We discuss upper bounds on the rate at which unitary evolution governed by a non-local Hamiltonian can generate entanglement in a bipartite system. Given a bipartite Hamiltonian H coupling two finite dimensional particles A and B, the entangling rate is shown to be upper bounded by c*log(d)*norm(H), where d is the smallest dimension of the interacting particles, norm(H) is the operator norm of H, and c is a constant close to 1. Under certain restrictions on the initial state we prove analogous upper bound for the ancilla-assisted entangling rate with a constant c that does not depend upon dimensions of local ancillas. The restriction is that the initial state has at most two distinct Schmidt coefficients (each coefficient may have arbitrarily large multiplicity). Our proof is based on analysis of a mixing rate -- a functional measuring how fast entropy can be produced if one mixes a time-independent state with a state evolving unitarily.
Entangling power and fidelity diagnostic for bipartite quantum channels
Jagiellonian University（雅盖隆大学）；University of Salerno（萨莱诺大学）；Institute of Nanotechnology of the National Research Council of Italy, CNR-NANOTEC（意大利国家研究委员会纳米技术研究所）；Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）
We compare three different characterizations of the globalness of bipartite unitary operations, namely, delocalization power, entanglement cost, and entangling power, to investigate global properties of unitary operations. We show that the globalness of the same unitary operation depends on whether input states are given by unknown states representing pieces of quantum information, or a set of known states for the characterization. We extend our analysis on the delocalization power in two ways. First, we show that the delocalization power differs whether the global operation is applied on one …
Nonlocal unitary operations can create quantum entanglement between distributed particles, and the quantification of created entanglement is a hard problem. It corresponds to the concepts of entangling and assisted entangling power when the input states are, respectively, product and arbitrary pure states. We analytically derive them for Schmidt-rank-two bipartite unitary and some complex bipartite permutation unitaries. In particular, the entangling power of permutation unitary of Schmidt rank three can take only one of two values: $\log_2 9 - 16/9$ or $\log_2 3$ ebits. The entangling power, …
It is known that any bipartite unitary operator of Schmidt rank three is equivalent to a controlled unitary under local unitaries. We propose a standard form of such operators. Using the form we improve the upper bound for the entanglement cost to implement such operators under local operations and classical communications (LOCC), and provide a corresponding protocol. A part of our protocol is based on a recursive-control protocol which is helpful for implementing other unitary operators. We show that any bipartite permutation unitary of Schmidt rank three can be implemented using LOCC and two…
We consider two capacity quantities associated with bipartite unitary gates: the entangling and the disentangling power. For two-qubit unitaries these two capacities are always the same. Here we prove that these capacities are different in general. We do so by constructing an explicit example of a qubit-qutrit unitary whose entangling power is maximal (2 ebits), but whose disentangling power is strictly less. A corollary is that there can be no unique ordering for unitary gates in terms of their ability to perform non-local tasks. Finally we show that in large dimensions, almost all bipartite …
This note proves that arbitrary local gates together with any entangling bipartite gate V are universal. Previously this was known only when access to both V and V^{-1} was given, or when approximate universality was demanded.
What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? We provide an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling n-qubit Hamiltonian and local unitaries. It follows that universal quantum computation can be performed using any entangling interaction and local unitary operations.
We characterize all maximally entangling bipartite unitary operators, acting on systems $A,B$ of arbitrary finite dimensions $d_A\le d_B$, when use of ancillary systems by both parties is allowed. Several useful and interesting consequences of this characterization are discussed, including an understanding of why the entangling and disentangling capacities of a given (maximally entangling) unitary can differ and a proof that these capacities must be equal when $d_A=d_B$.
What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? It has been shown that all two-body Hamiltonian evolutions can be simulated using \emph{any} fixed two-body entangling $n$-qubit Hamiltonian and fast local unitaries. By \emph{entangling} we mean that every qubit is coupled to every other qubit, if not directly, then indirectly via intermediate qubits. We extend this study to the case where interactions may involve more than two qubits at a time. We find necessary and sufficient conditions for an arbitrary $n$-qubit Hamiltonian to be \emph{dy…
When can a quantum system of finite dimension be used to simulate another quantum system of finite dimension? What restricts the capacity of one system to simulate another? In this paper we complete the program of studying what simulations can be done with entangling many-qudit Hamiltonians and local unitary control. By entangling we mean that every qudit is coupled to every other qudit, at least indirectly. We demonstrate that the only class of finite-dimensional entangling Hamiltonians that aren't universal for simulation is the class of entangling Hamiltonians on qubits whose Pauli operator…
We investigate the average bipartite entanglement, over all possible divisions of a multipartite system, as a useful measure of multipartite entanglement. We expose a connection between such measures and quantum-error-correcting codes by deriving a formula relating the weight distribution of the code to the average entanglement of encoded states. Multipartite entangling power of quantum evolutions is also investigated.
