Local Unitary Operations Research Articles

Boundary states for entanglement robustness under dephasing and bit flip channels**Project supported by the National Key Research and Development Program of China (Grant No. 2016YFE0200700) and the National Natural Science Foundation of China (Grant Nos. 61627820 and 61934003).

We investigate the robustness of entanglement for a multiqubit system under dephasing and bit flip channels. We exhibit the difference between the entanglement evolution of the two forms of special states, which are locally unitarily equivalent to each other and therefore possess precisely the same entanglement properties, and demonstrate that the difference increases with the number of qubits n. Moreover, those two forms of states are either the most robust genuine entangled states or the most fragile ones, which confirm that local unitary (LU) operations can greatly enhance the entanglement robustness of n-qubit states.

Multicast-based multiparty remote state preparation schemes of two-qubit states

We propose two novel schemes for multicast-based remote preparation of two kinds of two-qubit states with the aid of the eight-qubit entangled states as the quantum channels. In our scenario, the different encoded information is transmitted synchronously from one sender to many spatially separated receivers. Moreover, it is shown that the first scheme is realized successfully and efficiently, and only the four-qubit projective measurement and appropriate local unitary operations are needed. In particular, the eight-qubit partially entangled channel is utilized in the second.

Annular Controlled Teleportation

Extending the idea of bidirectional controlled teleportation, we propose a general framework to investigate the annular controlled teleportation (ACT). Under the control of the supervisor, each party transfers her single-qubit state to nearby one with the help of pre-shared entanglement and local unitary operations as well as classical communications. As the examples, we give two specific ACT protocols by using a seven-qubit cluster state and a nine-qubit entangled state as the quantum channel respectively to prove that this framework is possible.

Fidelity-based unitary operation-induced quantum correlation for continuous-variable systems

We propose a measure of nonclassical correlation [Formula: see text] in terms of local Gaussian unitary operations based on square of the fidelity [Formula: see text] for bipartite continuous-variable systems. This quantity is easier to be calculated or estimated and is a remedy for the local ancilla problem associated with the geometric measurement-induced nonlocality. A simple computation formula of [Formula: see text] for any [Formula: see text]-mode Gaussian states is presented and an estimation of [Formula: see text] for any [Formula: see text]-mode Gaussian states is given. For any [Formula: see text]-mode Gaussian states, [Formula: see text] does not increase after performing a local Gaussian channel on the unmeasured subsystem. Comparing [Formula: see text] in scale with other quantum correlations such as Gaussian geometric discord for two-mode symmetric squeezed thermal states reveals that [Formula: see text] is much better in detecting quantum correlations of Gaussian states.

A quantum walk induced by Hoffman graphs and its periodicity

In order to define the discrete-time quantum walks, we divide the vertex set of a graph by two partitions and give a time evolution operator by a product of two local unitary operators on each partition. In this paper, we make such partitions from a Hoffman graph, which gives generalization of line graphs, and study a staggered walk. We first show that the square of the time evolution operator of the Grover walk is unitarily equivalent to that of the staggered walk on the generalized line graph induced from a Hoffman graph. Furthermore, we focus on the periodicity of quantum walks and show that the staggered walk on the generalized line graph is periodic whenever the Grover walk on the original graph is periodic by using the unitary equivalence. This work is the first trial to develop the relation between quantum walks and Hoffman graphs.

Three-party quantum secret sharing against collective noise

In this paper, based on logical GHZ states and logical χ-states, we present four three-party quantum secret sharing protocols immune to the collective-dephasing noise and the collective-rotation noise, respectively. They make full use of the measurement correlation property of multi-particle entangled states and local unitary operations. Compared with existing three-party quantum secret sharing protocols against collective noise, our protocols are the most efficient. Furthermore, these protocols are congenitally free from the Trojan horse attacks.

Demonstration of the no-hiding theorem on the 5-Qubit IBM quantum computer in a category-theoretic framework

Quantum no-Hiding theorem, first proposed by Braunstein and Pati [Phys. Rev. Lett. 98, 080502 (2007)], was verified experimentally by Samal et al. [Phys. Rev. Lett. 186, 080401 (2011)] using NMR quantum processor. Till then, this fundamental test has not been explored in any of the experimental architecture. Here, we demonstrate the above no-hiding theorem using the IBM 5Q quantum processor. Categorical algebra developed by Coecke and Duncan [New J. Phys. 13, 043016 (2011)] has been used for better visualization of the no-hiding theorem by analyzing the quantum circuit using the ZX calculus. The experimental results confirm the recovery of missing information by the application of local unitary operations on the ancillary qubits.

