We study multi-qubit quantum channels that can be represented as a product of one-mode fermionic attenuation channels. An explicit formula for the classical capacity $C_1$ and for the minimum output entropy $S_{min}$ of these channels is proposed. We compute $S_{min}$ analytically for any number of qubits under assumption that the minimum is achieved on a Gaussian input. Apart from that, a simple numerical method for evaluating $S_{min}$ is developed. The method is applicable to any channels that are sufficiently noisy. For fermionic product channels the proposed formula for $S_{min}$ agrees with the numerical results with a precision about $10^{-9}$.
The spin can be described in the star product formalism by extending the bosonic Moyal product in the fermionic sector. The fermionic star product is then the Clifford product of geometric algebra and it is possible to formulate the fermionic star product formalism in analogy to the bosonic star product formalism. For the fermionic star product description of spin, one can then establish the relation to other approaches that describe spin with fermionic variables, i.e. the operator formalism and the path integral formalism. It is shown that the fermionic star product formalism and the fermioni…
Deformation quantization is a powerful tool for quantizing theories with bosonic and fermionic degrees of freedom. The star products involved generate the mathematical structures which have recently been used in attempts to analyze the algebraic properties of quantum field theory. In the context of quantum mechanics they provide a canonical quantization procedure for systems with either bosonic of fermionic degrees of freedom. We illustrate this procedure for a number a physical examples, including bosonic, fermionic and supersymmetric oscillators. We show how non-relativistic and relativistic…
We study the degradability of fermionic Gaussian channels. Fermionic quantum channels are a central building block of quantum information processing with fermions, and the family of Gaussian channels, in particular, is relevant in the emerging field of electron quantum optics and its applications for quantum information. Degradable channels are of particular interest since they have a simple formula that characterizes their quantum capacity. We derive a simple standard form for fermionic Gaussian channels. This allows us to fully characterize all degradable $n$-mode fermionic Gaussian channels…
We discuss the concept of unextendible product basis (UPB) and generalized UPB for fermionic systems, using Slater determinants as an analogue of product states, in the antisymmetric subspace $\wedge^ N \bC^M$. We construct an explicit example of generalized fermionic unextendible product basis (FUPB) of minimum cardinality $N(M-N)+1$ for any $N\ge2,M\ge4$. We also show that any bipartite antisymmetric space $\wedge^ 2 \bC^M$ of codimension two is spanned by Slater determinants, and the spaces of higher codimension may not be spanned by Slater determinants. Furthermore, we construct an example…
Capacity of entanglement (CoE), an information-theoretic measure of entanglement, defined as the variance of modular Hamiltonian, is known to capture the deviation from the maximal entanglement. We derive an exact expression for the average eigenstate CoE in fermionic Gaussian states as a finite series, valid for arbitrary bi-partition of the total system. Further, we consider the complex SYK$_2$ model in the thermodynamic limit and we obtain a closed-form expression of average CoE. In this limit, the variance of the average CoE becomes independent of the system size. Moreover, when the subsys…
The private quantum channel (PQC) maps any quantum state to the maximally mixed state for the discrete as well as the bosonic Gaussian quantum systems, and it has fundamental meaning on the quantum cryptographic tasks and the quantum channel capacity problems. In this paper, we introduce a notion of approximate private quantum channel ($\varepsilon$-PQC) on fermionic Gaussian systems (i.e., $\varepsilon$-FPQC), and construct its explicit form of the fermionic (Gaussian) private quantum channel. First of all, we suggest a general structure for $\varepsilon$-FPQC on the fermionic Gaussian system…
