What underlies all these different phenomena in a **quantum**-**many** **body** system is the pattern of entanglement individual constituents of the system have with each other. So, naturally, a paradigm of **representation** was needed which could encode the entanglement information of the system in a more explicit and efficient way than the traditional **representation** did. Traditionally, the **quantum** **states** have been represented as a superposition of conveniently chosen basis **states** in the Hilbert space of the system. Though it works well for small **quantum** systems such as an atom or small spin systems, it is not particularly useful for large **quantum**-systems with exponentially large Hilbert space. Broadly speaking, there are two problems with basis-superposition **representation**. First, the number of basis vectors needed for this **representation** scales exponentially with the system size. For example, for a system with N qubits, one needs 2 N basis **states** to represent a generic **quantum** wave function. To get an idea, a system with 1000 qubits would need 2 1000 ≈ 10 300 basis for description. This number is much larger than the number of atoms in the known universe. So we could not possibly hope to work with these **many** basis **states** in any present day computation system. Second, and more fundamentally, it is very difficult to read off entanglement patterns in a state from its basis-superposition **representation**. It just happens to be not the natural **representation** as far as describing entanglement in a system is concerned. From this need to extract information about entanglement patters, and various limitation of basis-superposition representations, emerged the development of a new paradigm of **representation** of **quantum** systems: the **Tensor** **Network** (TN) **representation**.

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The understanding of critical phenomena in classical mechanics owes a great deal to the spatial **representation** of critical **states**, whereby the order parameter experiences statistical fluctuations on all length scales due to a diverging correlation length [1,2] at the critical temperature. This scale invariance property was the starting point for one of the most powerful tools in theoretical physics, the renormalization group, which allowed rationalization of classical criticality in terms of trajectories in the space of coupling constants [3]. Today, one frontier of research in critical phenomena lies in the **quantum** realm, where criticality may govern some of the most fascinating and complex properties found in strongly correlated materials or cold atoms [4,5]. One very fruitful approach is to consider **quantum** criticality in light of an effective classical theory in higher dimensions [5], combining spatial and temporal fluctuations within the path integral formalism. **Quantum** phase transitions are then probed through physical response functions that display a diverging correlation length in space-time. However, this point of view does not provide a full picture of the physics at play, especially since **quantum** criticality pertains to a singular change in a **many**-**body** ground state. Developing wave-function-based approaches to strong correlations is indeed a blossoming field, ranging from **quantum** chemistry [6] to **quantum** information [7,8], so that hopes are high that **quantum** critical **states** may be rationalized in a simpler way.

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The **many**-**body** problem is a general name for a vast category of physical problems pertaining to the properties of microscopic systems made of a large number of interacting particles. In such a **quantum** system, the repeated interactions between particles create **quantum** correlations, or entanglement. As a consequence, the wave function of the system is a complicated object holding a large amount of information, which usually makes exact or analytical calculations impractical or even impossible. Thus, **many**-**body** theoretical physics most often relies on a set of approximations specific to the problem at hand, and ranks among the most computationally intensive fields of science. Recently, an idea that received a lot of attention from the scientific community consists in using neural networks as variational wave functions to approximate ground **states** of **many**-**body** **quantum** systems. In this direction, the networks are trained or optimized by the standard variational Monte Carlo (VMC) method while a few different neural-**network** architectures were tested [7–10], and the most promising results so far have been achieved with Boltzmann machines [10]. In particular, state-of-the-art numerical results have been obtained on popular models with restricted Boltzmann machines (RBM), and recent effort has demonstrated the power of deep Boltzmann machines to represent ground **states** of **many**-**body** Hamiltonians with polynomial-size gap and **quantum** **states** generated by any polynomial size **quantum** circuits [11,12]. Carleo and Troyer in [7] demonstrated the remarkable power of a reinforcement learning approach in calculating the ground state or simulating the **unitary** time evolution of complex **quantum** systems with strong interactions. Deng et al [13,14] show that this **representation** can be used to describe topological **states**. Besides, they have constructed exact representations for SPT **states** and intrinsic topologically ordered **states**. Very recently, Glasser et al [15] show that there are strong connections between neural **network** **quantum** **states** in the form of RBM and some classes of **tensor**-**network** **states** in arbitrary dimensions and obtain that neural **network** **quantum** **states** and their string-bond-state extension can describe a lattice fractional **quantum** Hall state exactly. In addition, there are a lot of related studies, such as [16–19].

