Bisimulation for quantum processes

Recently, the problem of discriminating multipartite unitary operations by local operations and classical communication (LOCC) has attracted significant attention. The latest work in the literature on this problem showed that two multipartite unitary operations can always be perfectly distinguished by LOCC when a finite number of runs are allowable. However, in these schemes, local entanglement (an entangled state holden by one party) was required, which seems to imply that local entanglement is necessary for perfect discrimination between unitary operations by LOCC. In this article, we show that a perfect discrimination between two unitary operations acting on a two-qudits can always be achieved without exploiting any entanglement. As a result, we conclude that local entanglement is not necessary for perfect discrimination between unitary operations acting on two-qudits by LOCC.

Entanglement is a key resource in the emerging field of Quantum Information. The strong correlations between systems described by an entangled state allow us to perform certain tasks more efficiently than it would be possible by using only classical resources. This is why the characterization of entanglement is one of the most important problems in Quantum Information. In this thesis, we analyze several aspects of entanglement. First, we introduce a new family of criteria to determine if a bipartite mixed state is entangled or not. This family consists of a sequence of tests that can be implemented efficiently, and has the property that all entangled states can be detected by some test in the sequence. Each test in the family can be stated as a semidefinite program, which is a class of convex optimization problems. The duality structure of these programs allows us to explicitly construct an entanglement witness that proves entanglement of a state, whenever the state fails one of the tests in the sequence. The entanglement witnesses constructed in this manner have well-defined algebraic properties that can be used to give a characterization of the interior of the set of all possible entanglement witnesses, as well as the set of strictly positive bihermitian forms and the set of strictly positive maps. We also study deterministic transformations of three-qubit pure state when only local operations and classical communication (LOCC) are allowed. We derive strong constraints that the operations and states involved must satisfy, and we apply these results to characterize the set of real states that can be obtained from the GHZ state by LOCC.

Bell nonlocality describes a manifestation of quantum mechanics that cannot be explained by any local hidden variable model. Its origin lies in the nature of quantum entanglement, although understanding the precise relationship between nonlocality and entanglement has been a notorious open problem. In this paper, we resolve this problem by developing a dynamical framework in which quantum Bell nonlocality emerges as special form of entanglement, and both are unified as resources under local operations and classical communication (LOCC). Our framework is built on the notion of quantum processes, which are abstract quantum channels mapping elements between fixed intervals in space and time. Entanglement is then identified as a quantum process that cannot be generated by LOCC while Bell nonlocality is the subset of these processes that have an instantaneous input-output delay time. LOCC pre-processing is a natural set of free operations in this theory, thereby enabling all entangled states to activate some form of Bell nonlocality. In addition, we generalize the CHSH witnesses from the state domain to the domain of entangled quantum measurements, and provide a systematic method to quantify the Bell nonlocality of a bipartite quantum channel.

Nonlocality is at the heart of quantum information processing. In this paper we investigate the minimum amount of classical communication required to simulate a nonlocal quantum measurement. We derive general upper bounds, which in turn translate to systematic classical simulations of quantum communication protocols.As a concrete application, we prove that any quantum communication protocol with shared entanglement for computing a Boolean function can be simulated by a classical protocol whose cost does not depend on the amount of the shared entanglement. This implies that if the cost of communication is a constant, quantum and classical protocols, with shared entanglement and shared coins, respectively, compute the same class of functions.Furthermore, we describe a new class of efficient quantum communication protocols based on fast quantum algorithms. While some of them have efficient classical simulations by our method, others appear to be good candidates for separating quantum v.s. classical protocols, and quantum protocols with v.s. without shared entanglement.Yet another application is in the context of simulating quantum correlations using local hidden variable models augmented with classical communications. We give a constant cost, approximate simulation of quantum correlations when the number of correlated variables is a constant, while the dimension of the entanglement and the number of possible measurements can be arbitrary.Our upper bounds are expressed in terms of some tensor norms on the measurement operator. Those norms capture the nonlocality of bipartite operators in their own way and may be of independent interest and further applications.

We present a simple and general simulation technique that transforms any black-box quantum algorithm (a la Grover's database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) two-party communication complexity models. In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separation for the zero-error model. The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model. In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of bounded-depth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present black-box algorithms that achieve near quadratic speed up for all such predicates.

