We prove that the entangling capacity of a two-qubit unitary operator without local ancillas, both with and without the restriction to initial product states, as quantified by the maximum attainable concurrence, is directly related to the distinguishability of a closely related pair of two-qubit unitary operators. These operators are the original operator transformed into its canonical form and the adjoint of this canonical form. The distinguishability of these operators is quantified by the minimum overlap of the output states over all possible input probe states. The entangling capacity of the original unitary operator is therefore directly related to the degree of non-Hermiticity of its canonical form, as quantified in an operationally satisfactory manner in terms of the extent to which it can be distinguished, by measurement, from its adjoint. Furthermore, the maximum entropy of entanglement, again without local ancillas, that a given two-qubit unitary operator can generate, is found to be closely related to the classical capacities of certain quantum channels.
The problem of distinguishing two unitary transformations, or quantum gates, is analyzed and a function reflecting their statistical distinguishability is found. Given two unitary operations, $U_1$ and $U_2$, it is proved that there always exists a finite number $N$ such that $U_1^{\otimes N}$ and $U_2^{\otimes N}$ are perfectly distinguishable, although they were not in the single-copy case. This result can be extended to any finite set of unitary transformations. Finally, a fidelity for one-qubit gates, which satisfies many useful properties from the point of view of quantum information theo…
We show that any two different unitary operations acting on an arbitrary multipartite quantum system can be perfectly distinguishable by local operations and classical communication when a finite number of runs is allowed. We then directly extend this result into the case when the number of unitary operations to be discriminated is more than two. Intuitively, our result means that the lost identity of a nonlocal (entangled) unitary operation can be recovered locally, without any use of entanglement or joint quantum 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 $d_A=d_B$.
The study of quantum systems evolving from initial states to distinguishable, orthogonal final states is important for information processing applications such as quantum computing and quantum metrology. However, for most unitary evolutions and initial states the system does not evolve to an orthogonal quantum state. Here, we ask what proportion of quantum states evolves to nearly orthogonal systems as a function of the dimensionality of the Hilbert space of the system, and numerically study the evolution of quantum states in low-dimensional Hilbert spaces. We find that, as well as the speed o…
A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output $p$-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is …
We study the entangling properties of multipartite unitary gates with respect to the measure of entanglement called one-tangle. Putting special emphasis on the case of three parties, we derive an analytical expression for the entangling power of an $n$-partite gate as an explicit function of the gate, linking the entangling power of gates acting on $n$-partite Hilbert space of dimension $d_1 \ldots d_n$ to the entanglement of pure states in the Hilbert space of dimension $(d_1 \ldots d_n)^2$. Furthermore, we evaluate its mean value averaged over the unitary and orthogonal groups, analyze the m…
We study the entangling power and perfect entangler nature of (SWAP)1/m, for m>=1, and controlled unitary (CU) gates. It is shown that (SWAP)1/2 is the only perfect entangler in the family. On the other hand, a subset of CU which is locally equivalent to CNOT is identified. It is shown that the subset, which is a perfect entangler, must necessarily possess the maximum entangling power.
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 …
Entangling and disentangling capacities are the key manifestation of the nonlocal content of a quantum operation. A lot of effort has been put recently into investigating (dis)entangling capacities of unitary operations, but very little is known about capacities of non-unitary operations. Here we investigate (dis)entangling capacities of unital CPTP maps acting on two qubits.
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 …
By using an ancilla qubit as a mediator, two distant qubits can undergo a non-local entangling unitary operation. This is desirable for when attempting to scale up or distribute quantum computation by combining fixed static local sets of qubits with ballistic mediators. Using a model driven by measurements on the ancilla, it is possible to generate a maximally entangling CZ gate while only having access to a less entangling gate between the pair qubits and the ancilla. However this results in a stochastic process of generating control phase rotation gates where the expected time for success do…
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…
Based on the matrix realignment and partial transpose, we develop an approach to entangling power and operator entanglement of quantum unitary operators. We demonstrate efficiency of the approach by studying several unitary operators on qudits, and indicate that these two matrix rearrangements are not only powerful for studying separability problem of quantum states, but also useful in studying entangling capabilities of quantum operators.
