TY - JOUR T1 - Pseudorandom States, Non-Cloning Theorems and Quantum Money JF - In: Shacham H., Boldyreva A. (eds) Advances in Cryptology – CRYPTO 2018. CRYPTO 2018. Lecture Notes in Computer Science. Y1 - 2018 A1 - Zhengfeng Ji A1 - Yi-Kai Liu A1 - Fang Song AB -

We propose the concept of pseudorandom states and study their constructions, properties, and applications. Under the assumption that quantum-secure one-way functions exist, we present concrete and efficient constructions of pseudorandom states. The non-cloning theorem plays a central role in our study—it motivates the proper definition and characterizes one of the important properties of pseudorandom quantum states. Namely, there is no efficient quantum algorithm that can create more copies of the state from a given number of pseudorandom states. As the main application, we prove that any family of pseudorandom states naturally gives rise to a private-key quantum money scheme.

VL - 10993 UR - https://arxiv.org/abs/1711.00385 U5 - https://doi.org/10.1007/978-3-319-96878-0_5 ER - TY - JOUR T1 - Quantum state tomography via reduced density matrices JF - Physical Review Letters Y1 - 2017 A1 - Tao Xin A1 - Dawei Lu A1 - Joel Klassen A1 - Nengkun Yu A1 - Zhengfeng Ji A1 - Jianxin Chen A1 - Xian Ma A1 - Guilu Long A1 - Bei Zeng A1 - Raymond Laflamme AB -

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.

VL - 118 U4 - 020401 UR - http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.020401 U5 - 10.1103/PhysRevLett.118.020401 ER - TY - JOUR T1 - Detecting Consistency of Overlapping Quantum Marginals by Separability JF - Physical Review A Y1 - 2016 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Nengkun Yu A1 - Bei Zeng AB - 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. VL - 93 U4 - 032105 UR - http://arxiv.org/abs/1509.06591 CP - 3 U5 - 10.1103/PhysRevA.93.032105 ER - TY - JOUR T1 - Joint product numerical range and geometry of reduced density matrices Y1 - 2016 A1 - Jianxin Chen A1 - Cheng Guo A1 - Zhengfeng Ji A1 - Yiu-Tung Poon A1 - Nengkun Yu A1 - Bei Zeng A1 - Jie Zhou AB - The reduced density matrices of a many-body quantum system form a convex set, whose three-dimensional projection Θ is convex in R3. The boundary ∂Θ 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 ∂Θ 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. UR - http://arxiv.org/abs/1606.07422 ER - TY - JOUR T1 - Pure-state tomography with the expectation value of Pauli operators JF - Physical Review A Y1 - 2016 A1 - Xian Ma A1 - Tyler Jackson A1 - Hui Zhou A1 - Jianxin Chen A1 - Dawei Lu A1 - Michael D. Mazurek A1 - Kent A.G. Fisher A1 - Xinhua Peng A1 - David Kribs A1 - Kevin J. Resch A1 - Zhengfeng Ji A1 - Bei Zeng A1 - Raymond Laflamme AB -

We examine the problem of finding the minimum number of Pauli measurements needed to uniquely determine an arbitrary n-qubit pure state among all quantum states. We show that only 11 Pauli measurements are needed to determine an arbitrary two-qubit pure state compared to the full quantum state tomography with 16 measurements, and only 31 Pauli measurements are needed to determine an arbitrary three-qubit pure state compared to the full quantum state tomography with 64 measurements. We demonstrate that our protocol is robust under depolarizing error with simulated random pure states. We experimentally test the protocol on two- and three-qubit systems with nuclear magnetic resonance techniques. We show that the pure state tomography protocol saves us a number of measurements without considerable loss of fidelity. We compare our protocol with same-size sets of randomly selected Pauli operators and find that our selected set of Pauli measurements significantly outperforms those random sampling sets. As a direct application, our scheme can also be used to reduce the number of settings needed for pure-state tomography in quantum optical systems.

VL - 93 U4 - 032140 UR - http://arxiv.org/abs/1601.05379 CP - 3 U5 - http://dx.doi.org/10.1103/PhysRevA.93.032140 ER - TY - JOUR T1 - Tomography is necessary for universal entanglement detection with single-copy observables JF - Physical Review Letters Y1 - 2016 A1 - Dawei Lu A1 - Tao Xin A1 - Nengkun Yu A1 - Zhengfeng Ji A1 - Jianxin Chen A1 - Guilu Long A1 - Jonathan Baugh A1 - Xinhua Peng A1 - Bei Zeng A1 - Raymond Laflamme AB - 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. VL - 116 U4 - 230501 UR - http://arxiv.org/abs/1511.00581 CP - 23 U5 - 10.1103/PhysRevLett.116.230501 ER - TY - JOUR T1 - Discontinuity of Maximum Entropy Inference and Quantum Phase Transitions JF - New Journal of Physics Y1 - 2015 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Chi-Kwong Li A1 - Yiu-Tung Poon A1 - Yi Shen A1 - Nengkun Yu A1 - Bei Zeng A1 - Duanlu Zhou AB - In this paper, we discuss the connection between two genuinely quantum phenomena --- the discontinuity of quantum maximum entropy inference and quantum phase transitions at zero temperature. It is shown that the discontinuity of the maximum entropy inference of local observable measurements signals the non-local type of transitions, where local density matrices of the ground state change smoothly at the transition point. We then propose to use the quantum conditional mutual information of the ground state as an indicator to detect the discontinuity and the non-local type of quantum phase transitions in the thermodynamic limit. VL - 17 U4 - 083019 UR - http://arxiv.org/abs/1406.5046v2 CP - 8 J1 - New J. Phys. U5 - 10.1088/1367-2630/17/8/083019 ER - TY - JOUR T1 - Symmetric Extension of Two-Qubit States JF - Physical Review A Y1 - 2014 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - David Kribs A1 - Norbert Lütkenhaus A1 - Bei Zeng AB - Quantum key distribution uses public discussion protocols to establish shared secret keys. In the exploration of ultimate limits to such protocols, the property of symmetric extendibility of underlying bipartite states $\rho_{AB}$ plays an important role. A bipartite state $\rho_{AB}$ is symmetric extendible if there exits a tripartite state $\rho_{ABB'}$, such that the $AB$ marginal state is identical to the $AB'$ marginal state, i.e. $\rho_{AB'}=\rho_{AB}$. For a symmetric extendible state $\rho_{AB}$, the first task of the public discussion protocol is to break this symmetric extendibility. Therefore to characterize all bi-partite quantum states that possess symmetric extensions is of vital importance. We prove a simple analytical formula that a two-qubit state $\rho_{AB}$ admits a symmetric extension if and only if $\tr(\rho_B^2)\geq \tr(\rho_{AB}^2)-4\sqrt{\det{\rho_{AB}}}$. Given the intimate relationship between the symmetric extension problem and the quantum marginal problem, our result also provides the first analytical necessary and sufficient condition for the quantum marginal problem with overlapping marginals. VL - 90 UR - http://arxiv.org/abs/1310.3530v2 CP - 3 J1 - Phys. Rev. A U5 - 10.1103/PhysRevA.90.032318 ER - TY - JOUR T1 - Symmetries of Codeword Stabilized Quantum Codes JF - 8th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2013) Y1 - 2013 A1 - Salman Beigi A1 - Jianxin Chen A1 - Markus Grassl A1 - Zhengfeng Ji A1 - Qiang Wang A1 - Bei Zeng AB - Symmetry is at the heart of coding theory. Codes with symmetry, especially cyclic codes, play an essential role in both theory and practical applications of classical error-correcting codes. Here we examine symmetry properties for codeword stabilized (CWS) quantum codes, which is the most general framework for constructing quantum error-correcting codes known to date. A CWS code Q can be represented by a self-dual additive code S and a classical code C, i.,e., Q=(S,C), however this representation is in general not unique. We show that for any CWS code Q with certain permutation symmetry, one can always find a self-dual additive code S with the same permutation symmetry as Q such that Q=(S,C). As many good CWS codes have been found by starting from a chosen S, this ensures that when trying to find CWS codes with certain permutation symmetry, the choice of S with the same symmetry will suffice. A key step for this result is a new canonical representation for CWS codes, which is given in terms of a unique decomposition as union stabilizer codes. For CWS codes, so far mainly the standard form (G,C) has been considered, where G is a graph state. We analyze the symmetry of the corresponding graph of G, which in general cannot possess the same permutation symmetry as Q. We show that it is indeed the case for the toric code on a square lattice with translational symmetry, even if its encoding graph can be chosen to be translational