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arXiv:1302.5902 · Physical Review A

Entanglement-assisted guessing of complementary measurement outcomes

Mario Berta, Patrick J. Coles, Stephanie Wehner

Published Dec 22, 2014• 70 citations
arXivHTMLPDF

Abstract

Heisenberg's uncertainty principle implies that if one party (Alice) prepares a system and randomly measures one of two incompatible observables, then another party (Bob) cannot perfectly predict the measurement outcomes. This implication assumes that Bob does not possess an additional system that is entangled to the measured one; indeed, the seminal paper of Einstein, Podolsky, and Rosen (EPR) showed that maximal entanglement allows Bob to perfectly win this guessing game. Although not in contradiction, the observations made by EPR and Heisenberg illustrate two extreme cases of the interplay between entanglement and uncertainty. On the one hand, no entanglement means that Bob's predictions must display some uncertainty. Yet on the other hand, maximal entanglement means that there is no more uncertainty at all. Here we follow an operational approach and give an exact relation---an equality---between the amount of uncertainty as measured by the guessing probability and the amount of entanglement as measured by the recoverable entanglement fidelity. From this equality, we deduce a simple criterion for witnessing bipartite entanglement and an entanglement monogamy equality.

DOI
10.1103/physreva.90.062127

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Heisenberg's uncertainty principle implies that if one party (Alice) prepares a system and randomly measures one of two incompatible observables, then another party (Bob) cannot perfectly predict the measurement outcomes. This implication assumes that Bob does not possess an additional system that is entangled to the measured one; indeed, the seminal paper of Einstein, Podolsky, and Rosen (EPR) showed that maximal entanglement allows Bob to perfectly win this guessing game. Although not in contradiction, the observations made by EPR and Heisenberg illustrate two extreme cases of the interplay between entanglement and uncertainty. On the one hand, no entanglement means that Bob's predictions must display some uncertainty. Yet on the other hand, maximal entanglement means that there is no more uncertainty at all. Here we follow an operational approach and give an exact relation---an equality---between the amount of uncertainty as measured by the guessing probability and the amount of entanglement as measured by the recoverable entanglement fidelity. From this equality, we deduce a simple criterion for witnessing bipartite entanglement and an entanglement monogamy equality.
The uncertainty principle can be understood as constraining the probability of winning a game in which Alice measures one of two conjugate observables, such as position or momentum, on a system provided by Bob, and he is to guess the outcome. Two variants are possible: either Alice tells Bob which observable she measured, or he has to furnish guesses for both cases. Here I derive uncertainty relations for both, formulated directly in terms of Bob's guessing probabilities. For the former these relate to the entanglement that can be recovered by action on Bob's system alone. This gives an explicit quantum circuit for approximate quantum error correction using the guessing measurements for ``amplitude'' and ``phase'' information, implicitly used in the recent construction of efficient quantum polar codes. I also find a relation on the guessing probabilities for the latter game, which has application to wave-particle duality relations.
We consider a game in which two separate laboratories collaborate to prepare a quantum system and are then asked to guess the outcome of a measurement performed by a third party in a random basis on that system. Intuitively, by the uncertainty principle and the monogamy of entanglement, the probability that both players simultaneously succeed in guessing the outcome correctly is bounded. We are interested in the question of how the success probability scales when many such games are performed in parallel. We show that any strategy that maximizes the probability to win every game individually is also optimal for the parallel repetition of the game. Our result implies that the optimal guessing probability can be achieved without the use of entanglement. We explore several applications of this result. Firstly, we show that it implies security for standard BB84 quantum key distribution when the receiving party uses fully untrusted measurement devices , i.e. we show that BB84 is one-sided device independent. Secondly, we show how our result can be used to prove security of a one-round position-verification scheme. Finally, we generalize a well-known uncertainty relation for the guessing probability to quantum side information.
Multipartite entanglement remains a challenging and not fully understood aspect of quantum information. Monogamy-of-Entanglement (MoE) games have been highly effective for studying limitations on the usefulness of entanglement imposed by monogamy constraints. To better reveal the extent to which multipartite entanglement can be useful, we introduce a class of quantum guessing games, termed One-at-a-Time Guessing (OTG) games. In these games, quantum players individually guess the outcomes of random measurements performed by a referee on a pre-shared entangled state. Unlike MoE games, OTG games select players individually at random according to a specified probability distribution, thereby probing each player's correlation with the referee. We show that, despite monogamy constraints, players sharing certain entangled states can moderately outperform those relying only on classical uncertainty. This advantage arises even in simple OTG games involving only Pauli measurements on qubits, where optimal entanglement increases the winning probability by at least 4%. This contrasts with MoE games, where shared entanglement has been shown in several settings to provide only limited (if any) advantage over classical strategies. We further establish a majorization property: the value of an OTG game respects the majorization ordering of the player-selection probability distribution. We also analyze in detail a two-player OTG game in which the referee measures one of the three Pauli observables on a qubit, and show that it is optimally played using a specific parameterized family of three-qubit WWW-like states. These results suggest that OTG games provide a useful framework for investigating the usefulness of multipartite entanglement in multiparty quantum correlations.
Complementary relationships exist among interference properties of particles such as pattern visibility, predictability, and distinguishability. Additionally relationships between average information gain Ḡ and measurement disturbance F for entangled spin pairs are well established. This article examines whether a similar complementary relationship exists between entanglement and measurement. For qubit systems, both measurements on a single system and measurements on a bipartite system are considered in regard to entanglement. It is proven that Ē + D ≤ 1 holds, where Ē is the average entanglement after a measurement is made and D is a measure of the measurement disturbance of a single measurement. Assuming measurements on a bipartite system shared by Alice and Bob, it is shown that Ē + Ḡ ≤ 1, where Ḡ is the maximum average information gain that Bob can obtain regarding Alice’s result. These results are generalized to arbitrary initial mixed states and non-Hermitian operators. In the case of maximally entangled initial states, it is found that D ≤ EL and Ḡ ≤ EL, where EL is the loss of entanglement due to measurement by Alice. We conclude that the amount of disturbance and average information gain one can achieve is strictly limited by entanglement.
According to Heisenberg's uncertainty relation, there is an ultimate limit to how precisely we may predict the outcome of position and momentum measurements on a quantum system. We show that this limit may be violated by an arbitrarily large factor if one aims, instead, to guess the unknown value of a past measurement. For experiments on a single quantum system, the precise assignment of past position and momentum measurement outcomes is accompanied by large uncertainty about their linear combinations, while we show that entanglement with an ancillary system permits accurate retrodiction of any such linear combination. Finally, we show that the outcomes of experiments that jointly measure multiple linear combinations of position and momentum observables by means of ancillary probe particles can also be guessed with no formal lower limit. We present quantitative results for projective measurements and for generalized measurements where all components are prepared in Gaussian states.
Abstract As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations.