For any quantum algorithm operating on pure states we prove that the presence of multi-partite entanglement, with a number of parties that increases unboundedly with input size, is necessary if the quantum algorithm is to offer an exponential speed-up over classical computation. Furthermore we prove that the algorithm can be classically efficiently simulated to within a prescribed tolerance ηeven if a suitably small amount of global entanglement (depending on η) is present. We explicitly identify the occurrence of increasing multi-partite entanglement in Shor's algorithm. Our results do not apply to quantum algorithms operating on mixed states in general and we discuss the suggestion that an exponential computational speed-up might be possible with mixed states in the total absence of entanglement. Finally, despite the essential role of entanglement for pure state algorithms, we argue that it is nevertheless misleading to view entanglement as a key resource for quantum computational power.
Using the approach offered by quantum speed limit, we show that geometric measure of multipartite entanglement for pure states [Phys. Rev. A 68, 042307(2003)] can be interpreted as the minimal time necessary to unitarily evolve a given quantum state to a separable one.
Entanglement speeds up evolution of a pure bipartite spin state, in line with the time energy uncertainty. However if the state is mixed this is not necessarily the case. We provide a counter example and point to other factors affecting evolution in mixed states, including classical correlations and entropy.
We investigate the issue of speed-up and the necessity of entanglement in Grover's quantum search algorithm. We find that in a pure state implementation of Grover's algorithm entanglement is present even though the initial and target states are product states. In pseudo-pure state implementations, the separability of the states involved defines an entanglement boundary in terms of a bound on the purity parameter. Using this bound we investigate the necessity of entanglement in quantum searching for these pseudo-pure state implementations. If every active molecule involved in the ensemble is `c…
Batle et al. [Phys. Rev. A {\bf 72}, 032337 (2005)] and Borrás et al. [Phys. Rev. A {\bf 74}, 022326 (2006)] studied the connection between entanglement and speed of quantum evolution for certain low-dimensional bipartite quantum states. However, their studies did not cover all possible cases. And the relation between entanglement and the maximum possible quantum evolution speed for these uncovered cases can very different from the ones that they have studied.
We analyze the role played by entanglement in the dynamical evolution of Grover's search algorithm in the space of qubits. We show that the algorithm can be equivalently described as an iterative change of the entanglement between the qubits, which governs the evolution of the initial state towards the target state, and where the entanglement can be quantified in terms a single entanglement monotone. We also provide a necessary and sufficient condition for the quadratic speedup, which illustrates how the change in the bipartite entanglement of the state after each iteration determines the corr…
The "speed" of unitary quantum evolution was recently shown to be connected to entanglement in multipartite quantum systems. Here, we discuss a tighter version of the Mandelstam-Tamm uncertainty relation that depends on the Fisher information. The passage time is estimated by a lower bound that depends inversely proportional to the square root of the Fisher information. This leads to a better understanding of the origin of a fast quantum time evolution of entangled states.
It has been recently pointed out [V. Giovanetti, S. Lloyd, and L. Maccone, Europhys. Lett., {\bf 62} pp. 615-621 (2003)] that, for certain classes of states, quantum entanglement enhances the "speed" of evolution of composite quantum systems, as measured by the time a given initial state requires to evolve to an orthogonal state. We provide here a systematic study of this effect for pure states of bipartite systems of low dimensionality, considering both distinguishable (two-qubits) subsystems, and systems constituted of two indistinguishable particles.
