For a graph G and a related symmetric matrix M , the continuous-time quantum walk on G relative to M is defined as the unitary matrix U (t) = exp(−itM ), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time τ if the (u, v)-entry of U (τ ) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer:• If a n-vertex graph has perfect state transfer at time τ relative to the Laplacian, then so does its complement if nτ ∈ 2πZ. As a corollary, the double cone over any m-vertex graph has perfect state transfer relative to the Laplacian if and only if m ≡ 2 (mod 4). This was previously known for a double cone over a clique (S. Bose, A. Casaccino, S. Mancini, S. Severini, Int. J. Quant. Inf., 7:11, 2009).• If a graph G has perfect state transfer at time τ relative to the normalized Laplacian, then so does the weak product G × H if for any normalized Laplacian eigenvalues λ of G and µ of H, we have µ(λ − 1)τ ∈ 2πZ. As a corollary, a weak product of P 3 with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of P 3 has perfect state transfer relative to the adjacency matrix.As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (C. Godsil, Discrete Math., 312:1, 2011).

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 ⊗ Φ. Equivalently, the non-mP dimension of Φ tells us the maximal number of negative eigenvalues that the adjoint map I m ⊗ 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 nontrivial 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.

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 prove a result 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 result implies a generalization of Kruskal's theorem. Several adaptations and generalizations of Kruskal's theorem have already been obtained, but most of these results still cannot certify uniqueness when the k-ranks are below a certain threshold. Our generalization contains several of these results, and can certify uniqueness below this threshold. As a corollary, we prove that if n product tensors form a circuit, then they have rank greater than one in at most n − 2 subsystems. This generalizes several recent results in this direction, and is sharp.

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 O(log n)-qubit high dimensional signals [1], whereas a recently-proposed optical protocol uses a tensor product of O(n) single-qubit signals, while maintaining the O(log n) information leakage of the original protocol [2]. We find a family of protocols which interpolate between the original and optical protocols while maintaining the 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 [3,4], 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 [5] by reducing the number of signals by a factor 1/2 and also reducing the information leakage.

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 ≥ 2, every (r + 1) × (r + 1) correlation matrix is r-decomposable, and we construct (2r + 1) × (2r + 1) correlation matrices that are not r-decomposable. One question this leaves open is whether every 4 × 4 correlation matrix is 2-decomposable, which we make partial progress toward resolving. We apply our results to an entanglement detection scenario.