Cryptanalysis of the Efficient Controlled Quantum Secure Direct Communication and Authentication by Using Four Particle Cluster States Protocol

It shows that there are some serious security issues in the controlled quantum secure direct communication (CQSDC) with authentication protocol based on four particle cluster states via quantum one-time pad and local unitary operations. Some information of the identity strings of the receiver and the controller can be stolen without being detected by the intercept-selectively-measure-resend (ISMR) attack. Furthermore, an eavesdropper (Eve) can eavesdrop on some information of the secret message that the sender transmits. In addition, the receiver can obtain half of information about the secret message without any active attack and the permission of the controller, which is not allowed in a secure CQSDC protocol. The receiver can take the ISMR attack to obtain more information about the secret message without the permission of the controller. If running the protocol many times, the identity strings of the receiver and the controller can be completely acquired by Eve. Finally, the original CQSDC protocol is improved to a secure one.

Robustness of the fractional topological phase to dephasing

In this work, we demonstrate theoretically via Kraus maps that the fractional topological phase in qudits is robust to phase noise. In our proposal, dephasing noise is inserted in an optical setup designed for measuring the fractional topological phase on photonic qudits encoded in path variables after local unitary operations are applied. Qudit states can be prepared with photon pairs generated by spontaneous parametric down-conversion crossing a multiple-slit array. Polarization as an additional degree of freedom and a two-photon longitudinal interferometer are the basic ingredients of the proposed setup. Although the visibility of the interference pattern decreases when noise is added, our calculation shows that the topological phase is preserved. This result stimulates further studies for the use of fractional topological phase in quantum gates due to its robustness in environments where phase noise is present.

Schemes to avoid entanglement sudden death of decohering two qubit system

We investigate the entanglement dynamics of two interacting qubits in a common vacuum environment. The inevitable environment interaction leads to entanglement sudden death (ESD) in a two qubit entangled system. The entanglement dynamics can be modified by the use of local unitary operations (quantum gates), applied on the system during its evolution. We show that these operations not only delays or avoids the ESD but also advances the entanglement revival with an enhanced entanglement depending on the time of operation. We have analytically found out different time windows for switching with different quantum gates so that the ESD can be completely avoided in the subsequent evolution of the system. Our result offers practical applications in the field of quantum information processing where the entanglement is a necessary resource.

Fractional topological phase measurement with a hyperentangled photon source

Pairs of photons simultaneously entangled in their path and polarization degrees of freedom are used to measure the topological phase acquired by bipartite entangled states. Conditional phase local unitary operations having the polarization degree of freedom as the control variable are applied. Qudits of arbitrary dimensions are encoded on the photons transverse positions while polarization entanglement is used as an auxiliary resource for quantum interference measurements. With this scheme the fractional phases predicted for dimensions d = 2, 3 and 4 could be measured with visibilities for the interference curves beyond the limit allowed for classical sources, which is expected for a source of quantum correlated photons. The strategy of perform a quantum interferometry experiment with photons entangled in an auxiliary degree of freedom and apply unitary local operations conditioned to this auxiliary variable shows an increase to the signal to noise ratio, simplifies alignment and can be used in different applications. This offers an interesting perspective for the efficient implementation of phase gates in quantum computing with hyperentangled photon sources in polarization and path degrees of freedom. Furthermore, one can conjecture whether the measured phase can serve as a dimensionality identifier of the Hilbert space dimension for an unknown state preparation.

