that the second law of thermodynamics holds. As a model system, we consider the free scalar quantum field on a static spacetime. Although the Weyl algebra does not satisfy our general assumptions, our abstract results do apply to a related algebra which we construct, following a general method which we carefully describe, in Hilbert-space representations induced by quasifree Hadamard states. We discuss the problem of reconstructing states on the Weyl algebra from states on the new algebra and give conditions under which this may be accomplished. Previous results for linear quantum fields show that, on one hand, quantum weak energy inequalities follow from the Hadamard condition (or microlocal spectrum condition) imposed on the states, and on the other hand, that the existence of passive states implies that there is a class of states fulfilling the microlocal spectrum condition. Thus, the results of this paper indicate that these three conditions of dynamical stability are essentially equivalent. This observation is significant because the three conditions become effective at different length scales: The microlocal spectrum condition constrains the short-distance behaviour of quantum states (microscopic stability), quantum weak energy inequalities impose conditions at finite distance (mesoscopic stability), and the existence of passive states is a statement on the global thermodynamic stability of the system (macroscopic stability).Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany. verch@mis.mpg.de     

Equivalence of the (Generalised) Hadamard and Microlocal Spectrum Condition for (Generalised) Free Fields in Curved Spacetime

We prove that the singularity structure of all n-point distributions of a state of a generalised real free scalar field in curved spacetime can be estimated if the two-point distribution is of Hadamard form. In particular this applies to the free field and the result has applications in perturbative quantum field theory, showing that the class of all Hadamard states is the state space of interest. In our proof we assume that the field is a generalised free field, i.e. that it satisfies scalar (c-number) commutation relations, but it need not satisfy an equation of motion. The same arguments also work for anti-commutation relations and for vector-valued fields. To indicate the strengths and limitations of our assumption we also prove the analogues of a theorem by Borchers and Zimmermann on the self-adjointness of field operators and of a weak form of the Jost-Schroer theorem. The original proofs of these results make use of analytic continuation arguments. In our case no analyticity is assumed, but to some extent the scalar commutation relations can take its place.     

Implementing the Deutsch-Jozsa algorithm by using Schrodinger cat states in cavity QED

We propose a scheme to implement the Deutsch-Jozsa algorithm by using Schroedinger cat states in cavity quantum electron-dynamics （QED）. The scheme is based on the Raman interaction of a degenerate three-level A-type atom with a coherent state in a cavity. By using Schroedinger cat states, the atomic spontaneous emission can be minimized and the Hadamard transformation in our scheme is not needed.