6 pages, accepted for publication in the special issue of IJQI "Noise, Information and Complexity at Quantum Scale", eds. S. Mancini and F. Marchesoni (2008)
发表信息：
International Journal of Quantum Information 6, 621 (2008)
We analyze qubit channels by exploiting the possibility of representing two-level quantum systems in terms of characteristic functions. To do so, we use functions of non-commuting variables (Grassmann variables), defined in terms of generalized displacement operators, following an approach which resemble the one adopted for continuous-variable (Bosonic) systems. It allows us to introduce the notion of qubit Gaussian channels and to show that they share similar properties with the corresponding continuous-variable counterpart. Some examples of qubit channels are investigated using this approach.
Classical capacity of unital qubit channels is well known, whereas that of nonunital qubit channels is not. We find lower and upper bounds on classical capacity of nonunital qubit channels by using a recently developed decomposition technique relating nonunital and unital positive qubit maps.
We propose channel matrices by using unfolding matrices from their reduced density matrices. These channel matrices can be a criterion for a channel whether the channel can teleport or not any qubit state. We consider a special case, teleportation of the arbitrary two-qubit state by using the four-qubit channel. The four-qubit channel can only teleport if the rank of the related channel matrix is four.
We consider the problem of discriminating qubit states that are sent over a quantum channel and derive a necessary and sufficient condition for an optimal measurement to be preserved by the channel. We apply the result to the characterization of optimal measurement preserving (OMP) channels for a given qubit ensemble, e.g., a set of two states or a set of multiple qubit states with equal a priori probabilities. Conversely, we also characterize qubit ensembles for which a given channel is OMP, such as unitary and depolarization channels. Finally, we show how the sets of OMP channels for a given…
Inspired by the Solovay-Kitaev decomposition for approximating unitary operations as a sequence of operations selected from a universal quantum computing gate set, we introduce a method for approximating any single-qubit channel using single-qubit gates and the controlled-NOT (CNOT). Our approach uses the decomposition of the single-qubit channel into a convex combination of "quasiextreme" channels. Previous techniques for simulating general single-qubit channels would require as many as 20 CNOT gates, whereas ours only needs one, bringing it within the range of current experiments.
In quantum error correction, it is an important assumption that errors on different qubits are independent. In our previous work [Phys. Rev. A {\bf 92}, 052320 (2015)], the generality of the concatenated five-qubit code has been investgated when noises of the principal system and the auxiliary environment are assumed to be the same. In the error correction with concatenated code, tiny differences (in fidelity or independent errors) between initial quantum channels may introduce different effective channels in the next level, and therefore, it is necessary to study a meaningful question: Does f…
We study entanglement dynamics of pure three-qubit Greenberger-Horne-Zeilinger-type (GHZ-type) entangled states when one, two or three qubits being subjected to general local noise. Employing a lower bound for three-qubit concurrence as an entanglement measure, we show that for some many-sided noisy channels the entanglement dynamics can be complete described by the evolution of the entangled states in single-sided channels.
We study the classical capacity of a forgetful quantum memory channel that switches between two qubit depolarizing channels according to an ergodic Markov chain. The capacity of this quantum memory channel depends on the parameters of the two depolarizing channels and the memory of the ergodic Markov chain. When the number of input qubit's is two, we show that depending on channel parameters either the maximally entangled input states or product input states achieve the classical capacity. Our conjecture based on numerics is that as the number of input qubits are increased the classical capaci…
The archetypal one-qubit noisy channels ---depolarizing, phase-damping and amplitude-damping channels--- describe both Markovian and non-Markovian evolution. Simple microscopic models for the depolarizing channel, both classical and quantum, are considered. Microscopic models which describe phase damping and amplitude damping channels are briefly reviewed.