We study the statistical behavior of entanglement in quantum bipartite systems over fermionic Gaussian states as measured by von Neumann entropy and entanglement capacity. The focus is on the variance of von Neumann entropy and the mean entanglement capacity that belong to the so-defined second-order statistics. The main results are the exact yet explicit formulas of the two considered second-order statistics for fixed subsystem dimension differences. We also conjecture the exact variance of von Neumann entropy valid for arbitrary subsystem dimensions. Based on the obtained results, we analyti…
In this article, we investigate the communication channel of fermionic system in an accelerated frame. We observe that at the infinite acceleration, the mutual information of single rail quantum channel coincides with that of double rail quantum channel, but those of classical ones reach different values. Furthermore, we find that at the infinite acceleration, the conditional entropy of single(or double) rail quantum channel vanishes, but those of classical ones may have finite values. In addition, we see that even when considering a method beyond the single mode approximation, the dual rail e…
We introduce the concept of fermionic matrix product operators, and show that they provide a natural representation of fermionic fusion tensor categories. This allows for the classification of two dimensional fermionic topological phases in terms of matrix product operator algebras. Using this approach we give a classification of fermionic symmetry protected topological phases with respect to a group $G$ in terms of three cohomology groups: $H^1(G,\mathbb{Z}_2)$, describing which matrix product operators are of Majorana type, $H^2(G,\mathbb{Z}_2)$, describing the fermionic nature of the fusion…
In this paper the entanglement of multi-qubit fermionic pseudo Hermitian coherent states (FPHCS) described by anticommutative Grassmann numbers is studied. The pseudo-Hermitian versions of the well known maximally entangled pure states such as Bell and GHZ, W and biseparable states are introduced through integrating over the tensor products of FPHCSs with suitable choice of Grassmannian weight functions. Meanwhile as an illustration, the method is applied to tensor product of 2 and 3 qubit pseudo Hermitian systems. Then the measures of concurrence and average entropy are applied to quantify th…
Freie Universität Berlin（柏林自由大学）；Technology Innovation Institute (TII)（技术创新研究所）；Tsinghua University（清华大学）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料能源中心）
We study matrix product unitary operators (MPUs) for fermionic one-dimensional (1D) chains. In stark contrast with the case of 1D qudit systems, we show that (i) fermionic MPUs do not necessarily feature a strict causal cone and (ii) not all fermionic Quantum Cellular Automata (QCA) can be represented as fermionic MPUs. We then introduce a natural generalization of the latter, obtained by allowing for an additional operator acting on their auxiliary space. We characterize a family of such generalized MPUs that are locality-preserving, and show that, up to appending inert ancillary fermionic de…
We study the capacity of a quantum channel where channel acts like controlled phase gate with the control being provided by a one-dimensional quantum spin chain environment. Due to the correlations in the spin chain, we get a quantum channel with memory. We derive formulas for the quantum capacity of this channel when the spin state is a matrix product state. Particularly, we derive exact formulas for the capacity of the quantum memory channel when the environment state is the ground state of the AKLT model and the Majumdar-Ghosh model. We find that the behavior of the capacity for the range o…
In this paper we investigate the entanglement of multi-qubit fermionic coherent states described by anticommutative Grassmann numbers. Choosing an appropriate weight function, we show that it is possible to construct some entangled pure states, consisting of {\bf{GHZ}}, {\bf{W}}, Bell and biseparable states, by tensor product of fermion coherent states. Moreover a comparison with maximal entangled bosonic coherent states is presented and it is shown that in some cases they have fermionic counterpart which are maximal entangled after integration with suitable weight functions.