invariant. VL - 22 U4 - 192-206 UR - http://arxiv.org/abs/1303.7020v2 U5 - 10.4230/LIPIcs.TQC.2013.192 ER - TY - JOUR T1 - Uniqueness of Quantum States Compatible with Given Measurement Results JF - Physical Review A Y1 - 2013 A1 - Jianxin Chen A1 - Hillary Dawkins A1 - Zhengfeng Ji A1 - Nathaniel Johnston A1 - David Kribs A1 - Frederic Shultz A1 - Bei Zeng AB - We discuss the uniqueness of quantum states compatible with given results for measuring a set of observables. For a given pure state, we consider two different types of uniqueness: (1) no other pure state is compatible with the same measurement results and (2) no other state, pure or mixed, is compatible with the same measurement results. For case (1), it is known that for a d-dimensional Hilbert space, there exists a set of 4d-5 observables that uniquely determines any pure state. We show that for case (2), 5d-7 observables suffice to uniquely determine any pure state. Thus there is a gap between the results for (1) and (2), and we give some examples to illustrate this. The case of observables corresponding to reduced density matrices (RDMs) of a multipartite system is also discussed, where we improve known bounds on local dimensions for case (2) in which almost all pure states are uniquely determined by their RDMs. We further discuss circumstances where (1) can imply (2). We use convexity of the numerical range of operators to show that when only two observables are measured, (1) always implies (2). More generally, if there is a compact group of symmetries of the state space which has the span of the observables measured as the set of fixed points, then (1) implies (2). We analyze the possible dimensions for the span of such observables. Our results extend naturally to the case of low rank quantum states. VL - 88 UR - http://arxiv.org/abs/1212.3503v2 CP - 1 J1 - Phys. Rev. A U5 - 10.1103/PhysRevA.88.012109 ER - TY - JOUR T1 - Comment on some results of Erdahl and the convex structure of reduced density matrices JF - Journal of Mathematical Physics Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Mary Beth Ruskai A1 - Bei Zeng A1 - Duan-Lu Zhou AB - In J. Math. Phys. 13, 1608-1621 (1972), Erdahl considered the convex structure of the set of $N$-representable 2-body reduced density matrices in the case of fermions. Some of these results have a straightforward extension to the $m$-body setting and to the more general quantum marginal problem. We describe these extensions, but can not resolve a problem in the proof of Erdahl's claim that every extreme point is exposed in finite dimensions. Nevertheless, we can show that when $2m \geq N$ every extreme point of the set of $N$-representable $m$-body reduced density matrices has a unique pre-image in both the symmetric and anti-symmetric setting. Moreover, this extends to the quantum marginal setting for a pair of complementary $m$-body and $(N-m)$-body reduced density matrices. VL - 53 U4 - 072203 UR - http://arxiv.org/abs/1205.3682v1 CP - 7 J1 - J. Math. Phys. U5 - 10.1063/1.4736842 ER - TY - JOUR T1 - Correlations in excited states of local Hamiltonians JF - Physical Review A Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Zhaohui Wei A1 - Bei Zeng AB - Physical properties of the ground and excited states of a $k$-local Hamiltonian are largely determined by the $k$-particle reduced density matrices ($k$-RDMs), or simply the $k$-matrix for fermionic systems---they are at least enough for the calculation of the ground state and excited state energies. Moreover, for a non-degenerate ground state of a $k$-local Hamiltonian, even the state itself is completely determined by its $k$-RDMs, and therefore contains no genuine ${>}k$-particle correlations, as they can be inferred from $k$-particle correlation functions. It is natural to ask whether a similar result holds for non-degenerate excited states. In fact, for fermionic systems, it has been conjectured that any non-degenerate excited state of a 2-local Hamiltonian is simultaneously a unique ground state of another 2-local Hamiltonian, hence is uniquely determined by its 2-matrix. And a weaker version of this conjecture states that any non-degenerate excited state of a 2-local Hamiltonian is uniquely determined by its 2-matrix among all the pure $n$-particle states. We construct explicit counterexamples to show that both conjectures are false. It means that correlations in excited states of local Hamiltonians could be dramatically different from those in ground states. We further show that any non-degenerate excited state of a $k$-local Hamiltonian is a unique ground state of another $2k$-local Hamiltonian, hence is uniquely determined by its $2k$-RDMs (or $2k$-matrix). VL - 85 UR - http://arxiv.org/abs/1106.1373v2 CP - 4 J1 - Phys. Rev. A U5 - 10.1103/PhysRevA.85.040303 ER - TY - JOUR T1 - From Ground States to Local Hamiltonians JF - Physical Review A Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Bei Zeng A1 - D. L. Zhou AB - Traditional quantum physics solves ground states for a given Hamiltonian, while quantum information science asks for the existence and construction of certain Hamiltonians for given ground states. In practical situations, one would be mainly interested in local Hamiltonians with certain interaction patterns, such as nearest neighbour interactions on some type of lattices. A necessary condition for a space $V$ to be the ground-state space of some local Hamiltonian with a given interaction pattern, is that the maximally mixed state supported on $V$ is uniquely determined by its reduced density matrices associated with the given pattern, based on the principle of maximum entropy. However, it is unclear whether this condition is in general also sufficient. We examine the situations for the existence of such a local Hamiltonian to have $V$ satisfying the necessary condition mentioned above as its ground-state space, by linking to faces of the convex body of the local reduced states. We further discuss some methods for constructing the corresponding local Hamiltonians with given interaction patterns, mainly from physical points of view, including constructions related to perturbation methods, local frustration-free Hamiltonians, as well as thermodynamical ensembles. VL - 86 UR - http://arxiv.org/abs/1110.6583v4 CP - 2 J1 - Phys. Rev. A U5 - 10.1103/PhysRevA.86.022339 ER - TY - JOUR T1 - Ground-State Spaces of Frustration-Free Hamiltonians JF - Journal of Mathematical Physics Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - David Kribs A1 - Zhaohui Wei A1 - Bei Zeng AB - We study the ground-state space properties for frustration-free Hamiltonians. We introduce a concept of `reduced spaces' to characterize local structures of ground-state spaces. For a many-body system, we characterize mathematical structures for the set $\Theta_k$ of all the $k$-particle reduced spaces, which with a binary operation called join forms a semilattice that can be interpreted as an abstract convex structure. The smallest nonzero elements in $\Theta_k$, called atoms, are analogs of extreme points. We study the properties of atoms in $\Theta_k$ and discuss its relationship with ground states of $k$-local frustration-free Hamiltonians. For spin-1/2 systems, we show that all the atoms in $\Theta_2$ are unique ground states of some 2-local frustration-free Hamiltonians. Moreover, we show that the elements in $\Theta_k$ may not be the join of atoms, indicating a richer structure for $\Theta_k$ beyond the convex structure. Our study of $\Theta_k$ deepens the understanding of ground-state space properties for frustration-free Hamiltonians, from a new angle of reduced spaces. VL - 53 U4 - 102201 UR - http://arxiv.org/abs/1112.0762v1 CP - 10 J1 - J. Math. Phys. U5 - 10.1063/1.4748527 ER - TY - JOUR T1 - Minimum Entangling Power is Close to Its Maximum Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - David W Kribs A1 - Bei Zeng AB - Given a quantum gate $U$ acting on a bipartite quantum system, its maximum (average, minimum) entangling power is the maximum (average, minimum) entanglement generation with respect to certain entanglement measure when the inputs are restricted to be product states. In this paper, we mainly focus on the 'weakest' one, i.e., the minimum entangling power, among all these entangling powers. We show that, by choosing von Neumann entropy of reduced density operator or Schmidt rank as entanglement measure, even the 'weakest' entangling power is generically very close to its maximal possible entanglement generation. In other words, maximum, average and minimum entangling powers are generically close. We then study minimum entangling power with respect to other Lipschitiz-continuous entanglement measures and generalize our results to multipartite quantum systems. As a straightforward application, a random quantum gate will almost surely be an intrinsically fault-tolerant entangling device that will always transform every low-entangled state to near-maximally entangled state. UR - http://arxiv.org/abs/1210.1296v1 ER - TY - JOUR T1 - Rank Reduction for the Local Consistency Problem JF - Journal of Mathematical Physics Y1 - 2012 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Alexander Klyachko A1 - David W. Kribs A1 - Bei Zeng AB - We address the problem of how simple a solution can be for a given quantum local consistency instance. More specifically, we investigate how small the rank of the global density operator can be if the local constraints are known to be compatible. We prove that any compatible local