The space of quantum Hamiltonians has a natural partition in classes of operators that can be adiabatically deformed into each other. We consider parametric families of Hamiltonians acting on a bi-partite quantum state-space. When the different Hamiltonians in the family fall in the same adiabatic class one can manipulate entanglement by moving through energy eigenstates corresponding to different value of the control parameters. We introduce an associated notion of adiabatic entangling power. This novel measure is analyzed for general $d\times d$ quantum systems and specific two-qubits exampl…
Adiabatic quantum computation starts from embedding a computational problem into a Hamiltonian whose ground state encodes the solution to the problem. This problem Hamiltonian, $H_{\rm p}$, is normally chosen to be diagonal in the computational basis, that is a product basis for qubits. We point out that $H_{\rm p}$ can be chosen to be non-diagonal in the computational basis. To be more precise, we show how to construct $H_{\rm p}$ in such a way that all its excited states are entangled with respect to the qubit tensor product structure, while the ground state is still of the product form and …
The entangling power of a bipartite unitary operation shows the maximum created entanglement with the product input states. For an arbitrary two-qubit unitary operation, it is sufficient to consider its normalized operation $U$ with parameters $c_0,c_1,c_2$ and $c_3$. We show how to compute the entangling power of $U$ when $c_2=c_3$. In particular we construct the analytical expressions of entangling power of such $U$ for two examples. We also construct the entangling power of bipartite unitary operations of Schmidt rank two for any dimensions.
We show that intrinsic fluctuations in system control parameters impose limits on the ability of two-qubit (exchange) Hamiltonians to generate entanglement starting from mixed initial states. We find three classes for Gaussian and Laplacian fluctuations. For the Ising and XYZ models there are qualitatively distinct sharp entanglement-generation transitions, while the class of Heisenberg, XY, and XXZ Hamiltonians is capable of generating entanglement for any finite noise level. Our findings imply that exchange Hamiltonians are surprisingly robust in their ability to generate entanglement in the…
We in theory proposed a hybrid system consisting of a mechanical resonator, an optical Fabry-Pérot cavity, and two superconducting microwave circuits to generate stationary continuous-variable quantum entanglement between two microwave modes. We show that the hybrid system can also achieve quantum entanglement of other bipartite subsystems in experimentally accessible parameter regimes, which has the potential to be useful in quantum information processing and quantum illumination radar.
Distinguishability is a fundamental and operational task generally connected to information applications. In quantum information theory, from the postulates of quantum mechanics it often has an intrinsic limitation, which then dictates and also characterises capabilities of related information tasks. In this work, we consider discrimination between bipartite two-qubit unitary transformations by local operations and classical communication (LOCC) and its relations to entangling capabilities of given unitaries. We show that a pair of entangling unitaries which do not contain local parts, if they…
The three-spin-$1/2$ decoherence-free subsystem defines a logical qubit protected from collective noise and supports exchange-only universal gates. Such logical qubits are well-suited for implementation with electrically-defined quantum dots. Exact exchange-only entangling logical gates exist but are challenging to construct and understand. We use a decoupling strategy to obtain straightforward approximate entangling gates. A benefit of the strategy is that if the physical spins are aligned, then it can implement evolution under entangling Hamiltonians. Hamiltonians expressible as linear combi…
A review is given on the thermodynamical structure of bipartite entanglement. By comparing it to the axiomatic formulation of thermodynamics presented by Giles it is shown that for finite dimensional systems the two theories are formally inequivalent. The same approach is used to demonstrate the full equivalence in the asymptotic limit for pure quantum states. For mixed states a different method for obtaining the second law is described applied to two different classes of operations, PPT-preserving and asymptotically non-entangling operations.