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There are **many** ways that **tensor** networks can aid the study of disordered systems. Although DMRG is in some ways imperfect for the modelling of disor- der, it is so efficient that much can still be learned by applying it. Beyond the Heisenberg and Bose-Hubbard models discussed in the thesis, there are still a myr- iad of possible Hamiltonians that can be examined with DMRG. A current area of intense research is **many**-**body** localisation (MBL), the generalisation of Anderson localisation to interacting **many**-**body** systems [177]. It is believed that the area law holds for all excited **states** in systems with MBL up to some mobility energy, unlike gapped **quantum** systems where only the ground state is area law satisfying [178]. This in principle should allow for an efficient MPS **representation**, and therefore ac- curate DMRG simulation, of any state in a one-dimensional MBL spectrum. Strong disorder renormalisation techniques such as tSDRG can be used as high precision methods when disorder is strong. The method should be accurate for use with the FM/AFM disordered spin-1/2 Heisenberg model where large effective spins would be created as the renormalisation progresses [124]. Beyond spin-1/2 there have been exciting discoveries in disordered spin-3/2 Heisenberg systems, where the rich phase diagram contains topological phases as well as spin doublet and triplet phases [179]. It would be fascinating to uncover the optimal **tensor** **network** geometries in these situations.

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The SDRG method was extended by Hikihara et. al. [30] to include higher **states** at each decimation, in the spirit of Wilson’s numerical renormalisation group [31] and DMRG [15]. This method therefore decomposes the system into blocks rather than larger spins allowing for more accurate computation of, e.g., spin-spin correlation functions. The more **states** that are kept at each decimation the more accurate the description and it is exact in the limit of all **states** kept. This numerical SDRG amounts to a coarse-graining mechanism that acts on the operator. We show in reference [26] that it is equivalent to view this as a multi-level **tensor** **network** wavefunction acting on the original operator. The **operators** that coarse-grain two sites to one can be seen as isometric tensors or isometries that satisfy ww † = 1 1 6= w † w. When viewed in terms of isometries, the algorithm can self-assemble a **tensor** **network** based on the disorder of the system. When written in full, it builds an inhomogeneous binary tree **tensor** **network** (TTN) as shown in fig. 4(b). We shall henceforth refer to this TTN approach to SDRG as tSDRG. For a full description of the algorithm see reference [26].

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The **representation** of **quantum** **many**-**body** **states** as **tensor** networks is connected to White’s density-matrix renormalization group [8], and in the case of one dimensional spin lattices is known as matrix product **states** (MPS) [9]. Among **many** useful properties of **tensor** networks, one which makes them well suited to the description of **states** with symmetries, is the ability to encode the symmetry on the level of a single **tensor** (or a few) describing the state. In the case of global symmetries, both for MPS and for certain classes of PEPS in 2D (Projected Entangled Pair **States** — the generalization of MPS to higher dimensional lattices), the relation between the symmetry of the state and the properties of the **tensor** is well understood [10]. **Tensor** networks studies of lattice gauge theories have so far included numerical works (e.g., mass spectra, thermal **states**, real time dynamics and string breaking, phase diagrams etc. for the Schwinger model and others) [11–30], furthermore, several theoretical formulations of classes of gauge invariant **tensor** **network** **states** have been proposed [31–35]. In all of the latter the construction method follows the ones common to conventional gauge theory formulations: symmetric tensors are used to describe the matter degree of freedom, and later on a gauge field degree of freedom is added, or, alternatively - a pure gauge field theory is considered. While the usefulness of **tensor** networks in lattice gauge theories has certainly been demonstrated by the above mentioned works, so far there were few attempts (e.g. [13]) to generally classify **tensor** **network** **states** with local symmetry.

