arXiv subjects

Runyao Duan

Publications and source records attributed to Runyao Duan.

At least 55 records · Page 3Linked to original sources

arXiv2011

Verification of Quantum Programs

This paper develops verification methodology for quantum programs, and the contribution of the paper is two-fold: 1. Sharir, Pnueli and Hart [SIAM J. Comput. 13(1984)292-314] presented a general method for proving properties of probabilistic programs, in which a probabilistic program is modeled by a Markov chain and an assertion on the output distribution is extended into an invariant assertion on all intermediate distributions. Their method is essentially a probabilistic generalization of the classical Floyd inductive assertion method. In this paper, we consider quantum programs modeled by quantum Markov chains which are defined by super-operators. It is shown that the Sharir-Pnueli-Hart method can be elegantly generalized to quantum programs by exploiting the Schrödinger-Heisenberg duality between quantum states and observables. In particular, a completeness theorem for the Sharir-Pnueli-Hart verification method of quantum programs is established. 2. As indicated by the completeness theorem, the Sharir-Pnueli-Hart method is in principle effective for verifying all properties of quantum programs that can be expressed in terms of Hermitian operators (observables). But it is not feasible for many practical applications because of the complicated calculation involved in the verification. For the case of finite-dimensional state spaces, we find a method for verification of quantum programs much simpler than the Sharir-Pnueli-Hart method by employing the matrix representation of super-operators and Jordan decomposition of matrices. In particular, this method enables us to compute easily the average running time and even to analyze some interesting long-run behaviors of quantum programs in a finite-dimensional state space.

cs.LO↗

arXiv2010

Tensor Rank and Stochastic Entanglement Catalysis for Multipartite Pure States

The tensor rank (also known as generalized Schmidt rank) of multipartite pure states plays an important role in the study of entanglement classifications and transformations. We employ powerful tools from the theory of homogeneous polynomials to investigate the tensor rank of symmetric states such as the tripartite state $\ket{W_3}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$ and its $N$-partite generalization $\ket{W_N}$. Previous tensor rank estimates are dramatically improved and we show that (i) three copies of $\ket{W_3}$ has rank either 15 or 16, (ii) two copies of $\ket{W_N}$ has rank $3N-2$, and (iii) $n$ copies of $\ket{W_N}$ has rank O(N). A remarkable consequence of these results is that certain multipartite transformations, impossible even probabilistically, can become possible when performed in multiple copy bunches or when assisted by some catalyzing state. This effect is impossible for bipartite pure states.

quant-ph↗

arXiv2010

Any $2\otimes n$ subspace is locally distinguishable

A subspace of a multipartite Hilbert space is called \textit{locally indistinguishable} if any orthogonal basis of this subspace cannot be perfectly distinguished by local operations and classical communication. Previously it was shown that any $m\otimes n$ bipartite system such that $m>2$ and $n>2$ has a locally indistinguishable subspace. However, it has been an open problem since 2005 whether there is a locally indistinguishable bipartite subspace with a qubit subsystem. We settle this problem by showing that any $2\otimes n$ bipartite subspace is locally distinguishable in the sense it contains a basis perfectly distinguishable by LOCC. As an interesting application, we show that any quantum channel with two Kraus operations has optimal environment-assisted classical capacity.

quant-ph↗

arXiv2010

Optimal Perfect Distinguishability between Unitaries and Quantum Operations

We study optimal perfect distinguishability between a unitary and a general quantum operation. In 2-dimensional case we provide a simple sufficient and necessary condition for sequential perfect distinguishability and an analytical formula of optimal query time. We extend the sequential condition to general d-dimensional case. Meanwhile, we provide an upper bound and a lower bound for optimal sequential query time. In the process a new iterative method is given, the most notable innovation of which is its independence to auxiliary systems or entanglement. Following the idea, we further obtain an upper bound and a lower bound of (entanglement-assisted) q-maximal fidelities between a unitary and a quantum operation. Thus by the recursion in [1] an upper bound and a lower bound for optimal general perfect discrimination are achieved. Finally our lower bound result can be extended to the case of arbitrary two quantum operations.

