The problem of conditions on the initial correlations between the system and the environment that lead to completely positive (CP) or not-completely positive (NCP) maps has been studied by various authors. Two lines of study may be discerned: one concerned with families of initial correlations that induce CP dynamics under the application of an arbitrary joint unitary on the system and environment; the other concerned with specific initial states that may be highly entangled. Here we study the latter problem, and highlight the interplay between the initial correlations and the unitary applied. In particular, for almost any initial entangled state, one can furnish infinitely many joint unitaries that generate CP dynamics on the system. Restricting to the case of initial, pure entangled states, we obtain the scaling of the dimension of the set of these unitaries and show that it is of zero measure in the set of all possible interaction unitaries.
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 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 …
We define the disentangling power of a unitary operator in a similar way as the entangling power defined by Zanardi, Zalka and Faoro [PRA, 62, 030301]. A general formula is derived and it is shown that both quantities are directly proportional. All results concerning the entangling power can simply be translated into similar statements for the disentangling power. In particular, the disentangling power is maximal for certain permutations derived from orthogonal latin squares. These permutations can therefore be interpreted as those that distort entanglement in a maximal way.
The notion of entangling power of unitary matrices was introduced by Zanardi, Zalka and Faoro [PRA, 62, 030301]. We study the entangling power of permutations, given in terms of a combinatorial formula. We show that the permutation matrices with zero entangling power are, up to local unitaries, the identity and the swap. We construct the permutations with the minimum nonzero entangling power for every dimension. With the use of orthogonal latin squares, we construct the permutations with the maximum entangling power for every dimension. Moreover, we show that the value obtained is maximum over…
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…
Maximally entangled bipartite unitary operators or gates find various applications from quantum information to being building blocks of minimal models of many-body quantum chaos, and have been referred to as "dual unitaries". Dual unitary operators that can create the maximum average entanglement when acting on product states have to satisfy additional constraints. These have been called "2-unitaries" and are examples of perfect tensors that can be used to construct absolutely maximally entangled states of four parties. Hitherto, no systematic method exists, in any local dimension, which resul…
We investigate the entangling capacity of dynamical operations when provided with local ancilla. A comparison is made between the entangling capacity with and without the assistance of prior entanglement. An analytic solution is found for the log-negativity entangling capacity of two-qubit gates, which equals the entanglement of the Choi matrix isomorphic to the unitary operator. Surprisingly, the availability of prior entanglement does not affect this result; a property we call resource independence of the entangling capacity. We prove several useful upper-bounds on the entangling capacity th…
In this work we extend the quantum channel detection method developed in [Phys. Rev. A 88, 042335 (2013)] and [Phys. Script. T153, 014044 (2013)] in order to detect other interesting convex sets of quantum channels. First we work out a procedure to detect non completely co-positive maps. Then we focus on the set of so-called bi-entangling operations and show how a map outside this set can be revealed. In both cases we provide explicit examples showing the theoretical technique and the corresponding experimental procedure.
We present a random-matrix analysis of the entangling power of a unitary operator as a function of the number of times it is iterated. We consider unitaries belonging to the circular ensembles of random matrices (CUE or COE) applied to random (real or complex) non-entangled states. We verify numerically that the average entangling power is a monotonic decreasing function of time. The same behavior is observed for the "operator entanglement" --an alternative measure of the entangling strength of a unitary. On the analytical side we calculate the CUE operator entanglement and asymptotic values f…
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.
Any evolution described by a completely positive trace-preserving linear map can be imagined as arising from the interaction of the evolving system with an initially uncorrelated ancilla. The interaction is given by a joint unitary operator, acting on the system and the ancilla. Here we study the properties such a unitary operator must have in order to force the choice of a physical- that is, positive-state for the ancilla if the end result is to be a physical-that is, completely positive-evolution of the system.
The information obtained from the operation of a quantum gate on only two complementary sets of input states is sufficient to estimate the quantum process fidelity of the gate. In the case of entangling gates, these conditions can be used to predict the multi qubit entanglement capability from the fidelities of two non-entangling local operations. It is then possible to predict highly non-classical features of the gate such as violations of local realism from the fidelities of two completely classical input-output relations, without generating any actual entanglement.