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In this paper, we introduce a new algorithm for efficiently producing an MPS **representation** for ground **states** of noninteracting fermion systems. Why is this useful, when DMRG is most useful in the opposite regime? This would be a valuable tool in a number of situations. For example, a powerful and widely used class of variational wavefunctions for strongly interacting systems begin with a mean-field fermionic wavefunction, and then one applies a Gutzwiller projection to reduce or eliminate double occupancy[7]. It could be very useful to find the overlaps of a DMRG ground state with a variety of such Gutzwiller **states** to help understand and classify the ground state. Once one has the MPS **representation** of the mean field state, the Gutzwiller projection is very easy, fast, and exact, whereas in other approaches it usually must be implemented with Monte Carlo. One might also begin a DMRG simulation with such a variational state, or in some cases with a mean field state without the Gutzwiller projection. Being able to represent fermion determinantal **states** as MPSs in a very efficient way also opens the door to using DMRG ground **states** as minus-sign constraints in determinantal **quantum** Monte Carlo, in particular in Zhang’s constrained path Monte Carlo (CPMC) method[8, 9]. In this case one would hope that, for systems too big for accurate DMRG, at least the qualitative structure of the ground state could be captured by DMRG, and then the results could be made quantitative with the Monte Carlo method.

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The recent wave of interest to modeling the process of decision making with the aid of the **quantum** formalism gives rise to the following question: ‘How can neurons generate **quantum**-like statistical data?’ (There is a plenty of such data in cognitive psychology and social science.) Our model is based on **quantum**-like **representation** of uncertainty in generation of action potentials. This uncertainty is a consequence of complexity of electrochemical processes in the brain; in particular, uncertainty of triggering an action potential by the membrane potential. **Quantum** information state spaces can be considered as extensions of classical information spaces corresponding to neural codes; e.g., 0/1, quiescent/firing neural code. The key point is that processing of information by the brain involves superpositions of such **states**. Another key point is that a neuronal group performing some psychological function F is an open **quantum** system. It interacts with the surrounding electrochemical environment. The process of decision making is described as decoherence in the basis of eigenstates of F. A decision state is a steady state. This is a linear **representation** of complex nonlinear dynamics of electrochemical **states**. Linearity guarantees exponentially fast convergence to the decision state.

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As an example of the second point consider the natu- ral realization of AF **states** by Fermions in an optical lat- tice. One of the main challenges for direct preparation by loading into an optical lattice consists in maintaining adi- abaticity with respect to the effective spin Hamiltonian as the lattice potential is raised. For the present proce- dure, on the other hand, we only require that initially a spin-polarized band insulator is prepared, which has been achieved [16]. It should be noted however, that current approaches to generating staggered magnetic fields with vector light shifts in the alkali’s [11] only work for the high-Z atoms Rb and Cs, for which there are no fermionic isotopes. On the other hand, spin dependent optical po- tentials are possible for alkaline earth atoms, including the fermionic isotopes [17].

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We consider the Magnus **representation** of the image of the braid group under the gener- alizations of the standard Artin **representation** discovered by M. Wada. We show that the images of the generators of the braid group under the Magnus **representation** are **unitary** relative to a Hermitian matrix. As a special case, we get that the Burau **representation** is **unitary**, which was known and proved by C. C. Squier.

We remark that correlation **operators** can be introduced by means of the cluster expansions of the density **operators** [2] (the kernel of a density operator is known as a density matrix) governed by a sequence of the von Neumann equations, and hence, they describe of the evolution of **states** by an equivalent method in comparison with the density **operators**. For **quantum** systems of fixed number of particles the state is described by finite sequence of correlation **operators** governed by a corresponding system of the von Neumann equations (6).