Entanglement of quantum variables is usually thought to be a prerequisite for obtaining quantum speed-ups of information processing tasks such as searching databases. This paper presents methods for quantum search that give a speed-up over classical methods, but that do not require entanglement. These methods rely instead on interference to provide a speed-up. Search without entanglement comes at a cost: although they outperform analogous classical devices, the quantum devices that perform the search are not universal quantum computers and require exponentially greater overhead than a quantum …
In the present paper, quantum speed limit (QSL) time of a bipartite V-type three-level atomic system under the effect of two-qutrit entanglement is investigated. Each party interacts with own independent reservoir. By considering two local unitarily equivalent Wener states and the Horodecki PPT state, as initial states, the QSL time is evaluated for each of them in the respective entangled regions. It is counterintu-itively observe that the effect of entanglement on the QSL time driven from each of the initial Werner states are completely different when the degree of non-Markovianity is consid…
The laws of quantum mechanics allow for the distribution of a secret random key between two parties. Here we analyse the security of a protocol for establishing a common secret key between N parties (i.e. a conference key), using resource states with genuine N-partite entanglement. We compare this protocol to conference key distribution via bipartite entanglement, regarding the required resources, achievable secret key rates and threshold qubit error rates. Furthermore we discuss quantum networks with bottlenecks for which our multipartite entanglement-based protocol can benefit from network c…
We investigate the speed of disentanglement in the multiqubit systems under the local depolarizing channel, in which each qubit is independently coupled to the environment. We focus on the bipartition entanglement between one qubit and the remaining qubits constituting the system, which is measured by the negativity. For the two-qubit system, the speed for the pure state completely depends on its entanglement. The upper and lower bounds of the speed for arbitrary two-qubit states, and the necessary conditions for a state achieving them, are obtained. For the three-qubit system, we study the sp…
In a quantum computation with pure states, the generation of large amounts of entanglement is known to be necessary for a speedup with respect to classical computations. However, examples of quantum computations with mixed states are known, such as the deterministic computation with one quantum qubit (DQC1) model [Knill and Laflamme, Phys. Rev. Lett. 81, 5672 (1998)], in which entanglement is at most marginally present, and yet a computational speedup is believed to occur. Correlations, and not entanglement, have been identified as a necessary ingredient for mixed-state quantum computation spe…
The concept of quantum speed limit-time (QSL) was initially introduced as a lower bound to the time interval that a given initial state $ψ_I$ may need so as to evolve into a state orthogonal to itself. Recently [V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. A {\bf 67}, 052109 (2003)] this bound has been generalized to the case where $ψ_I$ does not necessarily evolve into an orthogonal state, but into any other $ψ_F$. It was pointed out that, for certain classes of states, quantum entanglement enhances the evolution "speed" of composite quantum systems. In this work we provide an exhaust…
Entanglement is closely related to some fundamental features of the dynamics of composite quantum systems: quantum entanglement enhances the "speed" of evolution of certain quantum states, as measured by the time required to reach an orthogonal state. The concept of "speed" of quantum evolution constitutes an important ingredient in any attempt to determine the fundamental limits that basic physical laws impose on how fast a physical system can process or transmit information. Here we explore the relationship between entanglement and the speed of quantum evolution in the context of the quantum…
Recent work has studied entanglement between the spin and momentum components of a single spin-1/2 particle and showed that maximal entanglement is obtained only when boosts approach the speed of light. Here we extend the boost scenario to general geometries and show that, intriguingly, maximal entanglement can be achieved with boosts less than the speed of light. Boosts approaching the speed of light may even decrease entanglement. We also provide a geometric explanation for this behavior.
In this paper I argue that entanglement is a necessary component for any explanation of quantum speedup and I address some purported counter-examples that some claim show that the contrary is true. In particular, I address Biham et al.'s mixed-state version of the Deutsch-Jozsa algorithm, and Knill & Laflamme's deterministic quantum computation with one qubit (DQC1) model of quantum computation. I argue that these examples do not demonstrate that entanglement is unnecessary for the explanation of quantum speedup, but that they rather illuminate and clarify the role that entanglement does play.
We have studied dynamical properties of an exactly solvable quantum coupled double-well (DW) systems with Razavy's hyperbolic potential. With the use of four kinds of initial wavepackets, the correlation function $Γ(t)$ and the concurrence $C(t)$ which is a typical measure of the entanglement in two qubits, are calculated. We obtain the orthogonality time $τ$ which signifies a time interval for an initial state to evolve to its orthogonal state, and the temporal average of $C(t)$, $C_{av}$ $(=\sqrt{\langle C(t)^2 \rangle})$. The coupling dependence of $τ$ and the concurrence [$C_{av}$ or $C(0)…
We consider the simplest identical-fermion system that exhibits the phenomenon of entanglement (beyond exchange correlations) to analyze its speed of evolution towards an orthogonal state, and revisit the relation between this latter and the amount of fermionic entanglement. A characterization of the quantum speed limit and the orthogonality times is performed, throwing light into the general structure of the faster and the slower states. Such characterization holds not only for fermionic composites, but apply more generally to a wide family of 6-dimensional states, irrespective of the specifi…