Establishing the strong converse theorem for a communication channel confirms that the capacity of that channel, that is, the maximum achievable rate of reliable information communication, is the ultimate limit of communication over that channel. Indeed, the strong converse theorem for a channel states that coding at a rate above the capacity of the channel results in the convergence of the error to its maximum value 1 and that there is no trade-off between communication rate and decoding error. Here we prove that the strong converse theorem holds for the product-state capacity of quantum chan…
It is conjectured that the Holevo capacity of a product channel Ω\otimes Φis achieved when product states are used as input. Amosov, Holevo and Werner have also conjectured that the maximal p-norm of a product channel is achieved with product input states. In this paper we establish both of these conjectures in the case that Ωis arbitrary and Φis a CQ or QC channel (as defined by Holevo). We also establish the Amosov, Holevo and Werner conjecture when Ωis arbitrary and either Φis a qubit channel and p=2, or Φis a unital qubit channel and p is integer. Our proofs involve a new conjecture for th…
The information carrying capacity of the d-dimensional depolarizing channel is computed. It is shown that this capacity can be achieved by encoding messages as products of pure states belonging to an orthonormal basis of the state space, and using measurements which are products of projections onto this same orthonormal basis. In other words, neither entangled signal states nor entangled measurements give any advantage for information capacity. The result follows from an additivity theorem for the capacity of the product of the depolarizing channel with an arbitrary channel. We establish the A…
The projective construction (the slave-particle approach) has played an very important role in understanding strongly correlated systems, such as the emergence of fermions, anyons, and gauge theory in quantum spin liquids and quantum Hall states. Recently, fermionic Projected Entangled Pair States (fPEPS) have been introduced to effciently represent many-body fermionic states. In this paper, we show that the strongly correlated bosonic/fermionic states obtained both from the projective construction and the fPEPS approach can be represented systematically as Grassmann tensor product states. Thi…
This is a thesis submitted to university of Pune, India, for the Ph.D. degree. This work deals with entanglement production in two qubit, two qutrit and three qubit systems, entanglement in indistinguishable fermionic systems, quantum discord in a Heisenberg chain and geometric measure of quantum discord in an arbitrary state of a bipartite quantum system.
We extend the twin-space formulation of the hierarchical equations of motion approach in combination with the matrix product state representation (introduced in J. Chem. Phys. 150, 234102, [2019]) to nonequilibrium scenarios where the open quantum system is coupled to a hybrid fermionic and bosonic environment. The key ideas used in the extension are a reformulation of the hierarchical equations of motion for the auxiliary density matrices into a time-dependent Schrödinger-like equation for an augmented multi-dimensional wave function as well as a tensor decomposition into a product of low-ran…
Tensor network states and specifically matrix-product states have proven to be a powerful tool for simulating ground states of strongly correlated spin models. Recently, they have also been applied to interacting fermionic problems, specifically in the context of quantum chemistry. A new freedom arising in such non-local fermionic systems is the choice of orbitals, it being far from clear what choice of fermionic orbitals to make. In this work, we propose a way to overcome this challenge. We suggest a method intertwining the optimisation over matrix product states with suitable fermionic Gauss…
The capacity of quantum channel with product input states was formulated by the quantum coding theorem. However, whether entangled input states can enhance the quantum channel is still open. It turns out that this problem is reduced to aspecial case of the more general problem whether the capacity of product quantum channel exhibits additivity. In the present study, we apply one of the quantum Arimoto-Blahut type algorithms to the latter problem. The results suggest that the additivity of product quantum channel capacity always holds and that entangled input states cannot enhance the quantum c…
We develop the formalism of fermionic matrix product states (fMPS) and show how irreducible fMPS fall in two different classes, related to the different types of simple $\mathbb{Z}_2$ graded algebras, which are physically distinguished by the absence or presence of Majorana edge modes. The local structure of fMPS with Majorana edge modes also implies that there is always a two-fold degeneracy in the entanglement spectrum. Using the fMPS formalism we make explicit the correspondence between the $\mathbb{Z}_8$ classification of time-reversal invariant spinless superconductors and the modulo 8 pe…
