arXiv subjects

Nengkun Yu

Publications and source records attributed to Nengkun Yu.

At least 55 records · Page 3Linked to original sources

arXiv2019

Entirety of Quantum Uncertainty and Its Experimental Verification

As a foundation of modern physics, uncertainty relations describe an ultimate limit for the measurement uncertainty of incompatible observables. Traditionally, uncertain relations are formulated by mathematical bounds for a specific state. Here we present a method for geometrically characterizing uncertainty relations as an entire area of variances of the observables, ranging over all possible input states. We find that for the pair of position $x$ and momentum $p$ operators, Heisenberg's uncertainty principle points exactly to the area of the variances of $x$ and $p$. Moreover, for finite-dimensional systems, we prove that the corresponding area is necessarily semialgebraic; in other words, this set can be represented via finite polynomial equations and inequalities, or any finite union of such sets. In particular, we give the analytical characterization of the areas of variances of (a) a pair of one-qubit observables, (b) a pair of projective observables for arbitrary dimension, and give the first experimental observation of such areas in a photonic system.

quant-ph↗

arXiv2019

Quantum Temporal Logic

In this paper, we introduce a model of quantum concurrent program, which can be used to model the behaviour of reactive quantum systems and to design quantum compilers. We investigate quantum temporal logic, QTL, for the specification of quantum concurrent systems by suggesting the time-dependence of events. QTL employs the projections on subspaces as atomic propositions, which was established in the Birkhoff and von Neumann's classic treatise on quantum logic. For deterministic functional quantum program, We prove a quantum Böhm-Jacopini theorem which states that any such program is equivalent to a Q-While program. The decidability of basic QTL formulae for general quantum concurrent program is studied.

cs.LO↗

arXiv2019

Experimental Cryptographic Verification for Near-Term Quantum Cloud Computing

Recently, there are more and more organizations offering quantum-cloud services, where any client can access a quantum computer remotely through the internet. In the near future, these cloud servers may claim to offer quantum computing power out of reach of classical devices. An important task is to make sure that there is a real quantum computer running, instead of a simulation by a classical device. Here we explore the applicability of a cryptographic verification scheme that avoids the need of implementing a full quantum algorithm or requiring the clients to communicate with quantum resources. In this scheme, the client encodes a secret string in a scrambled IQP (instantaneous quantum polynomial) circuit sent to the quantum cloud in the form of classical message, and verify the computation by checking the probability bias of a class of output strings generated by the server. We provided a theoretical extension and implemented the scheme on a 5-qubit NMR quantum processor in the laboratory and a 5-qubit and 16-qubit processors of the IBM quantum cloud. We found that the experimental results of the NMR processor can be verified by the scheme with about $2.5\%$ error, after noise compensation by standard techniques. However, the fidelity of the IBM quantum cloud is currently too low to pass the test (about $42\%$ error). This verification scheme shall become practical when servers claim to offer quantum-computing resources that can achieve quantum supremacy.

quant-ph↗

arXiv2019

LOCC protocols with bounded width per round optimize convex functions

We start with the task of discriminating finitely many multipartite quantum states using LOCC protocols, with the goal to optimize the probability of correctly identifying the state. We provide two different methods to show that finitely many measurement outcomes in every step are sufficient for approaching the optimal probability of discrimination. In the first method, each measurement of an optimal LOCC protocol, applied to a $d_{\rm loc}$-dim local system, is replaced by one with at most $2d_{\rm loc}^2$ outcomes, without changing the probability of success. In the second method, we decompose any LOCC protocol into a convex combination of a number of "slim protocols" in which each measurement applied to a $d_{\rm loc}$-dim local system has at most $d_{\rm loc}^2$ outcomes. To maximize any convex functions in LOCC (including the probability of state discrimination or fidelity of state transformation), an optimal protocol can be replaced by the best slim protocol in the convex decomposition without using shared randomness. For either method, the bound on the number of outcomes per measurement is independent of the global dimension, the number of parties, the depth of the protocol, how deep the measurement is located, and applies to LOCC protocols with infinite rounds, and the "measurement compression" can be done "top-down" -- independent of later operations in the LOCC protocol. The second method can be generalized to implement LOCC instruments with finitely many outcomes: if the instrument has $n$ coarse-grained final measurement outcomes, global input dimension $D_0$ and global output dimension $D_i$ for $i=1,...,n$ conditioned on the $i$-th outcome, then one can obtain the instrument as a convex combination of no more than $R=\sum_{i=1}^n D_0^2 D_i^2 - D_0^2 + 1$ slim protocols; in other words, $\log_2 R$ bits of shared randomess suffice.

