One-way Local Operations Research Articles

Entanglement Monogamy via Multivariate Trace Inequalities

Entropy is a fundamental concept in quantum information theory that allows to quantify entanglement and investigate its properties, for example its monogamy over multipartite systems. Here, we derive variational formulas for relative entropies based on restricted measurements of multipartite quantum systems. By combining these with multivariate matrix trace inequalities, we recover and sometimes strengthen various existing entanglement monogamy inequalities. In particular, we give direct, matrix-analysis-based proofs for the faithfulness of squashed entanglement by relating it to the relative entropy of entanglement measured with one-way local operations and classical communication, as well as for the faithfulness of conditional entanglement of mutual information by relating it to the separably measured relative entropy of entanglement. We discuss variations of these results using the relative entropy to states with positive partial transpose, and multipartite setups. Our results simplify and generalize previous derivations in the literature that employed operational arguments about the asymptotic achievability of information-theoretic tasks.

Quantifying the performance of approximate teleportation and quantum error correction via symmetric 2-PPT-extendible channels

The ideal realization of quantum teleportation relies on having access to a maximally entangled state; however, in practice, such an ideal state is typically not available and one can instead only realize an approximate teleportation. With this in mind, we present a method to quantify the performance of approximate teleportation when using an arbitrary resource state. More specifically, after framing the task of approximate teleportation as an optimization of a simulation error over one-way local operations and classical communication (LOCC) channels, we establish a semi-definite relaxation of this optimization task by instead optimizing over the larger set of two-PPT-extendible channels. The main analytical calculations in our paper consist of exploiting the unitary covariance symmetry of the identity channel to establish a significant reduction of the computational cost of this latter optimization. Next, by exploiting known connections between approximate teleportation and quantum error correction, we also apply these concepts to establish bounds on the performance of approximate quantum error correction over a given quantum channel. Finally, we evaluate our bounds for various examples of resource states and channels.

Dense Coding with Locality Restriction on Decoders: Quantum Encoders versus Superquantum Encoders

We investigate practical dense coding by imposing locality restrictions on decoders and symmetry restrictions on encoders within the resource theory of asymmetry framework. In this task, the sender Alice and the helper Fred preshare an entangled state. Alice encodes a classical message into the quantum state using a symmetric preserving encoder and sends the encoded state to Bob through a noiseless quantum channel. The receiver Bob and helper Fred are limited to performing quantum measurements satisfying certain locality restrictions to decode the classical message. We are interested in the ultimate dense coding capacity under these constraints. Our contributions are summarized as follows. First, we derive both one-shot and asymptotic optimal achievable transmission rates of the dense coding task under different encoder and decoder combinations. Surprisingly, our results reveal that the transmission rate cannot be improved even when the decoder is relaxed from one-way local operations and classical communication (LOCC) to two-way LOCC, separable measurements, and partial transpose positive measurements of the bipartite system. Second, depending on the class of allowed decoders with certain locality restrictions, we relax the class of encoding operations to superquantum encoders in the general probability theory framework and derive dense coding transmission rates under this setting. For example, when the decoder is fixed to a separable measurement, theoretically, a positive operation is allowed as an encoding operation. Remarkably, even under this superquantum relaxation, the transmission rate still cannot be lifted. This fact highlights the universal validity of our analysis on practical dense coding beyond quantum theory.Received 10 October 2021Revised 6 May 2022Accepted 29 August 2022DOI:https://doi.org/10.1103/PRXQuantum.3.030346Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum channelsQuantum communicationQuantum entanglementResource theoriesQuantum Information

Complete classification of steerability under local filters and its relation with measurement incompatibility

Quantum steering is a central resource for one-sided device-independent quantum information. It is manipulated via one-way local operations and classical communication, such as local filtering on the trusted party. Here, we provide a necessary and sufficient condition for a steering assemblage to be transformable into another via local filtering. We characterize the equivalence classes with respect to filters in terms of the steering equivalent observables (SEO), first proposed to connect the problem of steerability and measurement incompatibility. We provide an efficient method to compute the extractable steerability that is maximal via local filters and show that it coincides with the incompatibility of the SEO. Moreover, we show that there always exists a bipartite state that provides an assemblage with steerability equal to the incompatibility of the measurements on the untrusted party. Finally, we investigate the optimal success probability and rates for transformation protocols (distillation and dilution) in the single-shot scenario together with examples.

