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arXiv:quant-ph/0510004

Reliably distinguishing states in qutrit channels using one-way LOCC

Christopher King, Daniel Matysiak

Published Oct 1, 2005• 1 citation
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Abstract

We present numerical evidence showing that any three-dimensional subspace of C^3 \otimes C^n has an orthonormal basis which can be reliably distinguished using one-way LOCC, where a measurement is made first on the 3-dimensional part and the result used to select an optimal measurement on the n-dimensional part. This conjecture has implications for the LOCC-assisted capacity of certain quantum channels, where coordinated measurements are made on the system and environment. By measuring first in the environment, the conjecture would imply that the environment-assisted classical capacity of any rank three channel is at least log 3. Similarly by measuring first on the system side, the conjecture would imply that the environment-assisting classical capacity of any qutrit channel is log 3. We also show that one-way LOCC is not symmetric, by providing an example of a qutrit channel whose environment-assisted classical capacity is less than log 3.

DOI
10.48550/arxiv.quant-ph/0510004

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We present numerical evidence showing that any three-dimensional subspace of C^3 \otimes C^n has an orthonormal basis which can be reliably distinguished using one-way LOCC, where a measurement is made first on the 3-dimensional part and the result used to select an optimal measurement on the n-dimensional part. This conjecture has implications for the LOCC-assisted capacity of certain quantum channels, where coordinated measurements are made on the system and environment. By measuring first in the environment, the conjecture would imply that the environment-assisted classical capacity of any rank three channel is at least log 3. Similarly by measuring first on the system side, the conjecture would imply that the environment-assisting classical capacity of any qutrit channel is log 3. We also show that one-way LOCC is not symmetric, by providing an example of a qutrit channel whose environment-assisted classical capacity is less than log 3.
We show that every three-dimensional subspace of qutrit-qudit complex or real systems has a distinguishable basis under one-way local projective measurements and classical communication (LPCC). This solves a long-standing open problem proposed in [J. Phys. A, 40, 7937, 2007]. We further construct a three-dimensional space whose locally distinguishable basis is unique and apply the uniqueness property to the task of state transformation. We also construct a three-dimensional locally distinguishable multipartite space assisted with entanglement. On the other hand, we show that four-dimensional indistinguishable bipartite subspace under one-way LPCC exists. Our work offers profound insights and introduces a theoretical tool for understanding the local distinguishability of subspace. As a consequence, every qutrit channel has optimal environment-assisting classical capacity, and the environment-assisted classical capacity of every rank-three channel is at leastlog⁡23\log _{2} 3log2​3. We also show that every two-qutrit state can be converted into a generalized classical state near the quantum-classical boundary by an entanglement-breaking channel.
We present numerical evidence showing that any three-dimensional subspace of has an orthonormal basis which can be reliably distinguished using one-way LOCC (local operations and classical communication), where a measurement is made first on the three-dimensional part and the result used to select an optimal measurement on the n -dimensional part. We also show that the order of measurement is essential, by providing an example of a three-dimensional subspace of which does not have any basis that can be distinguished by measuring first on the five-dimensional factor.
In a seminal work [PRL85.4972], Walgate, Short, Hardy, and Vedral prove in finite dimensions that for every pair of pure multipartite orthogonal quantum states, there exists a one-way local operations and classical communication (LOCC) protocol that perfectly distinguishes the pair. We extend this result to infinite dimensions with a simpler proof. For states on CdA×dA⊗CdB×dB\mathbb{C}^{d_A \times d_A} \otimes \mathbb{C}^{d_B \times d_B}CdA​×dA​⊗CdB​×dB​, we strengthen this existence result by constructing an O(dA2dB2)O(d_A^2 d_B^2)O(dA2​dB2​)-time algorithm that specifies such a perfect one-way LOCC protocol. Finally, we establish the equivalence between Walgate et al.'s result and the fact that the one-shot environment-assisted classical capacity of every quantum channel is at least 1 bit per channel use, thereby clarifying the literature on these notions. At the core of all of these results is the fact that every operator with vanishing trace admits a basis where its diagonal entries are all zero.
Abstract 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 C 3 ⊗ C n 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.
We study the local distinguishability of general multiqubit 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 multiqubit states can be decided efficiently. Another useful consequence is that a set of orthogonal nnn-qubit states is locally distinguishable only if the summation of their orthogonal Schmidt numbers is less than the total dimension 2n{2}^{n}2n. 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 444. We also explicitly construct locally indistinguishable subspaces with dimensions 333 and 555, respectively. Similar to the bipartite case, these results on multipartite locally indistinguishable subspaces can be used to estimate the one-shot environment-assisted classical capacity of a class of quantum broadcast channels.
Following initial work by Gregoratti and Werner [J. Mod. Optics 50, 913-933, 2003 and quant-ph/0403092] and Hayden and King [quant-ph/0409026], we study the problem of the capacity of a quantum channel assisted by a "friendly (channel) environment" that can locally measure and communicate classical messages to the receiver. Previous work [quant-ph/0505038] has yielded a capacity formula for the quantum capacity under this kind of help from the environment. Here we study the problem of the environment-assisted classical capacity, which exhibits a somewhat richer structure (at least, it seems to be the harder problem). There are several, presumably inequivalent, models of the permitted local operations and classical communications between receiver and environment: one-way, arbitrary, separable and PPT POVMs. In all these models, the task of decoding a message amounts to discriminating a set of possibly entangled states between the two receivers, by a class of operations under some sort of locality constraint. After introducing the operational capacities outlined above, we show that a lower bound on the environment-assisted classical capacity is always half the logarithm of the input space dimension. Then we develop a few techniques to prove the existence of channels which meet this lower bound up to terms of much smaller order, even when PPT decoding measurements are allowed (assuming a certain superadditivity conjecture).