quant-ph↗

arXiv2019

Quantum Earth mover's distance, No-go Quantum Kantorovich-Rubinstein theorem, and Quantum Marginal Problem

The earth mover's distance is a measure of the distance between two probabilistic measures. It plays a fundamental role in mathematics and computer science. The Kantorovich-Rubinstein theorem provides a formula for the earth mover's distance on the space of regular probability Borel measures on a compact metric space. In this paper, we investigate the quantum earth mover's distance. We show a no-go Kantorovich-Rubinstein theorem in the quantum setting. More precisely, we show that the trace distance between two quantum states can not be determined by their earth mover's distance. The technique here is to track the bipartite quantum marginal problem. Then we provide inequality to describe the structure of quantum coupling, which can be regarded as quantum generalization of Kantorovich-Rubinstein theorem. After that, we generalize it to obtain into the tripartite version, and build a new class of necessary criteria for the tripartite marginal problem.

quant-ph↗

arXiv2018

Characterization of multipartite entanglement in terms of local transformations

The degree of the generators of invariant polynomial rings of is a long standing open problem since the very initial study of the invariant theory in the 19th century. Motivated by its significant role in characterizing multipartite entanglement, we study the invariant polynomial rings of local unitary group---the tensor product of unitary group, and local general linear group---the tensor product of general linear group. For these two groups, we prove polynomial upper bounds on the degree of the generators of invariant polynomial rings. On the other hand, systematic methods are provided to to construct all homogenous polynomials that are invariant under these two groups for any fixed degree. Thus, our results can be regarded as a complete characterization of the invariant polynomial rings. As an interesting application, we show that multipartite entanglement is additive in the sense that two multipartite states are local unitary equivalent if and only if $r$-copies of them are LU equivalent for some $r$.

quant-ph↗

arXiv2018

Quantum Coupling and Strassen Theorem

We introduce a quantum generalisation of the notion of coupling in probability theory. Several interesting examples and basic properties of quantum couplings are presented. In particular, we prove a quantum extension of Strassen theorem for probabilistic couplings, a fundamental theorem in probability theory that can be used to bound the probability of an event in a distribution by the probability of an event in another distribution coupled with the first.

quant-ph↗

arXiv2018

Entanglement Verification, with or without tomography

Multipartite entanglement has been widely regarded as key resources in distributed quantum computing, for instance, multi-party cryptography, measurement based quantum computing, quantum algorithms. It also plays a fundamental role in quantum phase transitions, even responsible for transport efficiency in biological systems. Certifying multipartite entanglement is generally a fundamental task. Since an $N$ qubit state is parameterized by $4^N-1$ real numbers, one is interested to design a measurement setup that reveals multipartite entanglement with as little effort as possible, at least without fully revealing the whole information of the state, the so called "tomography", which requires exponential energy. In this paper, we study this problem of certifying entanglement without tomography in the constrain that only single copy measurements can be applied. This task is formulate as a membership problem related to a dividing quantum state space, therefore, related to the geometric structure of state space. We show that universal entanglement detection among all states can never be accomplished without full state tomography. Moreover, we show that almost all multipartite correlation, include genuine entanglement detection, entanglement depth verification, requires full state tomography. However, universal entanglement detection among pure states can be much more efficient, even we only allow local measurements. Almost optimal local measurement scheme for detecting pure states entanglement is provided.