quant-ph↗

arXiv2010

An Algebra of Quantum Processes

We introduce an algebra qCCS of pure quantum processes in which no classical data is involved, communications by moving quantum states physically are allowed, and computations is modeled by super-operators. An operational semantics of qCCS is presented in terms of (non-probabilistic) labeled transition systems. Strong bisimulation between processes modeled in qCCS is defined, and its fundamental algebraic properties are established, including uniqueness of the solutions of recursive equations. To model sequential computation in qCCS, a reduction relation between processes is defined. By combining reduction relation and strong bisimulation we introduce the notion of strong reduction-bisimulation, which is a device for observing interaction of computation and communication in quantum systems. Finally, a notion of strong approximate bisimulation (equivalently, strong bisimulation distance) and its reduction counterpart are introduced. It is proved that both approximate bisimilarity and approximate reduction-bisimilarity are preserved by various constructors of quantum processes. This provides us with a formal tool for observing robustness of quantum processes against inaccuracy in the implementation of its elementary gates.

quant-ph↗

arXiv2010

No-go Theorem for One-way Quantum Computing on Naturally Occurring Two-level Systems

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.

quant-ph↗

arXiv2010

Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function

We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain operator space as the quantum generalisation of the adjacency matrix, in terms of which the plain, quantum and entanglement-assisted capacity can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovasz' famous theta function, as the norm-completion (or stabilisation) of a "naive" generalisation of theta. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite programme, whose dual we write down explicitly, and that it is multiplicative with respect to the natural (strong) graph product. We explore various other properties of the new quantity, which reduces to Lovasz' original theta in the classical case, give several applications, and propose to study the operator spaces associated to channels as "non-commutative graphs", using the language of Hilbert modules.

quant-ph↗

arXiv2010

Quantum state reduction for universal measurement based computation

Measurement based quantum computation (MBQC), which requires only single particle measurements on a universal resource state to achieve the full power of quantum computing, has been recognized as one of the most promising models for the physical realization of quantum computers. Despite considerable progress in the last decade, it remains a great challenge to search for new universal resource states with naturally occurring Hamiltonians, and to better understand the entanglement structure of these kinds of states. Here we show that most of the resource states currently known can be reduced to the cluster state, the first known universal resource state, via adaptive local measurements at a constant cost. This new quantum state reduction scheme provides simpler proofs of universality of resource states and opens up plenty of space to the search of new resource states, including an example based on the one-parameter deformation of the AKLT state studied in [Commun. Math. Phys. 144, 443 (1992)] by M. Fannes et al. about twenty years ago.

quant-ph↗

arXiv2010

Multi-Error-Correcting Amplitude Damping Codes

We construct new families of multi-error-correcting quantum codes for the amplitude damping channel. Our key observation is that, with proper encoding, two uses of the amplitude damping channel simulate a quantum erasure channel. This allows us to use concatenated codes with quantum erasure-correcting codes as outer codes for correcting multiple amplitude damping errors. Our new codes are degenerate stabilizer codes and have parameters which are better than the amplitude damping codes obtained by any previously known construction.

quant-ph↗

arXiv2009

Local Unambiguous Discrimination with Remaining Entanglement

A bipartite state which is secretly chosen from a finite set of known entangled pure states cannot be immediately useful in standard quantum information processing tasks. To effectively make use of the entanglement contained in this unknown state, we introduce a new way to locally manipulate the original quantum system: either identify the state successfully or distill some pure entanglement. Remarkably, if many copies are available, we show that any finite set of entangled pure states, whatever orthogonal or not, can be locally distinguished in this way, which further implies that pure entanglement can be deterministically extracted from unknown pure entanglement. These results make it clear why a large class of entangled bipartite quantum operations including unitary operations and measurements that are globally distinguishable can also be locally distinguishable: they can generate pure entanglement consistently.

quant-ph↗

arXiv2009

When is there a multipartite maximum entangled state?

For a multipartite quantum system of the dimension $d_1\otimes d_2\otimes... d_n$, $d_1\ge d_2\ge...\ge d_n$, is there an entangled state {\em maximum} in the sense that all other states in the system can be obtained from the state through local quantum operations and classical communications (LOCC)? When $d_1\geΠ_{i=2}^n d_i$, such state exists. We show that this condition is also necessary. Our proof, somewhat surprisingly, uses results from algebraic complexity theory.