We analyze quantum states formed as superpositions of an initial pure product state and its image under local unitary evolution, using two measurement-based measures of superposition size: one based on the optimal quantum binary distinguishability of the branches of the superposition and another based on the ratio of the maximal quantum Fisher information of the superposition to that of its branches, i.e., the relative metrological usefulness of the superposition. A general formula for the effective sizes of these states according to the branch distinguishability measure is obtained and applie…
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 establish the entangling power of a unitary operator on a general finite-dimensional bipartite quantum system with and without ancillas, and give relations between the entangling power based on the von Neumann entropy and the entangling power based on the linear entropy. Significantly, we demonstrate that the entangling power of a general controlled unitary operator acting on two equal-dimensional qudits is proportional to the corresponding operator entanglement if linear entropy is adopted as the quantity representing the degree of entanglement. We discuss the entangling power and operator…
We consider the question of perfect local distinguishability of mutually orthogonal bipartite quantum states, with the property that every state can be specified by a unitary operator acting on the local Hilbert space of Bob. We show that if the states can be exactly discriminated by one-way LOCC where Alice goes first, then the unitary operators can also be perfectly distinguished by an orthogonal measurement on Bob's Hilbert space. We give examples of sets of N<=d maximally entangled states in $d \otimes d $ for d=4,5,6 that are not perfectly distinguishable by one-way LOCC. Interestingly fo…
The deterministic quantum computing with one qubit (DQC1) is a mixed-state quantum computation algorithm that evaluates the normalized trace of a unitary matrix and is more powerful than the classical counterpart. We find that the normalized trace of the unitary matrix can be directly described by the entangling power of the quantum circuit of the DQC1, so the nontrivial DQC1 is always accompanied with the non-vanishing entangling power. In addition, it is shown that the entangling power also determines the intrinsic complexity of this quantum computation algorithm, i.e., the larger entangling…
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, …
Optimal control techniques are applied for the decomposition of unitary quantum operations into a sequence of single-qubit gates and entangling operations. To this end, we modify a gradient-ascent algorithm developed for systems of coupled nuclear spins in molecules to make it suitable for trapped ion quantum computing. We decompose unitary operations into entangling gates that are based on a nonlinear collective spin operator and complemented by global spinflip and local light shift gates. Among others, we provide explicit decompositions of controlled-NOT and Toffoli gates, and a simple quant…
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…
For both unitary and open qubit dynamics, we compare asymmetry monotone-based bounds on the minimal time required for an initial qubit state to evolve to a final qubit state from which it is probabilistically distinguishable with fixed minimal error probability (i.e., the minimal error distinguishability time). For the case of unitary dynamics generated by a time-independent Hamiltonian, we derive a necessary and sufficient condition on two asymmetry monotones that guarantees that an arbitrary state of a two-level quantum system or a separable state of $N$ two-level quantum systems will unitar…
The computational problem of distinguishing two quantum channels is central to quantum computing. It is a generalization of the well-known satisfiability problem from classical to quantum computation. This problem is shown to be surprisingly hard: it is complete for the class QIP of problems that have quantum interactive proof systems, which implies that it is hard for the class PSPACE of problems solvable by a classical computation in polynomial space. Several restrictions of distinguishability are also shown to be hard. It is no easier when restricted to quantum computations of logarithmic d…
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…
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.
A $2\otimes 2$ unitary operation is called a perfect entangler if it can generate a maximally entangled state from some unentangled input. We study the following question: How many runs of a given two-qubit entangling unitary operation is required to simulate some perfect entangler with one-qubit unitary operations as free resources? We completely solve this problem by presenting an analytical formula for the optimal number of runs of the entangling operation. Our result reveals an entanglement strength of two-qubit unitary operations.
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 quantum baker map possesses two symmetries: a canonical "spatial" symmetry, and a time-reversal symmetry. We show that, even when these features are taken into account, the asymptotic entangling power of the baker's map does not always agree with the predictions of random matrix theory. We have verified that the dimension of the Hilbert space is the crucial parameter which determines whether the entangling properties of the baker are universal or not. For power-of-two dimensions, i.e., qubit systems, an anomalous entangling power is observed; otherwise the behavior of the baker is consiste…
It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate-typicality and study its properties. Analyzing multiple applications of any entangling operator interlaced with random local gates, we prove that both investigated quantities approach t…
We study the entanglement of unitary operators on $d_1\times d_2$ quantum systems. This quantity is closely related to the entangling power of the associated quantum evolutions. The entanglement of a class of unitary operators is quantified by the concept of concurrence.