Entanglement breaking channels play a significant role in quantum information theory. In this work we investigate qubit channels through their property of `non-locality breaking', defined in a natural way but within the purview of CHSH nonlocality. This also provides a different perspective on the relationship between entanglement and nonlocality through the dual picture of quantum channels instead of through states. For a channel to be entanglement breaking it is sufficient to `break' the entanglement of maximally entangled states. We provide examples to show that for CHSH nonlocality breakin…
It was shown [T.S. Cubitt et al., IEEE Trans. Inform. Theory 57, 8114 (2011)] that there exist quantum channels where a single use cannot transmit classical information perfectly yet two uses can. This phenomenon is called the superactivation of the zero-error classical capacity which does not occur in classical channels. In this paper, it is shown that qubit channels cannot generate the superactivation.
By preparing an input state and measuring an observable for the output state, we can measure a quantum channel. Following the formulation given by Xiao et al., we study an uncertainty relation for ancilla-free measurements of random unitary channels acting on a qubit. We obtain an explicit formula and give a necessary and sufficient condition for this formula to be nontrivial.
The Continuous-Variable (CV) quantum state of light allows for the teleportation of a Discrete-Variable (DV) photonic qubit. Such an operation is useful in the realm of hybrid quantum networks. However, it is known that the teleportation of a DV qubit via a Gaussian CV resource channel is severely limited under channel loss, with a teleportation fidelity beyond the classical limit unattainable for losses exceeding a small threshold of 0.5 dB. In this work, we present a new non-deterministic teleportation protocol that combines a Gaussian CV resource channel with a modified form of the Bell Sta…
It is known that maximally entangled Bell state and three-qubit W-type states are very useful in various quantum information processing task. Thus the problem of preparation of these type of states is very important in quantum information theory. But the factor which prohibit the generation of the above mentioned pure states shared between two and three distant partners is decoherence. When we send one qubit, from a two qubit state, through decoherence channel like amplitude damping channel, the purity of the qubit is lost and it ends up with a mixed state. Therefore it is very difficult to ke…
In secure quantum communication protocols, a set of single qubits prepared using 2 or more mutually unbiased bases or a set of $n$-qubit ($n\geq2$) entangled states of a particular form are usually used to form a verification string which is subsequently used to detect traces of eavesdropping. The qubits that form a verification string are referred to as decoy qubits, and there exists a large set of different quantum states that can be used as decoy qubits. In the absence of noise, any choice of decoy qubits provides equivalent security. In this paper, we examine such equivalence for noisy env…
We introduce a general odd qubit entangled system composed of GHZ and Bell pairs and explicate its usefulness for quantum teleportation, information splitting and superdense coding. After demonstrating the superdense coding protocol on the five qubit system, we prove that '2N +1' classical bits can be sent by sending 'N+1' quantum bits using this channel. It is found that the five-qubit system is also ideal for arbitrary one qubit and two qubit teleportation and quantum information splitting (QIS). For the single qubit QIS, three different protocols are feasible, whereas for the two qubit QIS,…
The action of qubit channels on projective measurements on a qubit state is used to establish an equivalence between channels and properties of generalized measurements characterized by bias and sharpness parameters. This can be interpreted as shifting the description of measurement dynamics from the Schrodinger to the Heisenberg picture. In particular, unital quantum channels are shown to induce unbiased measurements. The Markovian channels are found to be equivalent to measurements for which sharpness is a monotonically decreasing function of time. These results are illustrated by considerin…
The Solovay-Kitaev theorem allows us to approximate any single-qubit gate to arbitrary accuracy with a finite sequence of fundamental operations from a universal set of gates. Inspired by this decomposition, we present a quantum channel simulator capable of implementing any completely positive trace-preserving map. Our realization consists of one ancillary qubit, encoded in the transverse mode of a laser beam (orbital degree of freedom), one qubit system, encoded in its polarization (spin), one spin-orbit CNOT gate and four single-qubit operations performed with prisms and polarization compone…
We propose two controlled remote state preparation protocols via partially entangled channels. One prepares a single-qubit state and the other prepares a two-qubit state. Different from other controlled remote state preparation schemes which also utilize partially entangled channels, neither auxiliary qubits nor two-qubit unitary transformations are required in our schemes and the success probabilities are independent of the coefficients of the quantum channel. The success probabilities are 50% and 25% for arbitrary single-qubit states and two-qubit states, respectively. We also show that the …