Quantum correlations present in a broadband two-line squeezed microwave state can induce entanglement in a spatially separated bipartite system consisting of either two single qubits or two qubit ensembles. By using an appropriate master equation for a bipartite quantum system in contact with two separate but entangled baths, the generating entanglement process in spatially separated quantum systems is thoroughly characterized. Our results provide evidence that this entanglement transfer by dissipation is feasible yielding to a steady-state amount of entanglement in the bipartite quantum syste…
We provide upper bound on the maximal rate at which irreversible quantum dynamics can generate entanglement in a bipartite system. The generator of irreversible dynamics consists of a Hamiltonian and dissipative terms in Lindblad form. The relative entropy of entanglement is chosen as a measure of entanglement in an ancilla-free system. We provide an upper bound on the entangling rate which has a logarithmic dependence on a dimension of a smaller system in a bipartite cut. We also investigate the rate of change of quantum mutual information in an ancilla-assisted system and provide an upper bo…
The aim of this thesis is to advance the theory behind quantum information processing tasks, by deriving fundamental limits on bipartite quantum interactions and dynamics, which corresponds to an underlying Hamiltonian that governs the physical transformation of a two-body open quantum system. The goal is to determine entangling abilities of such arbitrary bipartite quantum interactions. Doing so provides fundamental limitations on information processing tasks, including entanglement distillation and secret key generation, over a bipartite quantum network. We also discuss limitations on the en…
Our work addresses the problem of generating maximally entangled two spin-1/2 (qubit) symmetric states using NMR, NQR, Lipkin-Meshkov-Glick Hamiltonians. Time evolution of such Hamiltonians provides various logic gates which can be used for quantum processing tasks. Pairs of spin-1/2's have modeled a wide range of problems in physics. Here we are interested in two spin-1/2 symmetric states which belong to a subspace spanned by the angular momentum basis {|j = 1, μ>; μ = +1, 0,-1}. Our technique relies on the decomposition of a Hamiltonian in terms of SU(3) generators. In this context, we defin…
Quantum correlations between identical particles are at the heart of quantum technologies. Several studies with two identical particles have shown that the spatial overlap and indistinguishability between the particles are necessary for generating bipartite entanglement. On the other hand, researches on the extension to more than two-particle systems are limited by the practical difficulty to control multiple identical particles in laboratories. In this work, we propose schemes to generate two fundamental classes of genuine tripartite entanglement, i.e., GHZ and W classes, which are experiment…
The recent experimental advances in capacitively coupled singlet-triplet qubits, particularly the demonstration of entanglement, opens the question of what type of entangling gates the system's Hamiltonian can produce directly via a single square pulse. We address this question by considering the system's Hamiltonian from first principles and using the representation of its nonlocal properties in terms of local invariants. In the analysis we include the three different ways in which the system can be biased and their effect on the generation of entangling gates. We find that, in one of the pos…
Our goal is to provide a new approach to the construction of geometry-induced entanglement between a pair of $Λ$ type atoms in a system consists of $N$ identical atoms by means of nonadiabatic quantum holonomies. By employing the quantum Zeno effect, we introduce a tripod type interaction Hamiltonian between two selected atoms trapped in an optical cavity, which allows arbitrary geometric entangling power. This would be a substantial step toward resolving the feasibility of realizing universal nonadiabatic holonomic entangling two-qubit gates.
We study families of dynamical maps generated from interactions with varying degrees of symmetry. For a family of time-independent Hamiltonians, we demonstrate the relationship between symmetry, strong-coupling, perfect entanglers, non-Markovian features, and non-time-locality. We show that by perturbing the initial environment state, effective time-local descriptions can be obtained that are non-singular yet capture essential non-unitary features of the reduced dynamics. We then consider a time-dependent Hamiltonian that changes the degree of symmetry by activating a dormant degree of freedom…
A Hamiltonian is presented, which can be used to convert any asymmetric state $|φ\rangle_{a}|ϕ\rangle_{b}$ of two oscillators $a$ and $b$ into an entangled state. Furthermore, with this Hamiltonian and local operations only, two oscillators, initially in any asymmetric initial states, can be entangled with a third oscillator. The prepared entangled states can be engineered with an arbitrary degree of entanglement. A discussion on the realization of this Hamiltonian is given. Numerical simulations show that, with current circuit QED technology, it is feasible to generate high-fidelity entangled…
What interactions are sufficient to simulate arbitrary quantum dynamics in a composite quantum system? Dodd et al. (quant-ph/0106064) provided a partial solution to this problem in the form of an efficient algorithm to simulate any desired two-body Hamiltonian evolution using any fixed two-body entangling N-qubit Hamiltonian, and local unitaries. We extend this result to the case where the component systems have D dimensions. As a consequence we explain how universal quantum computation can be performed with any fixed two-body entangling N-qudit Hamiltonian, and local unitaries.
