arXiv subjects

Benjamin Lovitz

Publications and source records attributed to Benjamin Lovitz.

23 records · Page 2Linked to original sources

arXiv2020

Toward a generalization of Kruskal's theorem on tensor decomposition

Kruskal's theorem states that a sum of product tensors constitutes a unique tensor rank decomposition if the so-called k-ranks of the product tensors are large. In this work, we propose a conjecture in which the k-rank condition of Kruskal's theorem is weakened to the standard notion of rank, and the conclusion is relaxed to a statement on the linear dependence of the product tensors. Our conjecture would imply a generalization of Kruskal's theorem. Several adaptations and generalizations of Kruskal's theorem have already been obtained, but these results still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization would contain several of these results, and could certify uniqueness below this threshold. We prove our conjecture over an arbitrary field $\mathbb{F}$ when the underlying multipartite vector space takes any one of three forms: ${\mathbb{F}^{d_1}\otimes \mathbb{F}^{d_2}}, \;{\mathbb{F}^{d_1}\otimes\mathbb{F}^{d_2}\otimes \mathbb{F}^2,}$ or $\mathbb{F}^{d_1}\otimes \mathbb{F}^2 \otimes\cdots \otimes \mathbb{F}^2$. As a corollary to the third case, we prove that if $n$ product tensors form a circuit, then they have rank greater than one in at most $n-2$ subsystems. This is a quadratic improvement over a recent bound obtained by Ballico, and is sharp.

math.CO↗

arXiv2019

The Non-m-Positive Dimension of a Positive Linear Map

We introduce a property of a matrix-valued linear map $Φ$ that we call its "non-m-positive dimension" (or "non-mP dimension" for short), which measures how large a subspace can be if every quantum state supported on the subspace is non-positive under the action of $I_m \otimes Φ$. Equivalently, the non-mP dimension of $Φ$ tells us the maximal number of negative eigenvalues that the adjoint map $I_m \otimes Φ^*$ can produce from a positive semidefinite input. We explore the basic properties of this quantity and show that it can be thought of as a measure of how good $Φ$ is at detecting entanglement in quantum states. We derive non-trivial bounds for this quantity for some well-known positive maps of interest, including the transpose map, reduction map, Choi map, and Breuer--Hall map. We also extend some of our results to the case of higher Schmidt number as well as the multipartite case. In particular, we construct the largest possible multipartite subspace with the property that every state supported on that subspace has non-positive partial transpose across at least one bipartite cut, and we use our results to construct multipartite decomposable entanglement witnesses with the maximum number of negative eigenvalues.

quant-ph↗

arXiv2018

On decomposable correlation matrices

Correlation matrices (positive semidefinite matrices with ones on the diagonal) are of fundamental interest in quantum information theory. In this work we introduce and study the set of $r$-decomposable correlation matrices: those that can be written as the Schur product of correlation matrices of rank at most $r$. We find that for all $r \geq 2$, every $(r+1) \times (r+1)$ correlation matrix is $r$-decomposable, and we construct ${(2r+1) \times (2r+1)}$ correlation matrices that are not $r$-decomposable. One question this leaves open is whether every $4 \times 4$ correlation matrix is $2$-decomposable, which we make partial progress toward resolving. We apply our results to an entanglement detection scenario.

quant-ph↗

arXiv2018

Practical Quantum Appointment Scheduling

We propose a protocol based on coherent states and linear optics operations for solving the appointment-scheduling problem. Our main protocol leaks strictly less information about each party's input than the optimal classical protocol, even when considering experimental errors. Along with the ability to generate constant-amplitude coherent states over two modes, this protocol requires the ability to transfer these modes back-and-forth between the two parties multiple times with low coupling loss. The implementation requirements are thus still challenging. Along the way, we develop new tools to study quantum information cost of interactive protocols in the finite regime.

quant-ph↗

arXiv2017

Families of Quantum Fingerprinting Protocols

We introduce several families of quantum fingerprinting protocols to evaluate the equality function on two $n$-bit strings in the simultaneous message passing model. The original quantum fingerprinting protocol uses a tensor product of a small number of $\mathcal{O}(\log n)$-qubit high dimensional signals [Buhrman et al. 2001], whereas a recently-proposed optical protocol uses a tensor product of $\mathcal{O}(n)$ single-qubit signals, while maintaining the $\mathcal{O}(\log n)$ information leakage of the original protocol [Arrazola and Lütkenhaus 2014]. We find a family of protocols which interpolate between the original and optical protocols while maintaining the $\mathcal{O}(\log n)$ information leakage, thus demonstrating a trade-off between the number of signals sent and the dimension of each signal. There has been interest in experimental realization of the recently-proposed optical protocol using coherent states [Xu et al. 2015, Guan et al. 2016], but as the required number of laser pulses grows linearly with the input size $n$, eventual challenges for the long-time stability of experimental set-ups arise. We find a coherent state protocol which reduces the number of signals by a factor $1/2$ while also reducing the information leakage. Our reduction makes use of a simple modulation scheme in optical phase space, and we find that more complex modulation schemes are not advantageous. Using a similar technique, we improve a recently-proposed coherent state protocol for evaluating the Euclidean distance between two real unit vectors [Kumar et al. 2017] by reducing the number of signals by a factor $1/2$ and also reducing the information leakage.

quant-ph↗