Operator and Graph Theoretic Techniques for Distinguishing Quantum States via One-Way LOCC

We bring together in one place some of the main results and applications from our recent work on quantum information theory, in which we have brought techniques from operator theory, operator algebras, and graph theory for the first time to investigate the topic of distinguishability of sets of quantum states in quantum communication, with particular reference to the framework of one-way local quantum operations and classical communication (LOCC). We also derive a new graph-theoretic description of distinguishability in the case of a single-qubit sender.

Local unitary classification of generalized Bell state sets in C5⊗C5

In order to study a large number of generalized Bell states (GBSs) efficiently, it is important to classify them by local unitary (LU) equivalence. For example, there are C254(=12650) sets of four GBSs in C5⊗C5, which greatly exceed the C164(=1820) sets of four GBSs in C4⊗C4, and it is inconceivable to explore these sets directly without prior classification. In this paper, we consider the classification of GBS sets in C5⊗C5. First, two simple and useful characterizations of LU-equivalence of GBS sets are given. Then, one characterization is used to divide all 12 650 sets of four GBSs into eight equivalence classes including six locally distinguishable classes and two classes that cannot be discriminated by one-way local operations and classical communication. The second characterization is powerful and can be used to write a program, by using which all C255(=53130) sets of five GBSs in C5⊗C5 are divided into 21 equivalence classes including 9 locally distinguishable classes and 12 locally indistinguishable classes.

Distinguishability of generalized Bell states in arbitrary dimension system via one-way local operations and classical communication

We investigate the distinguishability of generalized Bell states (GBSs) in arbitrary dimension system by one-way local operations and classical communication (LOCC). Firstly, by analyzing three local unitary transformations, we define two concepts for each set $$\mathcal {L}$$ of GBSs, i.e., an admissible solutions set $$\mathcal {S}_{\mathcal {A}}$$ and a genuinely nonadmissible solutions set $$\mathcal {S}_{\mathcal {L}}$$ . Then, we show that a set $${\mathcal {L}}$$ of any GBSs can be distinguished by one-way LOCC if $${\mathcal {S}}_{\mathcal {A}}\backslash {\mathcal {S}}_{\mathcal {L}}$$ is nonempty. Fan’s and Wang et al.’s results (Phys Rev Lett 92:177905, 2004: Phys Rev A 99:022307, 2019) can be covered by our result. Comparing with Wang et al.’s result, our successful ratio of local distinguishability improves markedly, especially for even dimension system. In some cases, it can increase by nearly $$42\%$$ . However, there are still some cases where our method cannot work. For the uncovered cases, we define an uncovered set and present that there is no method which is more powerful than ours for local discrimination of GBSs if the uncovered set is a subset of the different set of GBSs.

Locally distinguishable maximally entangled states by two-way LOCC

We concentrate on the differences between one-way local operations and classical communication (1-LOCC) and two-way local operations and classical communication (2-LOCC) in distinguishing maximally entangled states (MESs). We analyze the 2-LOCC distinguishability of k MESs which were constructed by using 1-LOCC indistinguishable qubit lattice states (LSs) (Nathanson in Phys Rev A 88:062316, 2013). We give a sufficient condition that illustrates which type of Nathanson’s constructed states can be distinguished by 2-LOCC. It partly solves an open question proposed by Nathanson. Furthermore, we answer the question in a simpler and more efficient way, at least to some extent. That is, the k constructed states can be distinguished by 2-LOCC but not by 1-LOCC if $$k(k-1)-m(m-1)< 2^{r+2}$$ , where the initial LSs are in $${\mathbb {C}}^{2^{r}}\otimes {\mathbb {C}}^{2^{r}}$$ and m is the largest number of pairwise commuting matrices corresponding to the initial LSs. Finally, we find an interesting phenomenon, i.e., 1-LOCC and 2-LOCC work differently when distinguishing four ququad-ququad LSs and distinguishing four constructed states derived from the four LSs. For the four ququad-ququad LSs, 2-LOCC has no advantage over 1-LOCC. However, for the four constructed states, 2-LOCC has advantages over 1-LOCC.

Constructions of locally distinguishable sets of maximally entangled states which require two-way LOCC

Most of known locally distinguishable sets of maximally entangled states (MESs) can be perfectly distinguished with one-way local operations and classical communication (1-LOCC), and there are not many known locally distinguishable sets which require two-way LOCC (2-LOCC). Recently some such sets of MESs have been built, that is, they can be locally distinguished only with 2-LOCC. In this work, we show that there exist a lot of locally distinguishable MESs which require 2-LOCC (denote such sets by 2-LOCC sets). Firstly generalized Bell states are employed to construct lots of 1-LOCC indistinguishable sets of MESs in the quantum systems and . Then we build a lot of 2-LOCC sets by using Weyl commutation relation and two constructive methods. Especially, it is shown that there are at least 8 and 105 2-LOCC sets in quantum systems and respectively.