The one clean qubit model of quantum computation (DQC1) efficiently implements a computational task that is not known to have a classical alternative. During the computation, there is never more than a small but finite amount of entanglement present, and it is typically vanishingly small in the system size. In this paper, we demonstrate that there is nothing unexpected hidden within the DQC1 model -- Grover's Search, when acting on a mixed state, provably exhibits a speed-up over classical with guarantees as to the presence of only vanishingly small amounts of quantum correlations (entanglemen…
Quantum Fisher Information as the Speed Limit for Multipartite Entanglement
Argonne National Laboratory（阿贡国家实验室）；Stevens Institute of Technology（史蒂文斯理工学院）；Southern Illinois University（南伊利诺伊大学）；Center for Nanoscale Materials, Argonne National Laboratory（阿贡国家实验室纳米材料中心）；Indian Institute of Science Education and Research Thiruvananthapuram（印度科学教育研究 institute 特里凡得琅）
We use a noisy signal with finite correlation time to drive a spin (dissipative impurity) in the quantum XY spin chain and calculate the dynamics of entanglement entropy of a bipartition of spins, for a stochastic quantum trajectory. We compute the noise averaged entanglement entropy of a bipartition of spins and observe that its speed of spreading decreases at strong dissipation, as a result of the Zeno effect. We recover the Zeno crossover and show that noise averaged entanglement entropy can be used as a proxy for the heating and Zeno regimes. Upon increasing the correlation time of the noi…
According to the Gottesman-Knill theorem, any quantum algorithm utilising operations chosen exclusively from a particular restricted set are efficiently simulable by a classical computer. Since some of these algorithms involve entangled states, it is commonly concluded that entanglement is insufficient to enable quantum speedup. As I explain, however, the operations belonging to this set are precisely those which will never yield a violation of the Bell inequalities. Thus it should be no surprise that entangled quantum states which only undergo operations in this set are efficiently simulable …
Exact computations of polarizations correlations probabilities are carried out in QED, to the leading order, for initially polarized as well as unpolarized particles. Quite generally they are found to be speed dependent and are in clear violation of Bells inequality of Local Hidden Variables (LHV) theories. This dynamical analysis shows how speed dependent entangled states are generated. These computations, based on QED are expected to lead to new experiments on polarization correlations monitoring speed in the light of Bells theorem. The paper provides a full QED treatment of the dynamics of …
We show that the two notions of entanglement: the maximum of the geometric measure of entanglement and the maximum of the nuclear norm is attained for the same states. We affirm the conjecture of Higuchi-Sudberry on the maximum entangled state of four qubits. We introduce the notion of d-density tensor for mixed d-partite states. We show that d-density tensor is separable if and only if its nuclear norm is $1$. We suggest an alternating method for computing the nuclear norm of tensors. We apply the above results to symmetric tensors. We give many numerical examples.
Entanglement or entanglement generating interactions permit to achieve the maximum allowed speed in the dynamical evolution of a composite system, when the energy resources are distributed among subsystems. The cases of pre-existing entanglement and of entanglement-building interactions are separately addressed. The role of classical correlations is also discussed.
We argue that entanglement is the essential non-classical ingredient which provides the computational speed-up in quantum algorithms as compared to algorithms based on the processes of classical physics.
The existence of entangled quantum states gives extra power to quantum computers over their classical counterparts. Quantum entanglement shows up qualitatively at the level of two qubits. We show that if no entanglement is envolved then whatever one can do with qubits can also be done with classical optical systems. We demonstrate that the one- and the two-bit Deutsch-Jozsa algorithm does not require entanglement and can be mapped onto a classical optical scheme. It is only for three and more input bits that the DJ algorithm requires the implementation of entangling transformations and in thes…
The maximal evolution speed of a quantum system can be represented by quantum speed limit time (QSLT).We investigate QSLT of a two-qubit system passing through a correlated channel (amplitude damping, phase damping, and depolarizing).By adjusting the correlation parameter of channel and the initial entanglement,a method to accelerate the evolution speed of the system for some specific channels is proposed.It is shown that, in amplitude damping channel and depolarizing channel,QSLT may be shortened in some cases by increasing correlation parameter of the channel and initial entanglement, which …
We discuss the fundamental role of entanglement as the essential nonclassical feature providing the computational speed-up in the known quantum algorithms. We review the construction of the Fourier transform on an Abelian group and the principles underlying the fast Fourier transform algorithm. We describe the implementation of the FFT algorithm for the group of integers modulo 2^n in the quantum context, showing how the group-theoretic formalism leads to the standard quantum network and identifying the property of entanglement that gives rise to the exponential speedup (compared to the classi…
Despite intensive research, the physical origin of the speed-up offered by quantum algorithms remains mysterious. No general physical quantity, like, for instance, entanglement, can be singled out as the essential useful resource. Here we report a close connection between the trace speed and the quantum speed-up in Grover's search algorithm implemented with pure and pseudo-pure states. For a noiseless algorithm, we find a one-to-one correspondence between the quantum speed-up and the polarization of the pseudo-pure state, which can be connected to a wide class of quantum statistical speeds. Fo…
Relativistic bipartite entangled quantum states is studied to show that Nature doesn't favor nonlocality for massive particles in the ultra-relativistic limit. We found that to an observer (Bob) in a moving frame S', the entangled Bell state shared by Alice and Bob appears as the superposition of the Bell bases in the frame S' due to the requirement of the special relativity. It is shown that the entangled pair satisfies the Bell's inequality when the boost speed approaches the speed of light, thus providing a counter example for nonlocality of Einstein-Podolsky-Rosen(EPR) paradox.