quant-ph↗

arXiv2009

The Tensor Rank of the Tripartite State $\ket{W}^{\otimes n}$}

Tensor rank refers to the number of product states needed to express a given multipartite quantum state. Its non-additivity as an entanglement measure has recently been observed. In this note, we estimate the tensor rank of multiple copies of the tripartite state $\ket{W}=\tfrac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$. Both an upper bound and a lower bound of this rank are derived. In particular, it is proven that the tensor rank of $\ket{W}^{\otimes 2}$ is seven, thus resolving a previously open problem. Some implications of this result are discussed in terms of transformation rates between $\ket{W}^{\otimes n}$ and multiple copies of the state $\ket{GHZ}=\tfrac{1}{\sqrt{2}}(\ket{000}+\ket{111})$.

quant-ph↗

arXiv2009

Optimal Simulation of a Perfect Entangler

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.

quant-ph↗

arXiv2009

Tripartite to Bipartite Entanglement Transformations and Polynomial Identity Testing

We consider the problem of deciding if a given three-party entangled pure state can be converted, with a non-zero success probability, into a given two-party pure state through local quantum operations and classical communication. We show that this question is equivalent to the well-known computational problem of deciding if a multivariate polynomial is identically zero. Efficient randomized algorithms developed to study the latter can thus be applied to the question of tripartite to bipartite entanglement transformations.

quant-ph↗

arXiv2009

The Perfect Distinguishability of Quantum Operations

We provide a feasible necessary and sufficient condition for when an unknown quantum operation (quantum device) secretely selected from a set of known quantum operations can be identified perfectly within a finite number of queries, and thus complete the characterization of the perfect distinguishability of quantum operations. We further design an optimal protocol which can achieve the perfect discrimination between two quantum operations by a minimal number of queries. Interestingly, employing the techniques from the theory of $q$-numerical range we find that an optimal perfect discrimination between two isometries is always achievable without using auxiliary systems or entanglement.

quant-ph↗

arXiv2009

Super-Activation of Zero-Error Capacity of Noisy Quantum Channels

We study various super-activation effects in the following zero-error communication scenario: One sender wants to send classical or quantum information through a noisy quantum channel to one receiver with zero probability of error. First we show that there are quantum channels of which a single use is not able to transmit classical information perfectly yet two uses can. This is achieved by employing entangled input states between different uses of the given channel and thus cannot happen for classical channels. Second we exhibit a class of quantum channel with vanishing zero-error classical capacity such that when a noiseless qubit channel or one ebit shared entanglement are available, it can be used to transmit $\log_2 d$ noiseless qubits, where 2d is the dimension of input state space. Third we further construct quantum channels with vanishing zero-error classical capacity when assisted with classical feedback can be used to transmit both classical and quantum information perfectly. These striking findings not only indicate both the zero-error quantum and classical capacities of quantum channels satisfy a strong super-additivity beyond any classical channels, but also highlight the activation power of auxiliary physical resources in zero-error communication.

quant-ph↗

arXiv2008

Nonlocal Entanglement Transformations Achievable by Separable Operations

For manipulations of multipartite quantum systems, it was well known that all local operations assisted by classical communication (LOCC) constitute a proper subset of the class of separable operations. Recently, Gheorghiu and Griffiths found that LOCC and general separable operations are equally powerful for transformations between bipartite pure states. In this letter we extend this comparison to mixed states and show that in general separable operations are strictly stronger than LOCC when transforming a mixed state to a pure entangled state. A remarkable consequence of our finding is the existence of entanglement monotone which may increase under separable operations.

quant-ph↗

arXiv2008

Locally Indistinguishable Subspaces Spanned by Three-Qubit Unextendible Product Bases

We study the local distinguishability of general multi-qubit states and show that local projective measurements and classical communication are as powerful as the most general local measurements and classical communication. Remarkably, this indicates that the local distinguishability of multi-qubit states can be decided efficiently. Another useful consequence is that a set of orthogonal $n$-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension $2^n$. When $n=2$ such a condition is also sufficient. Employing these results, we show that any orthonormal basis of a subspace spanned by arbitrary three-qubit orthogonal unextendible product bases (UPB) cannot be exactly distinguishable by local operations and classical communication. This not only reveals another intrinsic property of three-qubit orthogonal UPB, but also provides a class of locally indistinguishable subspaces with dimension 4. We also explicitly construct locally indistinguishable subspaces with dimensions 3 and 5, respectively. In particular, 3 is the minimal possible dimension of locally indistinguishable subspaces. Combining with the previous results, we conclude that any positive integer between 3 and 7 is the possible dimension of some three-qubit locally indistinguishable subspace.

quant-ph↗