We numerically study protocols consisting of repeated applications of two qubit gates used for generating random pure states. A necessary number of steps needed in order to generate states displaying bipartite entanglement typical of random states is obtained. For generic two qubit entangling gate the decay rate of purity is found to scale as $\sim n$ and therefore of order $\sim n^2$ steps are necessary to reach random bipartite entanglement. We also numerically identify the optimal two qubit gate for which the convergence is the fastest. Perhaps surprisingly, applying the same good two qubit…
The problem behind this paper is, if the number of queries to unitary operations is fixed, say $k$, then when do local operations and classical communication (LOCC) suffice for optimally distinguishing bipartite unitary operations? We consider the above problem for two-qubit unitary operations in the case of $k=1$, showing that for two two-qubit entangling unitary operations without local parties, LOCC achieves the same distinguishability as the global operations. Specifically, we obtain: (i) if such two unitary operations are perfectly distinguishable by global operations, then they are perfe…
Two dual questions in quantum information theory are to determine the communication cost of simulating a bipartite unitary gate, and to determine their communication capacities. We present a bipartite unitary gate with two surprising properties: 1) simulating it with the assistance of unlimited EPR pairs requires far more communication than with a better choice of entangled state, and 2) its communication capacity is far lower than its capacity to create entanglement. This suggests that 1) unlimited EPR pairs are not the most general model of entanglement assistance for two-party communication…
Bipartite Gaussian Boson Sampling for Hamiltonian Cycles in Directed Graphs
College of Computer Science and Technology, National University of Defense Technology（国防科技大学计算机学院）；College of Advanced Interdisciplinary Studies, National University of Defense Technology（国防科技大学前沿交叉学院）
Implementation of logical entangling gates is an important step towards realizing a quantum computer. We use a gradient-based optimization approach to find single-qubit rotations which can be interleaved between applications of a noisy nonlocal gate to dramatically suppress arbitrary logical errors, while steering the evolution operator towards the perfectly entangling subset of SU(4) gates. The modularity of the approach allows for application to any two-qubit system, regardless of the Hamiltonian or details of the experimental implementation. This approach is effective for both quasi-static …
Bosonic quantum error correction has proven to be a successful approach for extending the coherence of quantum memories, but to execute deep quantum circuits, high-fidelity gates between encoded qubits are needed. To that end, we present a family of error-detectable two-qubit gates for a variety of bosonic encodings. From a new geometric framework based on a "Bloch sphere" of bosonic operators, we construct $ZZ_L(θ)$ and $\text{eSWAP}(θ)$ gates for the binomial, 4-legged cat, dual-rail and several other bosonic codes. The gate Hamiltonian is simple to engineer, requiring only a programmable be…
The capacity of a quantum gate to produce entangled states on a bipartite system is quantified in terms of the entangling power. This quantity is defined as the average of the linear entropy of entanglement of the states produced after applying a quantum gate over the whole set of separable states. Here we focus on symmetric two-qubit quantum gates, acting on the symmetric two-qubit space, and calculate the entangling power in terms of the appropriate local-invariant. A geometric description of the local equivalence classes of gates is given in terms of the $\mathfrak{su}(3)$ Lie algebra root …
We propose practical and efficient protocols for verifying bipartite pure states for any finite dimension, which can also be applied to fidelity estimation. Our protocols are based on adaptive local projective measurements with either one-way or two-way communications, which are very easy to implement in practice. They can extract the key information much more efficiently than known protocols based on tomography or direct fidelity estimation, and their efficiencies are comparable to the best protocols based on entangling measurements. These protocols highlight the significance of mutually unbi…
Consider two quantum systems A and B interacting according to a product Hamiltonian H = H_A x H_B. We show that any two such Hamiltonians can be used to simulate each other reversibly (i.e., without efficiency losses) with the help of local unitary operations and local ancillas. Accordingly, all non-local features of a product Hamiltonian -- including the rate at which it can be used to produce entanglement, transmit classical or quantum information, or simulate other Hamiltonians -- depend only upon a single parameter. We identify this parameter and use it to obtain an explicit expression for…
Recently de La Torre et al. [1] reconstructed Quantum Theory from its local structure on the basis of local discriminability and the existence of a one-parameter group of bipartite transformations containing an entangling gate. This result relies on universality of an entangling gate for quantum computation. Here we prove universality of C-NOT with local gates for Real Quantum Theory (RQT), showing that such universality would not be sufficient for the result, whereas local discriminability and the qubit structure play a crucial role. For reversible computation, generally an extra rebit is nee…
The ability to characterise a Hamiltonian with high precision is crucial for the implementation of quantum technologies. In addition to the well-developed approaches utilising optimal probe states and optimal measurements, the method of optimal control can be used to identify time-dependent pulses applied to the system to achieve higher precision in the estimation of Hamiltonian parameters, especially in the presence of noise. Here, we extend optimally controlled estimation schemes for single qubits to non-commuting dynamics as well as two interacting qubits, demonstrating improvements in term…
Entanglement properties of bipartite unitary operators are studied via their local invariants, namely the entangling power $e_p$ and a complementary quantity, the gate typicality $g_t$. We characterize the boundaries of the set $K_2$ representing all two-qubit gates projected onto the plane $(e_p, g_t)$ showing that the fractional powers of the \textsc{swap} operator form a parabolic boundary of $K_2$, while the other bounds are formed by two straight lines. In this way a family of gates with extreme properties is identified and analyzed. We also show that the parabolic curve representing powe…
We consider interactions as bidirectional channels. We investigate the capacities for interaction Hamiltonians and nonlocal unitary gates to generate entanglement and transmit classical information. We give analytic expressions for the entanglement generating capacity and entanglement-assisted one-way classical communication capacity of interactions, and show that these quantities are additive, so that the asymptotic capacities equal the corresponding 1-shot capacities. We give general bounds on other capacities, discuss some examples, and conclude with some open questions.