'Tis said, to know others is to be learned, to know oneself, wise - I demonstrate that it could be more fundamental than knowing the rest of nature, by applying classical computational principles and engineering hindsight to derive and explain quantum entanglement, state space formalism and the statistical nature of quantum mechanics. I show that an entangled photon pair is literally no more than a 1-bit hologram, that the quantum state formalism is completely derivable from general considerations of representation of physical information, and that both the probabilistic aspects of quantum the…
Recently, a new method, which can significantly speed up adiabatic quantum state transfer by using dressed states, was proposed by Baksic \emph{et~al.} [Phys. Rev. Lett. \textbf{116}, 230503 (2016)]. Assisted by quantum Zeno dynamics, we develop this dressed-state method to achieve shortcuts to complete and fractional stimulated Raman adiabatic passage for speeding up adiabatic two-atom quantum state transfer and maximum entanglement generation, respectively. By means of some numerical simulations, we determine the parameters used in the scheme which can guarantee the feasibility and efficienc…
Computational entanglement distillation beyond pure states: efficient protocols and fundamental limitations
The University of Tokyo（东京大学）；Freie Universität Berlin（柏林自由大学）；Università degli Studi di Salerno（萨莱诺大学）；Istituto Nazionale di Fisica Nucleare (INFN)（意大利国家核物理研究所）
We discuss time displaced entanglement, produced by taking one member of a Bell pair on a round trip at relativistic speeds, thus inducing a time-shift between the pair. We show that decoherence with respect to Bell measurements on the pair is predicted. We then study a teleportation protocol, using time displaced entanglement as its resource, in which a time-like loop is apparently formed. The result is non-unitary, non-linear evolution of the teleported state.
We study entanglement harvesting for matter systems such as atoms, ions or molecules, whose center of mass degrees of freedom are quantum delocalized and which couple to a relativistic quantum field. We employ a generalized Unruh-deWitt detector model for the light-matter interaction, and we investigate how the coherent spreading of the quantum center of mass wave function of two delocalized detector systems impacts their ability to become entangled with one another, via their respective interaction with a quantum field. For very massive detectors with initially highly localized centers of mas…
Entanglement is widely believed to lie at the heart of the advantages offered by a quantum computer. This belief is supported by the discovery that a noiseless (pure) state quantum computer must generate a large amount of entanglement in order to offer any speed up over a classical computer. However, deterministic quantum computation with one pure qubit (DQC1), which employs noisy (mixed) states, is an efficient model that generates at most a marginal amount of entanglement. Although this model cannot implement any arbitrary algorithm it can efficiently solve a range of problems of significant…
We study dynamics of entanglement in the thermalization process of an isolated quantum many-body system. We propose a simple setup for measuring the propagation speed of entanglement entropy (EE) in numerical simulations and apply it to the integrable/non-integrable spin models in 1D - the transverse Ising (TI) model, the chaotic Ising (CI) model, and the extended chaotic Ising (ECI) model. We find that two distinct time-scales $t^\ast$ and $t_{\rm {diff}}$ arise in the dynamics of EE in the thermalization process: the former represents the time-scale for the saturation of EE and the latter ch…
The aim of this dissertation is to clarify the debate over the explanation of quantum speedup and to submit a tentative resolution to it. In particular, I argue that the physical explanation for quantum speedup is precisely the fact that the phenomenon of quantum entanglement enables a quantum computer to fully exploit the representational capacity of Hilbert space. This is impossible for classical systems, joint states of which must always be representable as product states. Chapter 2 begins with a discussion of the most popular of the candidate physical explanations for quantum speedup: the …
The speed-up provided by quantum algorithms with respect to their classical counterparts is at the origin of scientific interest in quantum computation. However, the fundamental reasons for such a speed-up are not yet completely understood and deserve further attention. In this context, the classical simulation of quantum algorithms is a useful tool that can help us in gaining insight. Starting from the study of general conditions for classical simulation, we highlight several important differences between two non-equivalent classes of quantum algorithms. We investigate their performance under…
We propose a bimetric space-time consisting of two vierbein bundles, in which a superluminal S-field mediates a causal exchange of quantum information associated with quantum entanglement. The resulting theory leaves the usual quantum field interactions, as well as the speed of light unchanged, but introduces quantum information effects related to the S-field vierbein bundle. We show that such S-field interactions with the Dirac field, affect its spin density, as we would expect for a field that would impart entanglement related information.
