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arXiv:1204.5589 · Physical Review A

Quantifying the noise of a quantum channel by noise addition

Antonella De Pasquale, Vittorio Giovannetti

Published Nov 1, 2012• 19 citations
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Abstract

In this paper we introduce a way to quantify the noise level associated to a given quantum transformation. The key mechanism lying at the heart of the proposal is noise addition: in other words we compute the amount of extra noise we need to add to the system through convex combination with a reference noisy map or by reiterative applications of the original map, before the resulting transformation becomes entanglement breaking. We also introduce the notion of entanglement-breaking channels of order $n$ (i.e., maps which become entanglement breaking after $n$ iterations), and the associated notion of amendable channels (i.e., maps which can be prevented from becoming entanglement breaking after iterations by interposing proper quantum transformations). Explicit examples are analyzed in the context of qubit and one-mode Gaussian channels.

DOI
10.1103/physreva.86.052302

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In this paper we introduce a way to quantify the noise level associated to a given quantum transformation. The key mechanism lying at the heart of the proposal is noise addition: in other words we compute the amount of extra noise we need to add to the system through convex combination with a reference noisy map or by reiterative applications of the original map, before the resulting transformation becomes entanglement breaking. We also introduce the notion of entanglement-breaking channels of order nnn (i.e., maps which become entanglement breaking after nnn iterations), and the associated notion of amendable channels (i.e., maps which can be prevented from becoming entanglement breaking after iterations by interposing proper quantum transformations). Explicit examples are analyzed in the context of qubit and one-mode Gaussian channels.
We study a set of new functionals (called entanglement–breaking indices) which characterize how many local iterations of a given (local) quantum channel are needed in order to completely destroy the entanglement between the system of interest over which the transformation is defined and an external ancilla. The possibility of contrasting the noisy effects introduced by the channel iterations via the action of intermediate (filtering) transformations is analyzed. We provide some examples in which our functionals can be exactly calculated. The differences between unitary and non-unitary filtering operations are analyzed showing that, at least for systems of dimension d larger than or equal to 3, the non-unitary choice is preferable (the gap between the performances of the two cases being divergent in some cases). For d = 2 (qubit case), on the contrary, no evidences of the presence of such gap is revealed: we conjecture that for this special case unitary filtering transformations are optimal. The scenario in which more general filtering protocols are allowed is also discussed in some detail. The case of a depolarizing noise acting on a two–qubit system is exactly solved in a general case.
Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry. We show that a channel and its conjugate have the same minimal output entropy and maximal output p-norm. It then follows that the additivity and multiplicativity conjectures for these measures of optimal output purity hold for a product of channels if and only if they also hold for the product of their conjugates. This allows us to reduce these conjectures to the special case of maps taking M_d to M_d' with a minimal representation of dimension at most d. We find explicit expressions for the conjugates for a number of well-known examples, including entanglement-breaking channels, unital qubit channels, the depolarizing channel, and a subclass of random unitary channels. For the entanglement-breaking channels, channels this yields a new class of channels for which additivity and multiplicativity of optimal output purity can be established. For random unitary channels using the generalized Pauli matrices, we obtain a new formulation of the multiplicativity conjecture. The conjugate of the completely noisy channel plays a special role and suggests a mechanism for using noise to transmit information.
Entanglement fidelity quantifies how well a quantum channel preserves the correlations between a transmitted system and an inaccessible reference system. We derive closed-form expressions for the entanglement fidelity associated with several standard quantum noise models, including the random Pauli-X, dephasing, depolarizing, Werner-Holevo, generalized Pauli (Weyl), and amplitude-damping channels. For each model, we express the entanglement fidelity in terms of a general input density operator ρρρ, using Schumacher's Kraus-operator approach, which provides a channel-agnostic recipe applicable to any completely positive trace-preserving (CPTP) map with a finite Kraus representation. We then specialize to a communication scenario in which the source emits a two-letter parametric alphabet, thereby making explicit the dependence of entanglement preservation on both channel and source parameters. The resulting expressions enable direct comparisons of channel performance and rankings for representative families of input states, including common qubit states.
ABSTRACT In this paper, we study the effects of additive Gaussian noise on Fock states. We put forth a simple analytical formula for the resulting output states. Additionally, we conduct numerical analysis into several key properties of noisy Fock states, including their purity, nonclassicality and non‐Gaussianness, with respect to the noise parameter. Finally, we examine the impact of noise on entangled N00N states, focusing on its entanglement‐breaking effects, which we quantify using the logarithmic negativity measure. Regarding purity, we observe that for a fixed noise level, the purity of a noisy Fock state decreases as the number of photons increases. In terms of non‐Gaussianity, we show that at any given noise level, higher Fock states exhibit higher non‐Gaussianity compared to lower ones. Although all noisy Fock states eventually lose their nonclassical characteristics under sufficient noise, higher Fock states exhibit more pronounced nonclassical features in low‐noise conditions, whereas lower Fock states show greater nonclassicality in high‐noise environments. We quantitatively show that the state is the most robust against additive Gaussian noise among the N00N states.
We characterise Gaussian quantum channels that are Gaussian incompatibility breaking, that is, transform every set of Gaussian measurements into a set obtainable from a joint Gaussian observable via Gaussian postprocessing. Such channels represent local noise which renders measurements useless for Gaussian EPR-steering, providing the appropriate generalisation of entanglement breaking channels for this scenario. Understanding the structure of Gaussian incompatibility breaking channels contributes to the resource theory of noisy continuous variable quantum information protocols.