It is well known that orthogonal quantum states can be distinguished perfectly. However, if we assume that these orthogonal quantum states are shared by spatially separated parties, the distinguishability of these shared quantum states may be completely different. We show that a set of linearly independent quantum states [formula: see text] where U(m,n) are generalized Pauli matrices, cannot be discriminated deterministically or probabilistically by local operations and classical communication. On the other hand, any l maximally entangled states from this set are locally distinguishable if l(l-1)< or =2d. The explicit projecting measurements are obtained to locally discriminate these states. As an example, we show that four Werner states are locally indistinguishable.

It is shown that while entanglement ensures difficulty in discriminating a set of mutually orthogonal states perfectly by local operations and classical communication (LOCC), entanglement content does not. In particular, for a class of entangled multiqubit states, the maximum number of perfectly LOCC distinguishable orthogonal states is shown to be independent of the average entanglement of the states, and the spatial configuration with respect to which LOCC operations may be carried out. It is also pointed out that for this class, the makeup of an ensemble, that is whether it consists only of entangled states or not, determines the maximum number of perfectly distinguishable states.

The paper studies unambiguous discrimination of fermionic states through local operations and classical communication (LOCC). In the task of unambiguous discrimination, no error is tolerated but an inconclusive result is allowed. We show that contrary to the quantum case, it is not always possible to distinguish two fermionic states through LOCC unambiguously with the same success probability as if global measurements were allowed. Furthermore, we prove that we can overcome such a limit through an ancillary system made of two fermionic modes, independently of the dimension of the system, prepared in a maximally entangled state: in this case, LOCC protocols achieve the optimal success probability.

In this paper, we consider the problem of discriminating quantum states by local operations and classical communication (LOCC) when an arbitrarily small amount of error is permitted. This paradigm is known as asymptotic state discrimination, and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic LOCC. We use this new criterion to prove a gap in the LOCC and separable distinguishability norms. We then turn to the operational advantage of using two-way classical communication over one-way communication in LOCC processing. With a simple two-qubit product state ensemble, we demonstrate a strict majorization of the two-way LOCC norm over the one-way norm.

Entanglement is a key resource in quantum communication, but bipartite schemes are often insufficient for advanced protocols like quantum secret sharing or distributed computing. Graph states offer a flexible way to represent and manage multipartite entanglement in quantum networks, enabling logical connectivity through local operations and classical communication (LOCC). In this work, we extend existing approaches based on bi-star configurations to more complex multi-star topologies. We analyze the maximum connectivity that can be achieved in networks of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$m$</tex> switches, each connected to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> clients, including asymmetric cases where the number of clients varies per switch. We also propose methods to enable logical communication between distant nodes. Our results support the development of scalable quantum networks with rich connectivity beyond traditional bipartite structures.

We present a procedure that uses a multipartite quantum state to communicate a secret spatial direction when there is no shared reference frame between the preparer and the group of recipients. The procedure guarantees that the recipients can determine the direction if they perform joint measurements on the state, but fail to do so if they restrict themselves to local operations and classical communication (LOCC). We calculate the fidelity for joint measurements, give bounds on the fidelity achievable by LOCC, and prove that there is a non-vanishing gap between the two of them, even in the limit of infinitely many copies. The robustness of the procedure under particle loss is also studied. Additionally, we find bounds on the probability of discriminating by LOCC between the invariant subspaces of total angular momentum N/2 and N/2−1 in a system of N elementary spins.

In this paper the density matrices derived from combinatorial Laplacian matrix of graphs is considered. More specifically, the paper places emphasis on the star-relevant graph, which means adding certain edges on peripheral vertices of star graph. Initially, we provide the spectrum of the density matrices corresponding to star-like graph (i.e., adding an edge on star graph) and present that the Von Neumann entropy increases under the graph operation (adding an edge on star graph) and the graph operation cannot be simulated by local operation and classical communication (LOCC). Subsequently, we illustrate the spectrum of density matrices corresponding to star-alike graph (i.e., adding one edge on star-like graph) and exhibit that the Von Neumann entropy increases under the graph operation (adding an edge on star-like graph) and the graph operation cannot be simulated by LOCC. Finally, the spectrum of density matrices corresponding to star-mlike graph (i.e., adding m nonadjacent edges on the peripheral vertices of star graph) is demonstrated and the relation between the graph operation and Von Neumann entropy, LOCC is revealed in this paper.