It is widely spread in the literature that non-Markovianity (NM) may be regarded as a resource in quantum mechanics. However, it is still unclear how and when this alleged resource may be exploited. Here, we study the relationship between NM and quantum optimal control under the objective of generating entanglement within M non-interacting subsystems, each one coupled to the same non-Markovian environment. Thus, we design a variety of entangling protocols that are only achievable due to the existence of the environment. We show that NM plays a crucial role in all the entangling protocols consi…
We show that randomly choosing the matrices in a completely positive map from the unitary group gives a quantum expander. We consider Hermitian and non-Hermitian cases, and we provide asymptotically tight bounds in the Hermitian case on the typical value of the second largest eigenvalue. The key idea is the use of Schwinger-Dyson equations from lattice gauge theory to efficiently compute averages over the unitary group.
The problem of the estimation of multiple phases (or of commuting unitaries) is considered. This is a sub-model of the estimation of a completely unknown unitary operation where it has been shown in recent works that there are considerable improvements by using entangled input states and entangled measurements. Here it is shown that when estimating commuting unitaries, there is practically no advantage in using entangled input states or entangled measurements.
We consider two independent bosonic oscillators immersed in a common bath, evolving in time with a completely positive, markovian, quasi-free (Gaussian) reduced dynamics. We show that an initially separated Gaussian state can become entangled as a result of a purely noisy mechanism. In certain cases, the dissipative dynamics allows the persistence of these bath induced quantum correlations even in the asymptotic equilibrium state.
We give analytical solutions for the time-optimal synthesis of entangling gates between indirectly coupled qubits 1 and 3 in a linear spin chain of three qubits subject to an Ising Hamiltonian interaction with equal coupling $J$ plus a local magnetic field acting on the intermediate qubit. The energy available is fixed, but we relax the standard assumption of instantaneous unitary operations acting on single qubits. The time required for performing an entangling gate which is equivalent, modulo local unitary operations, to the $\mathrm{CNOT}(1, 3)$ between the indirectly coupled qubits 1 and 3…
We study the quantum entanglement caused by unitary operators that have classical limits that can range from the near integrable to the completely chaotic. Entanglement in the eigenstates and time-evolving arbitrary states is studied through the von Neumann entropy of the reduced density matrices. We demonstrate that classical chaos can lead to substantially enhanced entanglement. Conversely, entanglement provides a novel and useful characterization of quantum states in higher dimensional chaotic or complex systems. Information about eigenfunction localization is stored in a graded manner in t…
We consider multiple collisions of quantum wave packets in one dimension. The system under investigation consists of an impenetrable wall and of two hard-core particles with very different masses. The lighter particle bounces between the heavier one and the wall. Both particles are initially represented by narrow Gaussian wave packets. A complete analytical solution of this problem is presented on the basis of a new method. The idea of the method is to decompose the two-particle wave function into a continuous superposition of terms (channels), such that the multiple collisions within each cha…
We classify the completely-positive maps acting on two $d$-dimensional systems which commute with all $U\otimes U$ unitaries, where $U\in SU(d)$. This set of operations map Werner states to Werner states. We find a simple condition for a map being implementable by stochastic local operations and classical communication (SLOCC). We show that all PPT-preserving maps can be implemented by SLOCC. This can be used to prove that the entanglement of Werner states cannot be stochastically increased, even if we allow PPT entanglement for free.
