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.
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.
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.
Walgate and Scott have determined the maximum number of generic pure quantum states that can be unambiguously discriminated by an LOCC measurement [Journal of Physics A: Mathematical and Theoretical, 41:375305, 08 2008]. In this work, we determine this number in a more general setting in which the local parties have access to pre-shared entanglement in the form of a resource state. We find that, for an arbitrary pure resource state, this number is equal to the Krull dimension of (the closure of) the set of pure states obtainable from the resource state by SLOCC. Surprisingly, a generic resource state maximizes this number.Local state discrimination is closely related to the topic of entangled subspaces, which we study in its own right. We introduce r-entangled subspaces, which naturally generalize previously studied spaces to higher multipartite entanglement. We use algebraic-geometric methods to determine the maximum dimension of an r-entangled subspace, and present novel explicit constructions of such spaces. We obtain similar results for symmetric and antisymmetric r-entangled subspaces, which correspond to entangled subspaces of bosonic and fermionic systems, respectively.
We propose a protocol based on coherent states and linear optics operations for solving the appointmentscheduling 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. communication channels (see, for example, [1, 2]). If we wish to limit Alice and Bob to quantum operations that should be experimentally accessible in the near future, can they still hope to achieve a quantum advantage in terms of information leakage?We show that indeed they can. More precisely, we focus on quantum protocols requiring coherent state messages over two optical modes that are manipulated with linear optics operations and do not require any pre-shared entanglement or any quantum memory from honest participants. We compare such protocols with the best classical protocols for which we allow both local and shared randomness for free in order to minimize the information leakage. We also allow these classical resources to be used in our quantum protocols, appropriately accounting for them while quantifying information leakage. We find that indeed, with experimental parameters that are challenging but should be reachable in the near future, it is possible to obtain such a quantum advantage in terms of information leakage. In fact, since we are mainly concerned with privacy here, Alice and Bob could be close to each other, in the same lab, and keep their inputs private but still have close-by set-ups which would perform much better than our data for clearly separated set-ups.The problem we focus on is that of appointment scheduling: Alice and Bob each hold a calendar of their availabilities, and they wish to find a date of common availability, or agree that no such date exists. Viewing their inputs x, y of available dates as subsets of a calendar [n] = {1, 2, · · · , n} on n dates, they wish to output an element i ∈ x ∩ y if such an i exists, or else output ∅ if x ∩ y = ∅. This problem, and in particular its binary variant, is one of the most well-studied problems in communication and information complexity.It is known that quantum protocols can provide a quadratic speed-up in terms of information leakage for this problem [3,4,5]. It is also known that interaction is necessary to get an advantage over classical protocols [6,7,8]. As it turns out, for our protocols, interaction poses a challenge in a realistic experimental setting: more interaction also implies more losses over the communication channels. We show that there is nevertheless some regime for which we can obtain a quantum advantage.Hence, our work is the first to propose an optica...
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