A note on the realignment criterion

The first part of the chapter is devoted to the discussion of density operators. We first introduce the definition of a density operator and the unitary evolution of a quantum state. We then examine the measurement of a quantum system using the density operator approach. We present the density operator associated with a qubit and a harmonic oscillator at thermal equilibrium. The second part of the chapter focuses on bipartite systems. We show that the quantum state of a subsystem is given by the partial trace of the density operator of the global system. Finally, we explain the concept of purification of a density operator

Search for an efficient entanglement witness operator for bound entangled states in bipartite quantum systems

Quantum state reconstruction, or state reconstruction for short, aims at identifying an unknown quantum state ( states in quantum mechanics) on the basis of experimentally accessible data. The Quantum Optics community usually refers to this inverse problem as quantum (state) tomography while the expression quantum state estimation is often used in the field of Quantum computation. Reconstruction procedures depend on the physical context defined by the system carrying the unknown state, the experimentally accessible observables, the size of the ensemble of systems prepared in the unknown state, and the precision of the measured data. A two-level system (such as a spin-1/2, a qubit, or the two polarizations of a photon) prepared in a state with density matrix ρ is sufficient to illustrate the idea of state reconstruction. With two non-negative eigenvalues summing to one, the density matrix is a positive operator, and it depends on three real parameters. In the Bloch representation, the parameters combine to a real vector n with length |n| 1,

Motivated by the Peres-Horodecki criterion and the realignment criterion we develop a more powerful method to identify entangled states for any bipartite system through a universal construction of the witness operator. The method also gives a new family of positive but non-completely positive maps of arbitrary high dimensions which provide a much better test than the witness operators themselves. Moreover, we find there are two types of positive maps that can detect 2xN and 4xN bound entangled states. Since entanglement witnesses are physical observables and may be measured locally our construction could be of great significance for future experiments.

In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this paper, I propose a framework for the distinguishability by one-way LOCC ($1$-LOCC) of sets of orthogonal bipartite states in a $d_A \otimes d_B$ bipartite system, where $d_A, d_B$ are the dimensions of both subsytems, labelled as $A$ and $B$. I show that if the $i$-th party (where $i=A,B$) can initiate a $1$-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a $1$-LOCC protocol lies in a subspace of $d_i \times d_i$ hermitian matrices, denoted by $\Tb^{(i)}$, and that the method to extract this information (of the existence of this $1$-LOCC protocol) from $\Tb^{(i)}$ depends on the value of $dim \Tb^{(i)}$. In this way one can give sweeping results for the $1$-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $dim \Tb^{(i)}$. Thus I propose that the value of $dim \Tb^{(i)}$ gives the common underlying property based on which sweeping results for the $1$-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.

We introduce a 3-parameter class of maps (1) acting on a bipartite system which are a natural generalisation of the depolarizing channel (and include it as a special case). Then, we find the exact regions of the parameter space that alternatively determine a positive, completely positive, entanglement-breaking, or entanglement-annihilating map. This model displays a much richer behaviour than the one shown by a simple depolarizing channel, yet it stays exactly solvable. As an example of this richness, positive partial transposition but not entanglement-breaking maps is found in Theorem 2. A simple example of a positive yet indecomposable map is provided (see the Remark at the end of Section IV). The study of the entanglement-annihilating property is fully addressed by Theorem 7. Finally, we apply our results to solve the problem of the entanglement annihilation caused in a bipartite system by a tensor product of local depolarizing channels. In this context, a conjecture posed in the work of Filippov [J. Russ. Laser Res. 35, 484 (2014)] is affirmatively answered, and the gaps that the imperfect bounds of Filippov and Ziman [Phys. Rev. A 88, 032316 (2013)] left open are closed. To arrive at this result, we furthermore show how the Hadamard product between quantum states can be implemented via local operations.

Density matrix, which characterizes a quantum state, plays an important role in quantum mechanics. Recently, a method which can directly measure the elements of a density matrix was proposed. Compared with the conventional quantum state tomography which is widely used to reconstruct the density matrix, this measurement method has the advantages of directness and simplicity. However, this direct measurement method relies on an extra pointer space. The addition of this extra pointer can increase the complexity of an experiment. In this paper, we first review previous work on direct measurement, then we propose a scheme to directly measure the density matrix based on <i>δ</i>-quench, which is also a direct measurement method but needs no additional pointer. This proposal reduces the complexity of the measuring system and further simplifies the measurement. We propose two schemes to realize this <i>δ</i>-quench measurement, then analyse their superiorities in different situations of measurement. An experiment to measure photon's density matrix is also designed.

The nonlocality of quantum states on a bipartite system is tested by comparing probabilistic outcomes of two local observables of different subsystems. For a fixed observable A of the subsystem its optimal approximate double A′ of the other system is defined such that the probabilistic outcomes of A′ are almost similar to those of the fixed observable A. The case of σ-finite standard von Neumann algebras is considered and the optimal approximate double A′ of an observable A is explicitly determined. The connection between optimal approximate doubles and quantum correlations is explained. Inspired by quantum states with perfect correlation, like Einstein–Podolsky–Rosen states and Bohm states, the nonlocality power of an observable A for general quantum states is defined as the similarity that the outcomes of A look like the properties of the subsystem corresponding to A′. As an application of optimal approximate doubles, maximal Bell correlation of a pure entangled state on is found explicitly.