quant-ph↗

arXiv2017

Sample-optimal tomography of quantum states

It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error $ε$ in trace distance required $O(dr^2/ε^2)$ copies for a $d$-dimensional density matrix of rank $r$. Here, we give a theoretical measurement scheme (POVM) that requires $O (dr/ δ) \ln (d/δ) $ copies of $ρ$ to error $δ$ in infidelity, and a matching lower bound up to logarithmic factors. This implies $O( (dr / ε^2) \ln (d/ε) )$ copies suffice to achieve error $ε$ in trace distance. We also prove that for independent (product) measurements, $Ω(dr^2/δ^2) / \ln(1/δ)$ copies are necessary in order to achieve error $δ$ in infidelity. For fixed $d$, our measurement can be implemented on a quantum computer in time polynomial in $n$.

quant-ph↗

arXiv2016

Exponential Separation of Quantum Communication and Classical Information

We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572], hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha. As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related Greater-Than function on n bits: say Bob communicates at most b bits, then Alice must send n/exp(O(b)) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.

quant-ph↗

arXiv2016

Joint product numerical range and geometry of reduced density matrices

The reduced density matrices of a many-body quantum system form a convex set, whose three-dimensional projection $Θ$ is convex in $\mathbb{R}^3$. The boundary $\partialΘ$ of $Θ$ may exhibit nontrivial geometry, in particular ruled surfaces. Two physical mechanisms are known for the origins of ruled surfaces: symmetry breaking and gapless. In this work, we study the emergence of ruled surfaces for systems with local Hamiltonians in infinite spatial dimension, where the reduced density matrices are known to be separable as a consequence of the quantum de Finetti's theorem. This allows us to identify the reduced density matrix geometry with joint product numerical range $Π$ of the Hamiltonian interaction terms. We focus on the case where the interaction terms have certain structures, such that ruled surface emerge naturally when taking a convex hull of $Π$. We show that, a ruled surface on $\partialΘ$ sitting in $Π$ has a gapless origin, otherwise it has a symmetry breaking origin. As an example, we demonstrate that a famous ruled surface, known as the oloid, is a possible shape of $Θ$, with two boundary pieces of symmetry breaking origin separated by two gapless lines.

quant-ph↗

arXiv2016

Physical origins of ruled surfaces on the reduced density matrices geometry

The reduced density matrices (RDMs) of many-body quantum states form a convex set. The boundary of low dimensional projections of this convex set may exhibit nontrivial geometry such as ruled surfaces. In this paper, we study the physical origins of these ruled surfaces for bosonic systems. The emergence of ruled surfaces was recently proposed as signatures of symmetry-breaking phase. We show that, apart from being signatures of symmetry-breaking, ruled surfaces can also be the consequence of gapless quantum systems by demonstrating an explicit example in terms of a two-mode Ising model. Our analysis was largely simplified by the quantum de Finetti's theorem---in the limit of large system size, these RDMs are the convex set of all the symmetric separable states. To distinguish ruled surfaces originated from gapless systems from those caused by symmetry-breaking, we propose to use the finite size scaling method for the corresponding geometry. This method is then applied to the two-mode XY model, successfully identifying a ruled surface as the consequence of gapless systems.

quant-ph↗

arXiv2016

Quantum State and Process Tomography via Adaptive Measurements

We investigate quantum state tomography (QST) for pure states and quantum process tomography (QPT) for unitary channels via $adaptive$ measurements. For a quantum system with a $d$-dimensional Hilbert space, we first propose an adaptive protocol where only $2d-1$ measurement outcomes are used to accomplish the QST for $all$ pure states. This idea is then extended to study QPT for unitary channels, where an adaptive unitary process tomography (AUPT) protocol of $d^2+d-1$ measurement outcomes is constructed for any unitary channel. We experimentally implement the AUPT protocol in a 2-qubit nuclear magnetic resonance system. We examine the performance of the AUPT protocol when applied to Hadamard gate, $T$ gate ($π/8$ phase gate), and controlled-NOT gate, respectively, as these gates form the universal gate set for quantum information processing purpose. As a comparison, standard QPT is also implemented for each gate. Our experimental results show that the AUPT protocol that reconstructing unitary channels via adaptive measurements significantly reduce the number of experiments required by standard QPT without considerable loss of fidelity.