Arbitrated quantum signature scheme based on quantum walks

<sec>Quantum signature is quantum counterpart of classical digital signature, which has been widely applied to modern communication, such as electronic payment, electronic voting and electronic medical, owing to its great implication in ensuring the authenticity and the integrity of the message and the non-repudiation. Arbitrated quantum signature (AQS) is an important and practical type of quantum signature. The AQS algorithm is a symmetric key cryptography-based quantum signature algorithm, and its implementation requires the trusted arbitrator to be directly involved. In this paper, employing the key-controlled chained CNOT (KCCC) operation as the appropriate encryption (decryption) algorithm, we suggest an AQS scheme based on teleportation of quantum walks with two coins on a four-vertex cycle, which is used to transfer the message copy from the sender to the receiver. In light of the model of teleportation of quantum walks, the sender encodes the message to be signed into her or his coin state, and the necessary entangled states can be created as a result of the conditional shift between the coin state and the position state. The measurements performed on the generated entangled states are the bases of signature production and message recovery. Then according to the classical measurement results from the sender, the receiver performs the appropriate local unitary operations (i.e., Pauli operations) on his own coin state to recover the original message and further verifies the validity of the completed signature by using the appropriate verification algorithms under the aid of the trustworthy arbitrator. </sec><sec>The suggested AQS scheme makes the following contributions: 1) the necessary entangled states for quantum teleportation of message copy do not need preparing in advance, and they can be produced automatically by the first step of quantum walks; 2) the scheme satisfies the features of non-repudiation, un-forgeability and non-disavowal due to the use of the KCCC operation; 3) the scheme may be achieved by linear optical elements such as beam splitters, wave plates, etc., because quantum walks have proven to be realizable in different physical systems and experiments.</sec><sec>Analysis and discussion show that the proposed AQS scheme possesses the impossibility of disavowals by the signer and the receiver and impossibility of forgeries by anyone. Comparisons reveal that the designed AQS protocol is favorable. Furthermore, it provides an idea by employing the quantum computing model into quantum communication protocols with a possible improvement with respect to its favorable properties, for example, the automatic generation of entangled states via the first step of quantum walks on different models. In the near future, we will further investigate the production of entanglement by quantum walks and its applications with some improvements in designing the quantum communication protocols.</sec>

Quantum Correlation Based on Uhlmann Fidelity for Gaussian States.

A quantum correlation for -mode continuous-variable systems is introduced in terms of local Gaussian unitary operations performed on Subsystem A based on Uhlmann fidelity F. This quantity is a remedy for the local ancilla problem associated with the geometric measurement-induced correlations; is local Gaussian unitary invariant; is non-increasing under any Gaussian quantum channel performed on Subsystem B;and is an entanglement monotone when restricted to pure Gaussian states in the -mode case. A concrete formula for -mode symmetric squeezed thermal states (SSTSs) is presented. We also compare with other quantum correlations in scale, such as Gaussian quantum discord and Gaussian geometric discord, for two-mode SSTSs, which reveals that has some advantage in detecting quantum correlations of Gaussian states.

Necessary and Sufficient Condition for Quantum Computing

A necessary and sufficient condition for quantum computing performed with, for example, the Deutsch-Jozsa algorithm or the Bernstein-Vazirani algorithm, has theoretically been investigated. Assume a 2N qubit-quantum computing which starts with the state $|\overbrace {0,0,...,0,1}^{N}\rangle |\overbrace {1,1,...,1}^{N}\rangle $ as follows: Uf|0,0,...,0,1〉|1,1,...,1〉 = |0,0,...,0,1〉 $ |\overline {f(0,0,...,0,1)}\rangle . $ Surprisingly the relation f(x) = f(−x) is the necessary and sufficient condition of holding this fundamental relation if local unitary operations can be used.

Comparing coherence and entanglement under resource non-generating unitary transformations

Quantum coherence is a fundamental property of quantum systems that can be studied within the framework of a quantum resource theory. From this resource-theoretic approach, many similarities emerge between coherence and quantum entanglement, the latter being another prominent quantum resource. In this paper, we study the relationship between coherence and entanglement from the perspective of resource loss/gain by unitary dynamics. Specifically we consider how much coherence can be either generated or destroyed on a bipartite system when the action is restricted to local unitary (LU) operations. For pure states, we find that the relative entropy of entanglement and the robustness of entanglement provide tight lower bounds on the amount of coherence that can be destroyed by LUs, as measured by the relative entropy and measures of coherence, respectively. This provides new operational interpretations of the entanglement entropy and robustness of entanglement in pure states as the minimal amount of coherence that persists when the system is subjected to LU transformations. We then study the amount of bipartite pure entanglement that can be either maximized or minimized when the action is restricted to a global incoherent operation. For two-qubit pure states, maximum and minimum are shown in terms of coefficients of given state with incoherent basis. Finally we consider a generalized version of the recently introduced CCP of a quantum channel.