In this paper, we introduce a discrete quantum walk model called bipartite walks. Bipartite walks include many known discrete quantum walk models, like arc-reversal walks, vertex-face walks. For the transition matrix of a quantum walk, there is a Hamiltonian associated with it. We will study the Hamiltonians of the bipartite walks. Let $S$ be a skew-symmetric matrix. We are mainly interested in the Hamiltonians of the form $iS$. We show that the Hamiltonian can be written as $iS$ if and only if the adjacency matrix of the bipartite graph is invertible. We show that arc-reversal walks and verte…
In this article we investigate the unitary dynamics of squashed entanglement and concurrence measures in Werner state and maximally entangled mixed states (MEMS) under two different Hamiltonians. The aim of the present study is two fold. The first part of the study deals with the dynamics under Heisenberg Hamiltonian and the second part deals under bi-linear bi-quadratic Hamiltonian which is the extension of first Hamiltonian. In both the parts we investigate the dynamical trade off and balancing points for squashed entanglement and concurrence. During the study, we also found the results of e…
A system of two or more quantum dots interacting with a dissipative plasmonic nanostructure is investigated in detail by using a cavity quantum electrodynamics approach with a model Hamiltonian. We focus on determining and understanding system configurations that generate multiple bipartite quantum entanglements between the occupation states of the quantum dots. These configurations include allowing for the quantum dots to be asymmetrically coupled to the plasmonic system. Analytical solution of a simplified limit for an arbitrary number of quantum dots and numerical simulations and optimizati…
We show that two interacting physical systems may admit entangled pure or non separable mixed states evolving in time as if the mutual interaction hamiltonian were absent. In this paper we define these states Interaction Free Evolving (IFE) states and characterize their existence for a generic binary system described by a time independent Hamiltonian. A comparison between IFE subspace and the decoherence free subspace is reported. The set of all pure IFE states is explicitly constructed for a non homogeneous spin star system model.
We show that entangling capacities based on the Jamiolkowski isomorphism may be used to place lower bounds on the communication capacities of arbitrary bipartite unitaries. Therefore, for these definitions, the relations which have been previously shown for two-qubit unitaries also hold for arbitrary dimensions. These results are closely related to the theory of the entanglement-assisted capacity of channels. We also present more general methods for producing ensembles for communication from initial states for entanglement creation.
We consider multi-qubit systems and relate quantitatively the problems of generating cluster states with high value of concurrence of assistance, and that of generating states with maximal bipartite entanglement. We prove an upper bound for the concurrence of assistance. We consider dynamics of spin-1/2 systems that model qubits, with different couplings and possible presence of magnetic field to investigate the appearance of the discussed entanglement properties. We find that states with maximal bipartite entanglement can be generated by an XY Hamiltonian, and their generation can be controll…
The notion of capacity of entanglement is the quantum information theoretic counterpart of the heat capacity which is defined as the second cumulant of the entanglement spectrum. Given any bipartite pure state, we can define the capacity of entanglement as the variance of the modular Hamiltonian in the reduced state of any of the subsystems. Here, we study the dynamics of this quantity under non-local Hamiltonian. Specifically, we address the question: Given an arbitrary non-local Hamiltonian what is the capacity of entanglement that the system can possess? As an useful application, we show th…
We study bipartite unitary operators which stay invariant under the local actions of diagonal unitary and orthogonal groups. We investigate structural properties of these operators, arguing that the diagonal symmetry makes them suitable for analytical study. As a first application, we construct large new families of dual unitary gates in arbitrary finite dimensions, which are important toy models for entanglement spreading in quantum circuits. We then analyze the non-local nature of these invariant operators, both in discrete (operator Schmidt rank) and continuous (entangling power) settings. …