quant-ph↗

arXiv2016

Dichotomy of entanglement depth for symmetric states

Entanglement depth characterizes the minimal number of particles in a system that are mutually entangled. For symmetric states, we show that there is a dichotomy for entanglement depth: an $N$-particle symmetric state is either fully separable, or fully entangled---the entanglement depth is either $1$ or $N$. This property is even stable under non-symmetric noise. We propose an experimentally accessible method to detect entanglement depth in atomic ensembles based on a bound on the particle number population of Dicke states, and demonstrate that the entanglement depth of some Dicke states, for example the twin Fock state, is very stable even under a large arbitrary noise. Our observation can be applied to atomic Bose-Einstein condensates to infer that these systems can be highly entangled with the entanglement depth that is of the order of the system size (i.e. several thousands of atoms).

quant-ph↗

arXiv2016

Quantum state tomography via reduced density matrices

Quantum state tomography via local measurements is an efficient tool for characterizing quantum states. However it requires that the original global state be uniquely determined (UD) by its local reduced density matrices (RDMs). In this work we demonstrate for the first time a class of states that are UD by their RDMs under the assumption that the global state is pure, but fail to be UD in the absence of that assumption. This discovery allows us to classify quantum states according to their UD properties, with the requirement that each class be treated distinctly in the practice of simplifying quantum state tomography. Additionally we experimentally test the feasibility and stability of performing quantum state tomography via the measurement of local RDMs for each class. These theoretical and experimental results advance the project of performing efficient and accurate quantum state tomography in practice.

quant-ph↗

arXiv2016

Quantum Capacities for Entanglement Networks

We discuss quantum capacities for two types of entanglement networks: $\mathcal{Q}$ for the quantum repeater network with free classical communication, and $\mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tensor network. We find that $\mathcal{Q}$ always equals $\mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities $\mathcal{Q}_1$ and $\mathcal{R}_1$ are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that $\mathcal{R}_1$ is a natural upper bound of $\mathcal{Q}_1$. We analyze the possible gap between $\mathcal{R}_1$ and $\mathcal{Q}_1$ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network.

quant-ph↗

arXiv2015

Tomography is necessary for universal entanglement detection with single-copy observables

Entanglement, one of the central mysteries of quantum mechanics, plays an essential role in numerous applications of quantum information theory. A natural question of both theoretical and experimental importance is whether universal entanglement detection is possible without full state tomography. In this work, we prove a no-go theorem that rules out this possibility for any non-adaptive schemes that employ single-copy measurements only. We also examine in detail a previously implemented experiment, which claimed to detect entanglement of two-qubit states via adaptive single-copy measurements without full state tomography. By performing the experiment and analyzing the data, we demonstrate that the information gathered is indeed sufficient to reconstruct the state. These results reveal a fundamental limit for single-copy measurements in entanglement detection, and provides a general framework to study the detection of other interesting properties of quantum states, such as the positivity of partial transpose and the $k$-symmetric extendibility.

quant-ph↗

arXiv2015

Detecting Consistency of Overlapping Quantum Marginals by Separability

The quantum marginal problem asks whether a set of given density matrices are consistent, i.e., whether they can be the reduced density matrices of a global quantum state. Not many non-trivial analytic necessary (or sufficient) conditions are known for the problem in general. We propose a method to detect consistency of overlapping quantum marginals by considering the separability of some derived states. Our method works well for the $k$-symmetric extension problem in general, and for the general overlapping marginal problems in some cases. Our work is, in some sense, the converse to the well-known $k$-symmetric extension criterion for separability.

quant-ph↗