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To model the **quantum**-impurity scenario, we go beyond mean-field theory and construct impurity ↓- polariton wave functions that include two- and three- point **quantum** **many**-**body** correlations. Such strong multi-point correlations can be continuously connected to the existence of multi-**body** bound **states** in vacuum, namely ↓↑ biexcitons and ↓↑↑ triexcitons [30] (higher- order bound **states** have not been observed, as far as we are aware). We calculate the ↓ linear transmission probe spectrum following resonant pumping of ↑ lower polari- tons, as illustrated in Fig. 1(b), and we expose how multi- point correlations emerge as additional splittings in the spectrum with increasing pump strength. There is thus the prospect of directly accessing polariton correlations from spectroscopic measurements performed in standard cryogenic experiments on GaAs-based structures [27, 28] — i.e., **many**-**body** correlations have measurable effects on transmission measurements, and do not require mea- surements of higher order coherence functions in order to be observed. Moreover, our theory requires few parame- ters which can be measured independently, and thus al- lows one to predict the pump-probe spectrum in other materials, such as transition metal dichalcogenides at room temperature [31, 32].

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of low lying **states** are similar to or less than collision times? Answering this question is a chal- lenge that is being addressed by performing experiments with weakly bound nuclei. It is indisputable that collisions of such nuclei must be described by a superposition of **quantum** **states** mentioned in Sec. 1. Understanding the role of low lying unbound **states** or short-lived resonances, which can lead to breakup of nuclei, is however a challenge [20–22]. Here the reaction dynamics becomes sensitive to both coupling e ff ects and the e ff ect of breakup [20–22]. One of the unambiguous observation of the consequences of the latter came through exploiting measurement of the fusion barrier distribution. Us- ing beams of 9 Be, it was shown unambigiously [23] that fusion cross-sections at above-barrier energies

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Definition 1.7: Suppose is a positive operator, then an operator is called an on if . If equality holds, that is , then is called an . Here is a self adjoint and invertible operator. Such **operators** were extensively studied by Suciu [19].

In the absence of interactions, we expect excitations of atoms to higher bands to have no effect on the imbalance, as they occur on even and odd sites with identical rates and cannot affect the remaining atoms in the noninteracting case. Figure 3 shows the noninteracting susceptibility χ as a function of disorder strength in the single-particle localized regime ( Δ > 2 J). The susceptibility strongly decreases for increasing disorder strength, which can be understood by considering a single particle localized around a site i: Deep in the localized phase, its time-averaged density distribu- tion will be almost identical to that of the Wannier state on site i, with almost no weight on neighboring sites. In this limit, a photon scattering event has negligible probability of moving the particle away from site i, resulting in a vanishing susceptibility. At weaker disorder strength, single-particle eigenstates are less localized and have finite overlap with the Wannier **states** of the neighboring sites. Hence, there is now a finite probability of scattering- induced hopping transferring the particle to a neighboring site and thereby relaxing the imbalance, giving rise to a finite susceptibility. This intuitive idea is also at the heart of a recently proposed rate model [31], which we compare to our data (Fig. 3). Since the rate model describes only dephasing events, its scattering rate has been rescaled by p dp to take their finite probability into account. We find very good agreement between experiment and theory. This demonstrates that atom losses and excited band populations cannot affect the imbalance in the absence of interactions. Our observable does not allow us to characterize the susceptibility at disorder strengths below Δ ≲ 3 J since, close to the phase transition point, the localization length becomes too large and the stationary imbalance of the closed system is already close to zero. However, we can derive a simple upper bound for the susceptibility based on the rate equation model: When the localization length diverges, each dephasing event has equal probability to project the atom onto an even or odd site, thereby canceling its contribution to the imbalance. In this limit, the FIG. 3. Noninteracting susceptibility vs disorder strength:

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Tenth Amendment (states and the people retain powers not delegated to the federal government). Eleventh Amendment (sovereign immunity)[r]