We consider spin-chain-star systems characterized by N-wise many-body interactions between the spins in each chain and the central one. We show that such systems can be exactly mapped into standard spin-star systems through unitary transformations. Such an approach allows the solution of the dynamic problem of an XX spin-chain-star model and transparently shows the emergence of quantum correlations in the system, based on the idea of entanglement between chains.
We draw attention to the existence of "genuine" fidelity gaps in an ancilla-assisted sequential decomposition of genuinely entangling isometry and unitary operations of quantum computing. The gaps arise upon a bipartite decomposition of a multiqubit operation in a one-way sequential recipe in which an ancillary system interacts locally and only once with each qubit in a row. Given the known "no-go" associated with such a theoretically and experimentally desirable decomposition, various figures of merit are introduced to analyze the optimal "fidelity" with which an arbitrary genuinely entanglin…
An important problem in quantum information is to construct multiqubit unextendible product bases (UPBs). By using the unextendible orthogonal matrices, we construct a 7-qubit UPB of size 11. It solves an open problem in [Quantum Information Processing 19:185 (2020)]. Next, we graph-theoretically show that the UPB is locally indistinguishable in the bipartite systems of two qubits and five qubits, respectively. It turns out that the UPB corresponds to a complete graph with 11 vertices constructed by three sorts of nonisomorphic graphs. Taking the graphs as product vectors, we show that they ar…
Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are "as random as a coin-toss". Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power $e_p(U)$ of the basic tw…
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…
We address the problem of unambiguous discrimination among oracle operators. The general theory of unambiguous discrimination among unitary operators is extended with this application in mind. We prove that entanglement with an ancilla cannot assist any discrimination strategy for commuting unitary operators. We also obtain a simple, practical test for the unambiguous distinguishability of an arbitrary set of unitary operators on a given system. Using this result, we prove that the unambiguous distinguishability criterion is the same for both standard and minimal oracle operators. We then show…
We analyze random unitary evolution of the qubit within memory kernel approach. We provide sufficient conditions which guarantee that the corresponding memory kernel generates physically legitimate quantum evolution. Interestingly, we are able to recover several well known examples and generate new classes of nontrivial qubit evolution. Surprisingly, it turns out that quantum evolution with memory kernel generated by our approach gives rise to vanishing non-Markovianity measure based on the distinguishability of quantum states.
A general method for implementing weakly entangling multipartite unitary operations using a small amount of entanglement and classical communication is presented. For the simple Hamiltonian σ_z\otimesσ_z this method requires less entanglement than previously known methods. In addition, compression of multiple operations is applied to reduce the average communication required.
P-divisibility conditions on subsystems in quantum dynamics
TCG Centres for Research and Education in Science and Technology (TCG CREST)（科学技术研究与教育中心）；Academy of Scientific and Innovative Research (AcSIR)（科学与创新研究学院）；Ramakrishna Mission Vivekananda Educational and Research Institute（罗摩克里希那使命维韦卡南达教育与研究所）
A general and an arbitrarily efficient scheme for entangling the spins (or any spin-like degree of freedom) of two independent uncorrelated identical particles by a combination of two particle interferometry and which way detection is formulated. It is shown that the same setup could be used to identify the quantum statistics of the incident particles from either the sign or the magnitude of measured spin correlations. Our setup also exhibits a curious complementarity between particle distinguishability and the amount of generated entanglement.
Notions of circuit complexity and cost play a key role in quantum computing and simulation where they capture the (weighted) minimal number of gates that is required to implement a unitary. Similar notions also become increasingly prominent in high energy physics in the study of holography. While notions of entanglement have in general little implications for the quantum circuit complexity and the cost of a unitary, in this note, we discuss a simple such relationship when both the entanglement of a state and the cost of a unitary take small values, building on ideas on how values of entangling…
Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $ρ=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study t…
We compare two approaches to non-Markovian quantum evolution: one based on the concept of divisible maps and the other one based on distinguishability of quantum states. The former concept is fully characterized in terms of local generator whereas it is in general not true for the latter one. A simple example of random unitary dynamics of a qubit shows the intricate difference between those approaches. Moreover, in this case both approaches are fully characterized in terms of local decoherence rates. As a byproduct it is shown that entropy might monotonically increase even for non-Markovian qu…
