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arXiv:quant-ph/0503092 · Physical Review Letters

Bipartite Subspaces Having No Bases Distinguishable by Local Operations and Classical Communication

John Watrous

Published Aug 18, 2005• 122 citations
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Abstract

It is proved that there exist subspaces of bipartite tensor product spaces that have no orthonormal bases that can be perfectly distinguished by means of local operations and classical communication. A corollary of this fact is that there exist quantum channels having suboptimal classical capacity even when the receiver may communicate classically with a third party that represents the channel's environment.

DOI
10.1103/physrevlett.95.080505

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It is proved that there exist subspaces of bipartite tensor product spaces that have no orthonormal bases that can be perfectly distinguished by means of local operations and classical communication. A corollary of this fact is that there exist quantum channels having suboptimal classical capacity even when the receiver may communicate classically with a third party that represents the channel's environment.
A subspace of a multipartite Hilbert space is said to be locally indistinguishable if any orthonormal basis of this subspace cannot be perfectly distinguished by local operations and classical communication. Previously it was shown that any $m\ensuremath{\bigotimes}n$ bipartite system with m>2m>2m>2 and n>2n>2n>2 has a locally indistinguishable subspace. However, it has been an open problem since 2005 whether there is a locally indistinguishable bipartite subspace with a qubit subsystem. We settle this problem in negative by showing that any $2\ensuremath{\bigotimes}n$ bipartite subspace contains a basis that is locally distinguishable. As an interesting application, we show that any quantum channel with two Kraus operators has optimal environment-assisted classical capacity.
In this paper, we find subspaces that have no locally distinguishable orthonormal bases. For a ${C}^{m}\ensuremath{\bigotimes}{C}^{n}$ $(m,n\ensuremath{\ge}3)$ bipartite tensor system, we show that the orthogonal complement of the space spanned by a Schmidt rank-3 maximally entangled state is such a subspace. We further demonstrate that, when both parties' dimensions are greater than three, the orthogonal complement of the space spanned by any entangled state whose Schmidt rank is greater than three has no locally distinguishable orthonormal bases. Although derived using a scheme simpler than previous methods, our results are much more general than known before.
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.
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.
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.
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.