One-shot assisted distillation of coherence via one-way local quantum-incoherent operations and classical communication

We obtain the general formula for the optimal rate on one of the subsystems that can be distilled from any bipartite quantum state by one-way local quantum-incoherent operations and classical communication. We provide lower and upper bounds on the assisted distillation of coherence using so-called one-shot entropies, and we establish a hierarchy of information quantities that can be used to calculate an assisted distillable rate in the asymptotical limit. We show that they asymptotically coincide with the quantum-incoherent relative entropy, and our findings imply a novel operational interpretation of quantum-incoherent relative entropy.

One-way LOCC indistinguishable lattice states via operator structures

Lattice states are a class of quantum states that naturally generalize the fundamental set of Bell states. We apply recent results from quantum error correction and from one-way local operations and classical communication (LOCC) theory that are built on the structure theory of operator systems and operator algebras, to develop a technique for the construction of relatively small sets of lattice states not distinguishable by one-way LOCC schemes. We also present examples, show the construction extends to generalized Pauli states, and compare the construction to other recent work.

A Note on the Relationship Between Genuinely Coherence and Generalized Entanglement Monotones

We find a one to one mapping between genuinely incoherent operations and special one-way local operations and classical communication(LOCC) for density matrices with full rank. We also define “generalized entanglement monotones” and “genuinely coherence monotones” under special one-way LOCC and genuinely incoherent operations respectively. Any entanglement monotone proposed by Vidal et al. is a generalized entanglement monotone. Any coherence monotone under incoherent operations is a genuinely coherence monotone. Furthermore, we clarify the relationship between generalized entanglement monotones and genuinely coherence monotones. We demonstrate that any generalized entanglement monotone of bipartite pure state is the lower bound of a suitable genuinely coherence monotone; any genuinely coherence monotone of a quantum state is the generalized entanglement monotone of the corresponding maximally correlated state.

Continuous-variable entanglement distillation over a pure loss channel with multiple quantum scissors

Entanglement distillation is a key primitive for distributing high-quality entanglement between remote locations. Probabilistic noiseless linear amplification based on the quantum scissors is a candidate for entanglement distillation from noisy continuous-variable (CV) entangled states. Being a non-Gaussian operation, quantum scissors is challenging to analyze. We present a derivation of the non-Gaussian state heralded by multiple quantum scissors in a pure loss channel with two-mode squeezed vacuum input. We choose the reverse coherent information (RCI)---a proven lower bound on the distillable entanglement of a quantum state under one-way local operations and classical communication (LOCC), as our figure of merit. We evaluate a Gaussian lower bound on the RCI of the heralded state. We show that it can exceed the unlimited two-way LOCCassisted direct transmission entanglement distillation capacity of the pure loss channel. The optimal heralded Gaussian RCI with two quantum scissors is found to be significantly more than that with a single quantum scissors, albeit at the cost of decreased success probability. Our results fortify the possibility of a quantum repeater scheme for CV quantum states using the quantum scissors.

Every entangled state provides an advantage in classical communication

We investigate the use of noisy entanglement as a resource in classical communication via a quantum channel. In particular, we are interested in the question whether for any entangled state, including bound entangled states, there exists a quantum channel, the classical capacity of which can be increased by providing the state as an additional resource. We partially answer this question by showing, for any entangled state, the existence of a quantum memory channel, the feedback-assisted classical capacity with product encodings of which can be increased by using the state as a resource. Using a different (memoryless) channel construction, we also provide a sufficient entropic condition for an advantage in classical communication (without feedback and for general encodings) and thus provide an example of a state that is not distillable by means of one-way local operations and classical communication but can provide an advantage in the classical capacity of a number of quantum channels. As separable states cannot provide an advantage in classical communication, our condition also provides an entropic entanglement witness.