We study the performance of two measures of non-Markovianity in detecting memory effects in two-qubit dephasing channels. By combining independent Markovian and non-Markovian noise on the qubits, our results show that the trace distance measure is able to detect the memory effects when at least one of the local channels displays non-Markovianity. A measure based on channel capacity, in turn, becomes non-zero when the global two-qubit dynamics shows memory effects. We apply these schemes to a well-known superdense coding protocol and demonstrate an optimal noise configuration to maximize the in…
Using the axiomatic definition of the coherence measure, such as the $l_{1}$ norm and the relative entropy, we study the phenomena of two-qubit system quantum coherence through quantum channels where successive uses of the channels are memory. Different types of noisy channels with memory, such as amplitude damping, phase-damping, and depolarizing channels effect on quantum coherence have been discussed in detail. The results show that, quantum channels with memory can efficiently protect coherence from noisy channels. Particularly, as channels with perfect memory, quantum coherence is unaffec…
We experimentally demonstrate a single-qubit decohering quantum channel using linear optics. We implement the channel, whose special cases include both the amplitude-damping channel and the bit-flip channel, using a single, static optical setup. Following a recent theoretical result [M. Piani et al., Phys. Rev. A, 84, 032304 (2011)], we realize the channel in an optimal way, maximizing the probability of success, i.e., the probability for the photonic qubit to remain in its encoding. Using a two-photon entangled resource, we characterize the channel using ancilla-assisted process tomography an…
We consider the estimation of two-qubit interactions when initial states of both qubit can be locally controlled, while the final state of only one qubit can be measured. This amounts to realize a model of quantum channel communication with environment assistance. In such a framework the unitaries' parameters space results a tetrahedron in $\mathbb{R}^3$. On its edges the problem, becoming of single parameter estimation, can be exactly solved and we derive optimal probe states and estimators. Our results show that the possibility of environment assistance is always beneficial, while the usage …
Incompatibility and nonlocality are not only of foundational interest but also act as important resources for quantum information theory. In the Clauser-Horne-Shimony-Holt (CHSH) scenario, the incompatibility of a pair of observables is known to be equivalent to Bell nonlocality. Here, we investigate these notions in the context of qubit channels. The Bell-CHSH inequality has a greater perspective--compared to any genuine tripartite nonlocality scenario--while determining the interplay between nonlocality breaking qubit channels and incompatibility breaking qubit channels. In the Bell-CHSH sce…
Quantum temporal correlations exhibited by violations of Leggett-Garg Inequality (LGI) and Temporal Steering Inequality (TSI) are in general found to be non-increasing under decoherence channels when probed on two-qubit pure entangled states. We study the action of decoherence channels, such as amplitude damping, phase-damping and depolarising channels when partial memory is introduced in a way such that two consecutive uses of the channels are time-correlated. We show that temporal correlations demonstrated by violations of the above temporal inequalities can be protected against decoherence …
Bidirectional quantum teleportation is a fundamental protocol for exchanging quantum information between two quantum nodes. All bidirectional quantum teleportation protocols till now have achieved a maximum efficiency of $40\%$. Here, we propose a new scheme for symmetric bidirectional quantum teleportation using a six-qubit cluster state as the quantum channel, for symmetric ($3\leftrightarrow3$) qubit bidirectional quantum teleportation of a special three-qubit entangled state. The novelty of our scheme lies in its generalization for ($N\leftrightarrow N$) qubit bidirectional quantum telepor…
In the present article, the robustness of entanglement in two qubits maximally entangled mixed states (MEME) have been studied under quantum decoherence channels. Here we consider bit flip, phase flip, bit-phase-flip, amplitude damping, phase damping and depolarization channels. To quantify the entanglement, the concurrence has been used as an entanglement measure. During this study interesting results have been found for sudden death and birth of entanglement under bit flip and bit-phase-flip channels. While amplitude damping channel produces entanglement sudden death and does not allow re-bi…
We study entanglement dynamics of the three-qubit system which is initially prepared in pure Greenberger-Horne- Zeilinger (GHZ) or W state and transmitted through one of the Pauli channels $σ_z, \, σ_x, \, σ_y$ or the depolarizing channel. With the help of the lower bound for three-qubit concurrence we show that the W state preserves more entanglement than the GHZ state in transmission through the Pauli channel $σ_z$. For the Pauli channels $σ_x, \, σ_y$ and the depolarizing channel, however, the entanglement of the GHZ state is more resistant against decoherence than the W-type entanglement. …
The Markovian dynamics of a qubit is investigated in the scheme of random unitary dynamics, where Kraus operators are changed by an extra noise. The behavior of Markovianity is explored in the perturbed scenario. We provide a new algorithm for checking CP-divisibility (Markovianity) of a dynamical map.
