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Register a new userPartial Distinguishability as a Coherence Resource in Boson Sampling
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Quantum coherence is a useful resource that is consumed to accomplish several tasks that classical devices are hard to fulfill. Especially, it is considered to be the origin of quantum speedup for many computational algorithms. In this work, we interpret the computational time cost of boson sampling with partially distinguishable photons from the perspective of coherence resource theory. With incoherent operations that preserve the diagonal elements of quantum states up to permutation, which we name emph{permuted genuinely incoherent operation} (pGIO), we present some evidence that the decrease of coherence corresponds to a computationally less complex system of partially distinguishable boson sampling. Our result shows that coherence is one of crucial resources for the computational time cost of boson sampling. We expect our work presents an insight to understand the quantum complexity of the linear optical network system.
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We develop a resource theory of symmetric distinguishability, the fundamental objects of which are elementary quantum information sources, i.e., sources that emit one of two possible quantum states with given prior probabilities. Such a source can be represented by a classical-quantum state of a composite system $XA$, corresponding to an ensemble of two quantum states, with $X$ being classical and $A$ being quantum. We study the resource theory for two different classes of free operations: $(i)$ ${rm{CPTP}}_A$, which consists of quantum channels acting only on $A$, and $(ii)$ conditional doubly stochastic (CDS) maps acting on $XA$. We introduce the notion of symmetric distinguishability of an elementary source and prove that it is a monotone under both these classes of free operations. We study the tasks of distillation and dilution of symmetric distinguishability, both in the one-shot and asymptotic regimes. We prove that in the asymptotic regime, the optimal rate of converting one elementary source to another is equal to the ratio of their quantum Chernoff divergences, under both these classes of free operations. This imparts a new operational interpretation to the quantum Chernoff divergence. We also obtain interesting operational interpretations of the Thompson metric, in the context of the dilution of symmetric distinguishability.
We study the distinguishability of a particular type of maximally entangled states -- the lattice states using a new approach of semidefinite program. With this, we successfully construct all sets of four ququad-ququad orthogonal maximally entangled states that are locally indistinguishable and find some curious sets of six states having interesting property of distinguishability. Also, some of the problems arose from cite{CosentinoR14} about the PPT-distinguishability of lattice maximally entangled states can be answered.
This paper systematically develops the resource theory of asymmetric distinguishability, as initiated roughly a decade ago [K. Matsumoto, arXiv:1010.1030 (2010)]. The key constituents of this resource theory are quantum boxes, consisting of a pair of quantum states, which can be manipulated for free by means of an arbitrary quantum channel. We introduce bits of asymmetric distinguishability as the basic currency in this resource theory, and we prove that it is a reversible resource theory in the asymptotic limit, with the quantum relative entropy being the fundamental rate of resource interconversion. The distillable distinguishability is the optimal rate at which a quantum box consisting of independent and identically distributed (i.i.d.) states can be converted to bits of asymmetric distinguishability, and the distinguishability cost is the optimal rate for the reverse transformation. Both of these quantities are equal to the quantum relative entropy. The exact one-shot distillable distinguishability is equal to the min-relative entropy, and the exact one-shot distinguishability cost is equal to the max-relative entropy. Generalizing these results, the approximate one-shot distillable distinguishability is equal to the smooth min-relative entropy, and the approximate one-shot distinguishability cost is equal to the smooth max-relative entropy. As a notable application of the former results, we prove that the optimal rate of asymptotic conversion from a pair of i.i.d. quantum states to another pair of i.i.d. quantum states is fully characterized by the ratio of their quantum relative entropies.
An orthonormal basis consisting of unentangled (pure tensor) elements in a tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In general, for $n$ qubits, we prove that in its natural structure as a real variety, the space of UOB is a bouquet of products of Riemann spheres parametrized by a class of edge colorings of hypercubes. Its irreducible components of maximum dimension are products of $2^n-1$ two-spheres. Using a theorem of Walgate and Hardy, we observe that the UOB whose elements are distinguishable by local operations and classical communication (called locally distinguishable or LOCC distinguishable UOB) are exactly those in the maximum dimensional components. Bennett et al, in their in-depth study of quantum nonlocality without entanglement, include a specific 3 qubit example UOB which is not LOCC distinguishable; we construct certain generalized counterparts of this UOB in $n$ qubits.
We show that the generalization of the relative entropy of a resource from states to channels is not unique, and there are at least six such generalizations. We then show that two of these generalizations are asymptotically continuous, satisfy a version of the asymptotic equipartition property, and their regularizations appear in the power exponent of channe
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