This work is devoted to the study, the development, and the application of a new systematic method yielding the dominant correlations that govern a quantum many-body state in an unbiased way. The dominant correlations between any two disjoint blocks of a system are extracted by performing a singular value decomposition of the correlation density matrix (CDM) between those blocks. We determine several mathematical properties and features of this method, in particular the consequences of the lattice symmetries or the symmetries intrinsic to the studied state on the singular values spectrum. We investigate the relation between the norm of the CDM – providing a natural measure of the total correlation between the two blocks – and the so-called mutual information, a quantity originally introduced in quantum information theory. This novel tool is utilized for sheding new light on the zero temperature physics of the spin-1/2 frustrated ferromagnetic J1–J2 Heisenberg chain in a magnetic field as well as on the low-energy physics of the spin-1/2 antiferromagnetic Heisenberg model on the two-dimensional kagome lattice. The states are computed using exact diagonalization and the density matrix renormalization group procedure in the first case, and exact diagonalization only in the second case. This work is introduced in Chapter 1. The first model is then presented in Chapter 2. Chapter 3 introduces the CDM method, and Chapter 4 is devoted to the study of the kagome antiferromagnet. In the J1–J2 chain, we reveal a vector chiral phase at low magnetic field and a sequence of multipolar Luttinger liquid phases at high field. We explicitly show that these multipolar phases result from the destabilization – driven by a locking mechanism – of the classical spiral ground state in the absence of magnetic field. This point of view is completely new: multipolar phases were known to be a possible destabilization of ferromagnetic phases, but they have never been reported as a destabilization of spiral states yet. Regarding the kagome antiferromagnet, we address for the first time the question of the nature of the singlet states forming its quite dense singlet spectrum above the ground state. We show that some of these low-lying singlet states have large dimer correlations which do not seem to significantly decrease with the distance, moreover our CDM studies confirm that the dominant correlations in those singlet states are of the dimer-dimer type. Studies of Von Neumann block entropies reveal a very short correlation length on the one hand, and entropies that are roughly independent on the energy of the state under consideration on the other hand. The scenario of a valence bond crystal phase is investigated and the relevance of different kinds of crystals (from the literature or ad hoc) for reproducing the dimer correlations in the 36-site sample is probed.

Let $h_\lambda$, $e_\lambda$, and $m_\lambda$ denote the homogeneous symmetric function, the elementary symmetric function and the monomial symmetric function associated with the partition $\lambda$ respectively. We give combinatorial interpretations for the coefficients that arise in expanding $m_\lambda$ in terms of homogeneous symmetric functions and the elementary symmetric functions. Such coefficients are interpreted in terms of certain classes of bi-brick permutations. The theory of Lyndon words is shown to play an important role in our interpretations.

The Landauer cost for erasing information demands that information about a physical system be included in the total entropy, as proposed by Zurek [Nature 341, 119 (1989); Phys. Rev. A 40, 4731 (1989)]. A consequence is that most system states---either classical phase-space distributions or quantum pure states---have total entropy much larger than thermal equilibrium. If total entropy is to be a useful concept, this must imply that work can be extracted in going from equilibrium to a typical system state. The work comes from randomization of a ``memory'' that holds a record of the system state.

With rapid progress across platforms for quantum systems, the problem of many-body quantum state reconstruction for noisy quantum states becomes an important challenge. There has been a growing interest in approaching the problem of quantum state reconstruction using generative neural network models. Here we propose the ‘attention-based quantum tomography’ (AQT), a quantum state reconstruction using an attention mechanism-based generative network that learns the mixed state density matrix of a noisy quantum state. AQT is based on the model proposed in ‘Attention is all you need’ by Vaswani et al (2017 NIPS) that is designed to learn long-range correlations in natural language sentences and thereby outperform previous natural language processing (NLP) models. We demonstrate not only that AQT outperforms earlier neural-network-based quantum state reconstruction on identical tasks but that AQT can accurately reconstruct the density matrix associated with a noisy quantum state experimentally realized in an IBMQ quantum computer. We speculate the success of the AQT stems from its ability to model quantum entanglement across the entire quantum system much as the attention model for NLP captures the correlations among words in a sentence.

With recent advances in nanophotonics, metasurfaces based on nano-resonators have facilitated novel types of optical devices. In particular, the interplay between different degrees of freedom, involving polarization and spatial modes, boosted classical polarization measurements and imaging applications. However, the use of metasurfaces for measuring the quantum states of light remains largely unexplored. Conventionally, the task of quantum state tomography is realized with several bulk optical elements, which need to be reconfigured multiple times. Such setups can suffer from decoherence, and there is a fundamental and practical interest in developing integrated solutions for measurement of multi-photon quantum states. We present a new concept and the first experimental realization of all-dielectric metasurfaces with no tuneable elements for imaging-based reconstruction of the full quantum state of entangled photons. Most prominently, we implement multi-photon interferometric measurements on a sub-wavelength thin optical element, which delivers ultimate miniaturization and extremely high robustness. Specifically, we realize a highly transparent all-dielectric metasurface, which spatially splits different components of quantum-polarization states. Then, a simple one-shot measurement of correlations with polarization-insensitive on-off click detectors enables complete reconstruction of multi-photon density matrices with high precision. In our experiment, we prepare sets of polarization states and reconstruct their density matrices with a high fidelity of over 99% for single photon states and above 95% for two-photon states. Our work provides a fundamental advance in the imaging of quantum states, where multi-photon quantum interference takes place at sub-wavelength scale.

Although the origin of quantum optics can be traced back to the beginning of the 20th century, when the fundamental ideas about the quantum nature of the interaction between light and matter were put forward, the splendid blossoming of this part of physics began half a century later, after the invention of masers and lasers. It is remarkable that after another half a century the tree of quantum optics is not only very strong and spreading, but all its branches continue to grow, showing new beautiful blossoms and giving very useful fruits.

One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements since it requires a global reconstruction. Here, we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states. The scheme relies on measuring a sequence of three observables, each complementary to the last. The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong. We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix. The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degrees of freedom of the photons. This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states.