We explore a general feature of the interaction mediated by the gravitational fields of spatially superposed masses. For this purpose, based on quantum information theory, we characterize the evolution of two particles each in a superposition state of paths. The evolution is assumed to be given by a completely positive trace-preserving (CPTP) map. We further assume that the probability of particle being on each path is unchanged during the evolution. This property is called population-preserving. We examine when a population-preserving CPTP map can create entanglement in terms of separable ope…
We investigate the quantum nature of gravity in terms of the coherence of quantum objects. As a basic setting, we consider two gravitating objects each in a superposition state of two paths. The evolution of objects is described by the completely positive and trace-preserving (CPTP) map with a population-preserving property. This property reflects that the probability of objects being on each path is preserved. We use the $\ell_1$-norm of coherence to quantify the coherence of objects. In the present paper, the quantum nature of gravity is characterized by an entangling map, which is a CPTP ma…
A quantum system composed of two or more subsystems can be in an entangled state, i.e. a state in which the properties of the global system are well defined but the properties of each subsystem are not. Entanglement is at the heart of quantum physics, both for its conceptual foundations and for applications in information processing and quantum communication. Remarkably, entanglement can be "swapped": if one prepares two independent entangled pairs A1-A2 and B1-B2, a joint measurement on A1 and B1 (called a "Bell-State Measurement", BSM) has the effect of projecting A2 and B2 onto an entangled…
When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state's projective Hilbert space. A correction term to this geometric phase in addition to the local subsystem phases may appear from correlations between the subsystems. We find this correction term can be characterized completely either by the entanglement or completely by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ st…
We obtain new types of exponential decay laws for solutions of density-matrix master equations in the weak-coupling limit: after comparing with results already present in the literature and developing the necessary techniques, we study the crucial aspect of complete positivity under fairly general conditions. We propose a new type of time average that guarantees complete positivity and approximates, in markovian fashion, the exact dynamics for a plethora of physical applications, no matter which are the spectral properties of the subsystem, or its dimensions. We shall comment on some interesti…
An evolution between a system and its environment which leads to pure dephasing of the system may either be a result of entanglement building up between the system and the environment or not (the second option is only possible for initially mixed environmental states). We find a way of distinguishing between an entangling and non-entangling evolutions for systems which are larger than a single qubit. The generalization of the single qubit separability criterion to larger systems is not sufficient to make this distinction (it constitutes a necessary condition of separability). A set of addition…
One way to diagnose chaos in bipartite unitary channels is via the tripartite information of the corresponding Choi state, which for certain choices of the subsystems reduces to the negative conditional mutual information (CMI). We study this quantity from a quantum information-theoretic perspective to clarify its role in diagnosing scrambling. When the CMI is zero, we find that the channel has a special normal form consisting of local channels between individual inputs and outputs. However, we find that arbitrarily low CMI does not imply arbitrary proximity to a channel of this form, although…
We analyze an unusual class of bosonic dynamical instabilities that arise from dissipative (or non-Hermitian) pairing interactions. We show that, surprisingly, a completely stable dissipative pairing interaction can be combined with simple hopping or beam-splitter interactions (also stable) to generate instabilities. Further, we find that the dissipative steady state in such a situation remains completely pure up until the instability threshold (in clear distinction from standard parametric instabilities). These pairing-induced instabilities also exhibit an extremely pronounced sensitivity to …
In this paper we initiate the study of entanglement-breaking (EB) superchannels. These are processes that always yield separable maps when acting on one side of a bipartite completely positive (CP) map. EB superchannels are a generalization of the well-known EB channels. We give several equivalent characterizations of EB supermaps and superchannels. Unlike its channel counterpart, we find that not every EB superchannel can be implemented as a measure-and-prepare superchannel. We also demonstrate that many EB superchannels can be superactivated, in the sense that they can output non-separable c…
Most states in the Hilbert space are maximally entangled. This fact has proven useful to investigate - among other things - the foundations of statistical mechanics. Unfortunately, most states in the Hilbert space of a quantum many body system are not physically accessible. We define physical ensembles of states by acting on random factorized states by a circuit of length k of random and independent unitaries with local support. We study the typicality of entanglement by means of the purity of the reduced state. We find that for a time k=O(1) the typical purity obeys the area law. Thus, the up…
Continuously monitoring the environment of a quantum many-body system reduces the entropy of (purifies) the reduced density matrix of the system, conditional on the outcomes of the measurements. We show that, for mixed initial states, a balanced competition between measurements and entangling interactions within the system can result in a dynamical purification phase transition between (i) a phase that locally purifies at a constant system-size-independent rate, and (ii) a "mixed" phase where the purification time diverges exponentially in the system size. The residual entropy density in the m…