density operators can be satisfied by a low rank global density operator. Then we study both fermionic and bosonic versions of the N-representability problem as applications. After applying the channel-state duality, we prove that any compatible local channels can be obtained through a global quantum channel with small Kraus rank. VL - 53 U4 - 022202 UR - http://arxiv.org/abs/1106.3235v2 CP - 2 J1 - J. Math. Phys. U5 - 10.1063/1.3685644 ER - TY - JOUR T1 - No-go Theorem for One-way Quantum Computing on Naturally Occurring Two-level Systems JF - Physical Review A Y1 - 2011 A1 - Jianxin Chen A1 - Xie Chen A1 - Runyao Duan A1 - Zhengfeng Ji A1 - Bei Zeng AB - One-way quantum computing achieves the full power of quantum computation by performing single particle measurements on some many-body entangled state, known as the resource state. As single particle measurements are relatively easy to implement, the preparation of the resource state becomes a crucial task. An appealing approach is simply to cool a strongly correlated quantum many-body system to its ground state. In addition to requiring the ground state of the system to be universal for one-way quantum computing, we also want the Hamiltonian to have non-degenerate ground state protected by a fixed energy gap, to involve only two-body interactions, and to be frustration-free so that measurements in the course of the computation leave the remaining particles in the ground space. Recently, significant efforts have been made to the search of resource states that appear naturally as ground states in spin lattice systems. The approach is proved to be successful in spin-5/2 and spin-3/2 systems. Yet, it remains an open question whether there could be such a natural resource state in a spin-1/2, i.e., qubit system. Here, we give a negative answer to this question by proving that it is impossible for a genuinely entangled qubit states to be a non-degenerate ground state of any two-body frustration-free Hamiltonian. What is more, we prove that every spin-1/2 frustration-free Hamiltonian with two-body interaction always has a ground state that is a product of single- or two-qubit states, a stronger result that is interesting independent of the context of one-way quantum computing. VL - 83 UR - http://arxiv.org/abs/1004.3787v1 CP - 5 J1 - Phys. Rev. A U5 - 10.1103/PhysRevA.83.050301 ER - TY - JOUR T1 - Principle of Maximum Entropy and Ground Spaces of Local Hamiltonians Y1 - 2010 A1 - Jianxin Chen A1 - Zhengfeng Ji A1 - Mary Beth Ruskai A1 - Bei Zeng A1 - Duanlu Zhou AB - The structure of the ground spaces of quantum systems consisting of local interactions is of fundamental importance to different areas of physics. In this Letter, we present a necessary and sufficient condition for a subspace to be the ground space of a k-local Hamiltonian. Our analysis are motivated by the concept of irreducible correlations studied by [Linden et al., PRL 89, 277906] and [Zhou, PRL 101, 180505], which is in turn based on the principle of maximum entropy. It establishes a better understanding of the ground spaces of local Hamiltonians and builds an intimate link of ground spaces to the correlations of quantum states. UR - http://arxiv.org/abs/1010.2739v4 ER - TY - JOUR T1 - Existence of Universal Entangler JF - Journal of Mathematical Physics Y1 - 2008 A1 - Jianxin Chen A1 - Runyao Duan A1 - Zhengfeng Ji A1 - Mingsheng Ying A1 - Jun Yu AB - A gate is called entangler if it transforms some (pure) product states to entangled states. A universal entangler is a gate which transforms all product states to entangled states. In practice, a universal entangler is a very powerful device for generating entanglements, and thus provides important physical resources for accomplishing many tasks in quantum computing and quantum information. This Letter demonstrates that a universal entangler always exists except for a degenerated case. Nevertheless, the problem how to find a universal entangler remains open. VL - 49 U4 - 012103 UR - http://arxiv.org/abs/0704.1473v2 CP - 1 J1 - J. Math. Phys. U5 - 10.1063/1.2829895 ER - TY - JOUR T1 - The LU-LC conjecture is false Y1 - 2007 A1 - Zhengfeng Ji A1 - Jianxin Chen A1 - Zhaohui Wei A1 - Mingsheng Ying AB - The LU-LC conjecture is an important open problem concerning the structure of entanglement of states described in the stabilizer formalism. It states that two local unitary equivalent stabilizer states are also local Clifford equivalent. If this conjecture were true, the local equivalence of stabilizer states would be extremely easy to characterize. Unfortunately, however, based on the recent progress made by Gross and Van den Nest, we find that the conjecture is false. UR - http://arxiv.org/abs/0709.1266v2 J1 - Quantum Inf. Comput. ER -