A quantum observable and a channel are considered compatible if they form parts of the same measurement device, otherwise they are incompatible. Constrains on compatibility between observables and channels can be quantified via relations highlighting the necessary trade-offs between noise and disturbance within quantum measurements. In this paper we shall discuss the general properties of these compatibility relations, and then fully characterize the compatibility conditions for an unbiased qubit observable and a Pauli channel. The implications of the characterization are demonstrated on some …
A protocol based on quantum error correction based characterization of quantum dynamics (QECCD) is developed for quantum process tomography on a two-qubit system interacting dissipatively with a vacuum bath. The method uses a 5-qubit quantum error correcting code that corrects arbitrary errors on the first two qubits, and also saturates the quantum Hamming bound. The dissipative interaction with a vacuum bath allows for both correlated and independent noise on the two-qubit system. We study the dependence of the degree of the correlation of the noise on evolution time and inter-qubit separatio…
In this work, a quantum error correction (QEC) procedure with the concatenated five-qubit code is used to construct a near-perfect effective qubit channel (with a error below $10^{-5}$) from arbitrary noise channels. The exact performance of the QEC is characterized by a Choi matrix, which can be obtained via a simple and explicit protocol. In a noise model with five free parameters, our numerical results indicate that the concatenated five-qubit code is general: To construct a near-perfect effective channel from the noise channels, the necessary size of the concatenated five-qubit code depend…
We suggest and demonstrate an all-optical quantum simulator for single-qubit noisy channels originating from the interaction with a fluctuating field. The simulator employs the polarization degree of freedom of a single photon, and exploits its spectral components to average over the realizations of the stochastic dynamics. As a proof of principle, we run simulations of dephasing channels driven either by Gaussian (Ornstein-Uhlenbeck) or non-Gaussian (random telegraph) stochastic processes.
The concept of divisibility of dynamical maps is used to introduce an analogous concept for quantum channels by analyzing the \textit{simulability} of channels by means of dynamical maps. In particular, this is addressed for Lindblad divisible, completely positive divisible and positive divisible dynamical maps. The corresponding L-divisible, CP-divisible and P-divisible subsets of channels are characterized (exploiting the results by Wolf et al., Comm. Math. Phys., 279(1):147-168, 2008) and visualized for the case of qubit channels. We discuss the general inclusions among divisibility sets an…
We consider two qubit teleportation via quantum channel affected by amplitude damping noise. Addressing the same problem, X. Hu, Y. Gu, Q. Gong and G. Guo [Phys. Rev. A 81, 054302, (2010)] recently showed that in presence of noise, subjecting more qubits in quantum channel to amplitude damping can increase the fidelity of teleportation protocol. However, in this paper, by making some adjustments on quantum channel, we obtain teleportation fidelity which is even higher than one in the case of X. Hu et al. Moreover, our strategy is simpler than quantum distillation and compared to using weak mea…
We address the issue of the discrimination between two-qubit amplitude damping channels by exploring several strategies. For the single-shot, we show that the excited state does not always give the optimal input, and that side entanglement assistance has limited benefit. On the contrary, feedback assistance from the environment is more beneficial. For the two-shot, we prove the in-utility of entangled inputs. Then focusing on individual (local) measurements, we find the optimal adaptive strategy.