Quantifying Discriminating Strength for Gaussian States**Supported by Natural Science Foundation of China under Grant Nos. 11671006, 11671294, and Outstanding Youth Foundation of Shanxi Province (201701D211001)

The discriminating strength induced by local Gaussian unitary operators for any (n + m)-mode Gaussian state is introduced in [Phys. Rev. A 83 (2011) 042325]. In this paper, we further discuss the quantity by restricting to Hilbert-Schmidt norm. The analytic formulas of for two-mode squeezed thermal states and mixed thermal states are given. Then, the relationship between and for some special Gaussian channels Φ is discussed. In addition, is compared with Gaussian entanglement for symmetric squeezed thermal states.

Deterministic Quantum Controlled Teleportation of Arbitrary Multi-qubit States via Partially Entangled States

In this paper, an efficient proposal for quantum controlled teleporatation of arbitrary multi-qubit states is presented via three-qubit non-maximally entangled states. The successful probability is viewed as one of the most important performance parameters for quantum teleportation. The significant advantage of our scheme is that the successful probability is independent of the coefficients of partially entangled states, and is always equal to 100% in spite of using non-maximally quantum channel. From the concrete implementation processes of this proposal, it could be found that only the usual Bell-state measurement, simple single-qubit projective measurement and common single-qubit local unitary operations, of which the physical realization has been widely explored, need to be performed without the introduction of auxiliary particles.

Interacting Floquet topological phases in three dimensions

In two dimensions, interacting Floquet topological phases may arise even in the absence of any protecting symmetry, exhibiting chiral edge transport that is robust to local perturbations. We explore a similar class of Floquet topological phases in three dimensions, with translational invariance but no other symmetry, which also exhibit anomalous transport at a boundary surface. By studying the space of local 2D unitary operators, we show that the boundary behavior of such phases falls into equivalence classes, each characterized by an infinite set of reciprocal lattice vectors. In turn, this provides a classification of the 3D bulk, which we argue is complete. We demonstrate that such phases may be generated by exactly-solvable `exchange drives' in the bulk. In the process, we show that the edge behavior of a general exchange drive in two or three dimensions can be deduced from the geometric properties of its action in the bulk, a form of bulk-boundary correspondence.

Stochastic local operations and classical communication (SLOCC) and local unitary operations (LU) classifications of n qubits via ranks and singular values of the spin-flipping matrices

We construct $\ell $-spin-flipping matrices from the coefficient matrices of pure states of $n$ qubits and show that the $\ell $-spin-flipping matrices are congruent and unitary congruent whenever two pure states of $n$ qubits are SLOCC and LU equivalent, respectively. The congruence implies the invariance of ranks of the $\ell $-spin-flipping matrices under SLOCC and then permits a reduction of SLOCC classification of n qubits to calculation of ranks of the $\ell $-spin-flipping matrices. The unitary congruence implies the invariance of singular values of the $\ell $-spin-flipping matrices under LU and then permits a reduction of LU classification of n qubits to calculation of singular values of the $\ell $-spin-flipping matrices. Furthermore, we show that the invariance of singular values of the $\ell $-spin-flipping matrices $\Omega _{1}^{(n)}$ implies the invariance of the concurrence for even $n$ qubits and the invariance of the n-tangle for odd $n$ qubits. Thus, the concurrence and the n-tangle can be used for LU classification and computing the concurrence and the n-tangle only performs additions and multiplications of coefficients of states.

Nonclassicality by Local Gaussian Unitary Operations for Gaussian States.

A measure of nonclassicality in terms of local Gaussian unitary operations for bipartite Gaussian states is introduced. is a faithful quantum correlation measure for Gaussian states as product states have no such correlation and every non product Gaussian state contains it. For any bipartite Gaussian state , we always have , where the upper bound 1 is sharp. An explicit formula of for -mode Gaussian states and an estimate of for -mode Gaussian states are presented. A criterion of entanglement is established in terms of this correlation. The quantum correlation is also compared with entanglement, Gaussian discord and Gaussian geometric discord.