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arXiv:1210.3125 · Electronic Proceedings in Theoretical Computer Science

Distinguishability, Ensemble Steering, and the No-Signaling Principle

Joonwoo Bae

Published Dec 26, 2014• 8 citations
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Abstract

We consider a fundamental operational task, distinguishing systems in different states, in the framework of generalized probabilistic theories and provide a general formalism of minimum-error discrimination of states in convex optimization. With the formalism established, we show that the distinguishability is generally a global property assigned to the ensemble of given states rather than other details of a given state space or pairwise relations of given states. Then, we consider bipartite systems where ensemble steering is possible, and show that show that with two operational tasks, ensemble steering and the no-signaling condition, the distinguishability is tightly determined. The result is independent to the structure of the state space. This concludes that the distinguishability is generally determined by the compatibility between two tasks, ensemble steering on states and the non-signaling principle on probability distributions of outcomes.

DOI
10.4204/eptcs.171.3

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We consider a fundamental operational task, distinguishing systems in different states, in the framework of generalized probabilistic theories and provide a general formalism of minimum-error discrimination of states in convex optimization. With the formalism established, we show that the distinguishability is generally a global property assigned to the ensemble of given states rather than other details of a given state space or pairwise relations of given states. Then, we consider bipartite systems where ensemble steering is possible, and show that show that with two operational tasks, ensemble steering and the no-signaling condition, the distinguishability is tightly determined. The result is independent to the structure of the state space. This concludes that the distinguishability is generally determined by the compatibility between two tasks, ensemble steering on states and the non-signaling principle on probability distributions of outcomes.
We consider optimal state discrimination in a general convex operational framework, so-called generalized probabilistic theories (GPTs), and present a general method of optimal discrimination by applying the complementarity problem from convex optimization. The method exploits the convex geometry of states but not other detailed conditions or relations of states and effects. We also show that properties in optimal quantum state discrimination are shared in GPTs in general: (i) no measurement sometimes gives optimal discrimination, and (ii) optimal measurement is not unique.
A central question in quantum information is whether a given set of states can provide an advantage in some information-processing task. We establish a universal connection between two such questions: how well a set of states can be discriminated, and whether it can power tasks whose advantage stems from generalized contextuality. Working in the framework of generalized probabilistic theories (GPTs), which includes quantum and classical systems as special cases, we show that any set of states able to provide a nonclassical advantage in a contextuality-powered task must obey nontrivial upper bounds on the success probability of every state discrimination task using the full set. Contextual advantage therefore implies limited distinguishability, exposing a trade-off between two basic operational resources. Importantly, this is more than a statement about the idealized limit: perfect distinguishability is not needed to preclude contextuality. Our thresholds lie strictly below unity, so any set whose discrimination performance exceeds them while still imperfectly distinguishable is already guaranteed to admit a noncontextual explanation. The bounds follow from a simple geometric property, linear dependence of the state set, take a closed analytical form, and depend only on the prior of the task and the convex geometry of the states. As an illustration, we apply them to generalized parity-oblivious multiplexing, where sufficiently high success in the task implies that at least one sub-ensemble of the codebook cannot power any contextual advantage.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
For distinguishing quantum states sampled from a fixed ensemble, the gap in bipartite and single-party distinguishability can be interpreted as a nonlocality of the ensemble. In this paper, we consider bipartite state discrimination in a composite system consisting of NNN subsystems, where each subsystem is shared between two parties and the state of each subsystem is randomly sampled from a particular ensemble comprising the Bell states. We show that the success probability of perfectly identifying the state converges to 111 as N→∞N\rightarrow\inftyN→∞ if the entropy of the probability distribution associated with the ensemble is less than 1/21/21/2. In other words, the nonlocality of the NNN-fold ensemble asymptotically disappears if the probability distribution associated with each ensemble is concentrated. Furthermore, we show that the discrimination task is a remarkable counterexample of a fundamental open question in theoretical computer science, called a parallel repetition conjecture of interactive games with two classically communicating players. Measurements for the discrimination task include a projective measurement of one party represented by stabilizer states, which enable the other party to perfectly distinguish states that are sampled with high probability.
Within finite-dimensional generalized probabilistic theories (GPTs), we distinguish between the geometric transition probability tau(psi,phi), defined as the maximum probability of accepting phi when the state is psi, and the predictive probability P(phi|psi) assigned to measurement outcomes. We ask what functional relationship P = Phi(tau) is compatible with relativistic causality. We prove that in any GPT satisfying purification, and therefore admitting steering, the only such relationship consistent with no-signaling is the identity Phi(p) = p. Any strictly convex or concave deviation from linearity enables superluminal signaling through steering scenarios. We provide an explicit qubit example showing how nonlinear probability rules generate detectable signaling channels. Combined with standard reconstruction results, this yields the Born rule |<phi|psi>|^2 as the unique causally consistent probability assignment. Our analysis clarifies the distinction between geometric structure and probabilistic prediction in quantum theory, and identifies steering as the mechanism enforcing the Born rule.
In this paper, we consider the conditions under which a given ensemble of two-qubit states can be optimally distinguished by local operations and classical communication (LOCC). We begin by completing the perfect distinguishability problem of two-qubit ensembles-both for separable operations and LOCC-by providing necessary and sufficient conditions for the perfect discrimination of one pure and one mixed state. Then, for the well-known task of minimum error discrimination, it is shown that almost all two-qubit ensembles consisting of three pure states cannot be optimally discriminated using LOCC. This is surprising considering that any two pure states can be distinguished optimally by LOCC. Special attention is given to ensembles that lack entanglement, and we prove an easy sufficient condition for when a set of three product states cannot be optimally distinguished by LOCC, thus providing new examples of the phenomenon known as non-locality without entanglement. We next consider an example of N parties who each share the same state but who are ignorant of its identity. The state is drawn from the rotationally invariant trine ensemble, and we establish a tight connection between the N-copy ensemble and Shor's lifted single-copy ensemble. For any finite N, we prove that optimal identification of the states cannot be achieved by LOCC; however, as N→∞, LOCC can indeed discriminate the states optimally. This is the first result of its kind. Finally, we turn to the task of unambiguous discrimination and derive new lower bounds on the LOCC inconclusive probability for symmetric states. When applied to the double trine ensemble, this leads to a rather different distinguishability character than when the minimum error probability is considered.