In the task of quantum state exclusion, we consider a quantum system prepared in a state chosen from a known set. The aim is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures cannot have taken place. We ask what conditions the set of states must obey in order for this to be possible and how well we can complete the task when it is not. The task of quantum state discrimination forms a subclass of this set of problems. Within this paper, we formulate the general problem as a semidefinite program (SDP), enabling us to derive sufficient and necessary conditions for a measurement to be optimal. Furthermore, we obtain a necessary condition on the set of states for exclusion to be achievable with certainty, and we give a construction for a lower bound on the probability of error. This task of conclusively excluding states has gained importance in the context of the foundations of quantum mechanics due to a result from Pusey, Barrett, and Rudolph (PBR). Motivated by this, we use our SDP to derive a bound on how well a class of hidden variable models can perform at a particular task, proving an analog of Tsirelson's bound for the PBR experiment and the optimality of a measurement given by PBR in the process. We also introduce variations of conclusive exclusion, including unambiguous state exclusion, and state exclusion with worst-case error.
Quantum state exclusion is the task of determining which states from a given set a system was not prepared in. We provide a complete solution to optimal quantum state exclusion for arbitrary sets of pure states generated by finite groups, establishing necessary and sufficient conditions for perfect (zero-error conclusive) exclusion. When perfect exclusion is impossible, we introduce two natural extensions: minimum-error and unambiguous exclusion. For both, we derive the optimal protocols and present analytical expressions for the corresponding failure probabilities and measurements, providing additional insight into how quantum states encode information.
Part 1: Why can we not distinguish between pure, non-orthogonal quantum states? Regarding the quantum state as a state of knowledge rather than something physically real, has the potential to answer this question and explain other quantum properties but such interpretations have recently been undermined. This important no-go result, due to Pusey, Barrett and Rudolph, makes use of a specific example of a task we term state exclusion. Here, a system is prepared in a state chosen from a known set and the aim is to determine a preparation that has not taken place. We formulate state exclusion as a semidefinite program, using it to investigate when exclusion is conclusively possible and how it can be achieved. Based on state exclusion, we construct a communication task which exhibits drastic, `infinite', separations between a variety of classical and quantum information and communication complexity measures. This serves to requisition the aforementioned foundational result for use in information theoretic protocols. Part 2: What does thermodynamics look like in the absence of the thermodynamic limit? In recent years there has been a concerted effort to apply techniques from quantum information theory to study the laws of thermodynamics at the nano-scale. This has led to the resource theory of thermal operations for determining when single-copy transformations are possible. However, if a deterministic transition is forbidden, can it occur probabilistically? Here we compute and bound the maximum probability with which nano-scale thermodynamical transformations can occur. Thermal operations assume that one can precisely manipulate all of the degrees of freedom in a very large heat bath. While this enables the derivation of ultimate limits on nano-scale thermodynamics, it does not make them feasible to perform in reality. We show how allowed transitions can be implemented whilst manipulating only a single bath-qubit, making thermal operations more experimentally palatable.
Secret sharing is ubiquitous throughout cryptography. All possible access structures are classically feasible, and in the case of threshold access structures, the protocols are even very efficient. However, when moving to the quantum setting, the no-cloning theorem shows that many access structures are impossible. In fact, no-cloning exactly characterizes feasibility: for thresholds, quantum secret sharing is possible if and only if the threshold
t is strictly more than
n/2.
In this work, we propose a variant of quantum secret sharing (QSS) where two or more identical copies of the input state are provided, but the output is only required to recover one copy. This notion circumvents the simple one-copy no-cloning obstruction, though the natural
k-copy generalization still gives much milder obstructions. For thresholds using
k copies, no-cloning implies that QSS is impossible whenever
t≤n/(k+1).
It is tempting to hypothesize that no-cloning continues to exactly characterize the many-copy case. However, we show that this is not the case. We give positive results showing that multiple copies allow for going slightly beyond the single-copy obstruction: for thresholds, we construct QSS whenever
t>(n−k+1)/2. On the other hand, we give a novel obstruction we call the Clique Path obstruction, which applies to arbitrary access structures. For thresholds, it shows that QSS is impossible whenever
t≤(n−1)/k. Our upper and lower bounds exactly match for
k=2. Both our results leverage connections to the
k-colorability of certain graphs derived from the access structure. We leave closing the gap for
k≥3 copies as a fascinating direction for future work.
Secret sharing is closely related to error correction, which can also be considered in the many-copy setting. Our results imply similar obstructions for quantum error correction for erasure channels.
Abstract Consider a quantum system prepared in a state chosen from a known set, the aim of quantum state discrimination is to perform a measurement on the system which can determine that a subset of the possible preparation procedures have been prepared. However, the aim of quantum state exclusion is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures have not taken place. For classical states, probabilities of error in discrimination and exclusion are always equal. For quantum states, we point out that probabilities of error in discrimination and exclusion are still equal in two-state ensembles, however they are no longer equal in multiple-state ensembles. More specifically, the error-probability of discrimination is not less than the error-probability of exclusion. We also show that the error-probability of exclusion is continuous which is similar with the error-probability of discrimination.
A set of quantum states is said to be antidistinguishable if, upon being given a randomly chosen state from the set, it is possible to exclude with certainty a state that was definitely not prepared. We study quantum nonlocality within the framework of local state exclusion and prove that any set of multipartite pure states containing a state that is orthogonal to all the others is locally antidistinguishable. Next, we introduce the task of
local state antimarking, where, instead of a single state, a non-repetitive sequence of states from a known set of multipartite states is randomly selected and distributed to spatially separated parties, whose objective is to identify, using LOCC, at least one sequence that was not supplied. We present an ensemble of product states that is not globally antidistinguishable; yet, by choosing states from it without replacement, one can construct sequences that can be perfectly excluded globally but, importantly, not locally. This reveals a hitherto unexplored form of `nonlocality without entanglement'. Finally, for a given ensemble, we compare its local antidistinguishability and antimarkability with its conclusive local distinguishability and markability. We demonstrate that no strict hierarchy exists between these paradigms: there exist product-state ensembles that permit one task while strictly forbidding the other, and vice versa.