Constructions of one-way LOCC indistinguishable sets of generalized Bell states

In this paper, we mainly consider the local indistinguishability of the set of bipartite generalized Bell states (GBSs). We systematically show constructions of small sets of GBSs with cardinalities greatly smaller than d which are not distinguishable by one-way local operations and classical communication (1-LOCC) in \(d\otimes d\). The constructions, based on linear system and Vandermonde matrix, are different for odd dimensions than for even dimensions. The results give a unified upper bound for the minimum cardinality of 1-LOCC indistinguishable set of GBSs and greatly improve previous results in Zhang et al. (Phys Rev A 91:012329, 2015) and Wang et al. (Quantum Inf Process 15:1661, 2016). Among others, the case that d is odd of the results shows that the set of 4 GBSs in \(5\otimes 5\) in Fan (Phys Rev A 75:014305, 2007) is indeed a 1-LOCC indistinguishable set which cannot be distinguished by Fan’s method, and the minimum cardinality of 1-LOCC indistinguishable set of GBSs in \(7\otimes 7\) is 5. The case that d is even of the results shows that there exists a 1-LOCC indistinguishable set of 18 GBSs in \(100\otimes 100\).

One-way local distinguishability of generalized Bell states in arbitrary dimension

In this paper, we study the distinguishability of arbitrary dimensional generalized Bell states under one-way local operations and classical communication (LOCC). We introduce an admissible solutions set and a nonadmissible solutions set for each set of generalized Bell states. Any element that exists in the admissible solution set but not in the nonadmissible one implies the one-way LOCC distinguishablity of a given set. This criterion is very convenient in the following two aspects: both solution sets can be calculated easily and, once the criterion can be applied successfully for a given set, it immediately presents an explicit distinguishing strategy. Furthermore, Fan's results in Phys. Rev. Lett. 92, 177905 (2004) can also be deduced as special cases of our results. For the case no such element exists, we disprove a conjecture proposed by Yang et al. in Quantum Inf. Process. 17, 29 (2018) by presenting a counterexample.

Locally indistinguishable generalized Bell states with one-way local operations and classical communication

In this paper, we investigate the indistinguishability of generalized Bell states (GBSs) by one-way local operations and classical communication (LOCC). We first introduce two sets, i.e., the difference set and the testing set, and show that the GBSs cannot always be distinguished by one-way LOCC if the testing set is a subset of the difference set of GBSs. Then we give a unified upper bound about indistinguishable GBSs by one-way LOCC. The newly derived bound is smaller than those presented in Zhang et al. [Phys. Rev. A 91, 012329 (2015)] and Wang et al. [Quantum Inf. Process. 15, 1661 (2016).]. More precisely, the minimum number of one-way locally indistinguishable GBSs is not more than $2\ensuremath{\lceil}\sqrt{d}\ensuremath{\rceil}+\ensuremath{\lceil}\frac{\sqrt{d}}{4}\ensuremath{\rceil}+1$.

On the locally distinguishable three-dimensional subspace of bipartite systems

It has been conjectured and numerically supported in King and Matysiak (2007 J. Phys. A: Math. Theor. 40 7939) that any three-dimensional subspace of has an orthonormal basis that can be reliably distinguished using one-way local operations and classical communications (LOCC). Since then there has been little progress towards the conjecture. We propose and prove a few cases of the conjecture. We also present a few equivalent or sufficient conditions for the conjecture, and thus moderately simplify the conjecture.

Operator structures and quantum one-way LOCC conditions

We conduct the first detailed analysis in quantum information of recently derived operator relations from the study of quantum one-way local operations and classical communications (LOCC). We show how operator structures such as operator systems, operator algebras, and Hilbert C*-modules naturally arise in this setting. We make use of these structures to derive new results and bounds in the study of one-way LOCC, and we use the approach to uncover new derivations of some previously established results.

Tight Asymptotic Bounds on Local Hypothesis Testing Between a Pure Bipartite State and the White Noise State

We consider asymptotic hypothesis testing (or state discrimination with asymmetric treatment of errors) between an arbitrary fixed bipartite pure state $ | \Psi \rangle $ and the white noise state (the completely mixed state) under one-way local operations and classical communications (LOCC), two-way LOCC, and separable Positive Operator Valued Measures (POVMs). As a result, we derive the Hoeffding bounds under two-way LOCC POVMs and separable POVMs. Further, we derive Stein’s lemma type of optimal error exponents under one-way LOCC, two-way LOCC, and separable POVMs up to the third order, which clarifies the difference between one-way and two-way LOCC POVM. Our results clarify the relationship between the entanglement of Renyi entropy and the hypothesis testing under LOCC, since the entanglement of Renyi entropy appears in the formula of both the Hoeffding bounds and Stein’s lemma type of error exponents. This paper gives a very rare example in which the optimal performance under the infinite-round two-way LOCC is also equal to that under separable operations and can be attained with two-round communication, but not with the one-way LOCC.