One-time memories (OTM's) are simple tamper-resistant cryptographic devices,\nwhich can be used to implement one-time programs, a very general form of\nsoftware protection and program obfuscation. Here we investigate the\npossibility of building OTM's using quantum mechanical devices. It is known\nthat OTM's cannot exist in a fully-quantum world or in a fully-classical world.\nInstead, we propose a new model based on "isolated qubits" -- qubits that can\nonly be accessed using local operations and classical communication (LOCC).\nThis model combines a quantum resource (single-qubit measurements) with a\nclassical restriction (on communication between qubits), and can be implemented\nusing current technologies, such as nitrogen vacancy centers in diamond. In\nthis model, we construct OTM's that are information-theoretically secure\nagainst one-pass LOCC adversaries that use 2-outcome measurements.\n Our construction resembles Wiesner's old idea of quantum conjugate coding,\nimplemented using random error-correcting codes; our proof of security uses\nentropy chaining to bound the supremum of a suitable empirical process. In\naddition, we conjecture that our random codes can be replaced by some class of\nefficiently-decodable codes, to get computationally-efficient OTM's that are\nsecure against computationally-bounded LOCC adversaries.\n In addition, we construct data-hiding states, which allow an LOCC sender to\nencode an (n-O(1))-bit messsage into n qubits, such that at most half of the\nmessage can be extracted by a one-pass LOCC receiver, but the whole message can\nbe extracted by a general quantum receiver.\n

Any set of states which cannot be perfectly distinguished by local operations\nand classical communication (LOCC) alone, can always be locally distinguished\nusing quantum teleportation with enough entanglement resource. However, in\nquantum information theory, entanglement is a very valuable resource, so it\nleaves the following open question: how to accomplish this task more\nefficiently than teleportation, that is, design the local discrimination\nprotocol using less entanglement resource. In this paper, we first present two\nprotocols to locally distinguish a set of unextendible product bases (UPB) in\n$5\\otimes 5$ by using different entanglement resource. Then, we generalize the\ndistinguishing methods for a class of UPB in $d\\otimes d$, where $d$ is odd and\n$d\\geqslant 3$. Furthermore, for a class of UPB in $d\\otimes d$, where $d$ is\neven and $d\\geqslant 4$, we prove that these states can also be distinguished\nby LOCC with multiple copies of low-dimensional entanglement resource. These\nresults offer rather general insight into how to use entanglement resource more\nefficiently, and also reveal the phenomenon of less nonlocality with more\nentanglement.\n

This chapter unifies all of the channel coding theorems that we have studied in this book. One of the most general information-processing tasks that a sender and receiver can accomplish is to transmit classical and quantum information and generate entanglement with many independent uses of a quantum channel and with the assistance of classical communication, quantum communication, and shared entanglement.1 The resulting rates for communication are net rates that give the generation rate of a resource less its consumption rate. Since we have three resources, all achievable rates are rate triples ( C,Q,E ) that lie in a three-dimensional capacity region, where C is the net rate of classical communication, Q is the net rate of quantum communication, and E is the net rate of entanglement consumption/generation. The capacity theorem for this general scenario is known as the quantum dynamic capacity theorem, and it is the main theorem that we prove in this chapter. All of the rates given in the channel coding theorems of previous chapters are special points in this three-dimensional capacity region. The proof of the quantum dynamic capacity theorem comes in two parts: the direct coding theorem and the converse theorem. The direct coding theorem demonstrates that the strategy for achieving any point in the three-dimensional capacity region is remarkably simple: we just combine the protocol from Corollary 22.5.2 for entanglement-assisted classical and quantum communication with the three unit protocols of teleportation, super-dense coding, and entanglement distribution. The interpretation of the achievable rate region is that it is the unit resource capacity region from Chapter 8 translated along the points achievable with the protocol from Corollary 22.5.2. In the proof of the converse theorem, we analyze the most general protocol that can consume and generate classical communication, quantum communication, and entanglement along with the consumption of many independent uses of a quantum channel, and we show that the net rates for such a protocol are bounded by the regularization of the achievable rate region.