Tensor network methods have progressed from variational techniques based on matrix-product states able to compute properties of one-dimensional condensed-matter lattice models into methods rooted in more elaborate states such as projected entangled pair states aimed at simulating the physics of two-dimensional models. In this work, we advocate the paradigm that for two-dimensional fermionic models, matrix-product states are still applicable to significantly higher accuracy levels than direct embeddings into one-dimensional systems allow for. To do so, we exploit schemes of fermionic mode trans…
While general quantum many-body systems require exponential resources to be simulated on a classical computer, systems of non-interacting fermions can be simulated exactly using polynomially scaling resources. Such systems may be of interest in their own right, but also occur as effective models in numerical methods for interacting systems, such as Hartree-Fock, density functional theory, and many others. Often it is desirable to solve systems of many thousand constituent particles, rendering these simulations computationally costly despite their polynomial scaling. We demonstrate how this sca…
We extend to a non-Hermitian fermionic quantum field theory with PT symmetry our previous discussion of second quantization, discrete symmetry transformations, and inner products in a scalar field theory [arXiv:2006.06656]. For illustration, we consider a prototype model containing a single Dirac fermion with a parity-odd, anti-Hermitian mass term. In the phase of unbroken PT symmetry, this Dirac fermion model is equivalent to a Hermitian theory under a similarity transformation, with the non-Hermitian nature of the model residing only in the spinor structure, whereas the algebra of the creati…
The capacity of a quantum channel for transmission of classical information depends in principle on whether product states or entangled states are used at the input, and whether product or entangled measurements are used at the output. We show that when product measurements are used, the capacity of the channel is achieved with product input states, so that entangled inputs do not increase capacity. We show that this result continues to hold if sequential measurements are allowed, whereby the choice of successive measurements may depend on the results of previous measurements. We also present …
Under an appropriate symmetric bulk bipartition in a one-dimensional symmetry protected topological phase with the Affleck-Kennedy-Lieb-Tasaki matrix product state wave function for the odd integer spin chains, a bulk critical entanglement spectrum can be obtained, describing the excitation spectrum of the critical point separating the topological phase from the trivial phase with the same symmetry. Such a critical point is beyond the standard Landau-Ginzburg-Wilson paradigm for symmetry breaking phase transitions. Recently, the framework of matrix product states for topological phases with Ma…
Gaussian fermionic matrix product states (GfMPS) form a class of ansatz quantum states for 1d systems of noninteracting fermions. We show, for a simple critical model of free hopping fermions, that: (i) any GfMPS approximation to its ground state must have bond dimension scaling superpolynomially with the system size, whereas (ii) there exists a non-Gaussian fermionic MPS approximation to this state with polynomial bond dimension. This proves that, in general, imposing Gaussianity at the level of the tensor network may significantly alter its capability to efficiently approximate critical Gaus…
Approximation theorems for fermionic Gaussian states
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）；Department of Physics, University of Helsinki（赫尔辛基大学物理系）；HUN-REN Wigner Research Centre for Physics（匈牙利研究与创新网络维格纳物理研究中心）；Algorithmiq Ltd（Algorithmiq有限公司）；Center for Theoretical Physics – a Leinweber Institute, Massachusetts Institute of Technology（麻省理工学院莱因韦伯理论物理中心）；Leinweber Institute for Theoretical Physics, Stanford University（斯坦福大学莱因韦伯理论物理研究所）；Google DeepMind（谷歌DeepMind）；Helmholtz-Zentrum Berlin für Materialien und Energie（柏林亥姆霍兹材料与能源中心）
Quantum queue-channels arise naturally in the context of buffering in quantum networks, wherein the noise suffered by the quantum states depends on the time spent waiting in the buffer. It has been shown that the upper-bound on the classical capacity of an additive queue-channel has a simple expression, and is achievable for the erasure and depolarizing channels [IEEE JSAIT, 1(2):432-444]. In this paper, we characterise the classical capacity for the class of unital qubit queue-channels, and show that a simple product (non-entangled) decoding strategy is capacity-achieving. As an intermediate …
Perimeter Institute for Theoretical Physics（佩里尔研究所）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；Dahlem Center for Complex Quantum Systems, Freie Universität Berlin（柏林自由大学达勒姆复杂量子系统中心）；Department of Applied Mathematics, University of Waterloo（滑铁卢大学应用数学系）；Department of Combinatorics and Optimization, University of Waterloo（滑铁卢大学组合优化系）