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According to that theory, the process of dispersion should be continuous, but Compton electrons are emitted in a random way. The authors assumed both processes of wave dispersion and Compton electrons dispersion were not connected with each other (?). The main idea was to lay a bridge between **quantum** theory of the atom and classical emission theory. There were introduced specially so called “virtual” oscillators which generate in accordance with classical theory waves (non **quantum** one) enable to induce the transition from the state with lower energy to the state with higher energy. These waves did not carry the energy, but power necessary for atom transition from lower to the higher state was generated within the atom itself. Along with that the inverse process of the atom transition from excited state to the lower one could take place, but the energy was not taken away by waves but should disappear inside the atom. In other words, the increase of one atom energy was not connected with energy decrease in another one. Authors considered that these processes compensated each other on average only and that compensation was the better the more events are participated. Energy conservation law has statistical character according to that interpretation, and there is no law of conservation for single events, but they appear in processes involving large number of particles, i.e. at transition to Newton mechanics. But then it should be acknowledged that in the case of Compton effect the changes of motion direction of the light **quantum** and its energy to be appeared in the result of collision were happening apart from the changes of electron’s state. The unfoundedness of such an approach was lately experimentally proved by Bote and Geiger. To say the truth, the authors abandoned that point of view later; moreover at that time this idea did not follow from **quantum** theory equations. And to get out of the tight spot it was declared that **quantum** mechanics did not describe single events at all. Thus the most striking paradox was removed by a simple prohibition just to think about it! But genius idea that laws of conservation are not valid for individual processes and appear in **quantum** mechanics after statistical averaging does not become less genius even if those for whom it “has come to mind” rejected it. May be, this idea was a little premature and should have a somewhat different shape.

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Embedding efficiency describes how **many** bits (qubits) of a secret message can be embedded in a single qubit **quantum** particle for a steganography scheme. The hiding channel of our protocol can be built up within the QSDC channel to transmit a secret message without consuming any extra **quantum** communications or **quantum** **states** besides the QSDC. By transferring a χ-type **quantum** state that is encoded with 8-bit of information in the QSDC process, Alice can hide an 8-bit secret message under the cover of normal information transmission. Bob decodes the information that Alice has sent to him by decode the information gets from performing measurements on the χ-type **quantum** state based on FBM. So we can say that The embedding capacity of the proposed protocol depends on the source capacity of the QSDC channel.

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Summary and outlook.— To summarize, we have con- structed a constrained spin model which exhibits nearly perfect QMBS. The remarkably long-lived oscillatory dy- namics suggests that **quantum** scars remain stable in the thermodynamic limit. We showed that the dynamics can be understood in terms of a large, precessing SU(2) spin, and used this intuition to introduce a family of toy mod- els with perfect scarring. In future work, it would be highly desirable to find an analytical mapping between the toy models and the constrained spin model. More- over, the approach developed here may be applied to sta- bilize other types of **quantum** scars, in particular the ones originating from the |Z 3 i state in the model (1) [15], as

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The collapse has to possess three defining characteristics. First of all it must be non-**unitary** and lead to evolution of **quantum** **states** into everyday **states**, like the ones that can be observed all around us. In particular the final **states** of the collapse process must not include macroscopic superpositions. The proper way to formalize this requirement is by the introduction of a pointer basis, as was first pointed out by Zurek [127,130]. The second requirement is that **quantum** mechanics must survive unharmed for microscopic particles. In that regime **quantum** mechanics has been very thoroughly tested and is certainly correct. One way to e ﬀ ectively make the collapse process act on macroscopic bodies, but not on microscopic particles would be to make it act on both, but to let the timescale over which it becomes noticeable depend on the number of constituent particles or, equivalently, on the total mass of an object. That way the process would take too long to notice for small systems, but would be almost instantaneous for classical objects. The final requirement on the collapse process is that it must reproduce Born’s rule [38]. The probability for ending up in a certain macroscopic state after doing a measurement must be equal to the squared amplitude of that part of the microscopic wavefunction that corresponded to the measured value. This implies the introduction of uncertainty and stochasticity into the time evolution in one way or another.

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