Optimal signal states for quantum detectors

Quantum detectors provide information about quantum systems by establishing correlations between certain properties of those systems and a set of macroscopically distinct states of the corresponding measurement devices. A natural question of fundamental significance is how much information a quantum detector can extract from the quantum system it is applied to. In the present paper we address this question within a precise framework: given a quantum detector implementing a specific generalized quantum measurement, what is the optimal performance achievable with it for a concrete information readout task, and what is the optimal way to encode information in the quantum system in order to achieve this performance? We consider some of the most common information transmission tasks - the Bayes cost problem (of which minimal error discrimination is a special case), unambiguous message discrimination, and the maximal mutual information. We provide general solutions to the Bayesian and unambiguous discrimination problems. We also show that the maximal mutual information has an interpretation of a capacity of the measurement, and derive various properties that it satisfies, including its relation to the accessible information of an ensemble of states, and its form in the case of a group-covariant measurement. We illustrate our results with the example of a noisy two-level symmetric informationally complete measurement, for whose capacity we give analytical proofs of optimality. The framework presented here provides a natural way to characterize generalized quantum measurements in terms of their information readout capabilities.Comment: 13 pages, 1 figure, example section extende

Universal field equations for metric-affine theories of gravity

We show that almost all metric--affine theories of gravity yield Einstein equations with a non--null cosmological constant $\Lambda$. Under certain circumstances and for any dimension, it is also possible to incorporate a Weyl vector field $W_\mu$ and therefore the presence of an anisotropy. The viability of these field equations is discussed in view of recent astrophysical observations.Comment: 13 pages. This is a copy of the published paper. We are posting it here because of the increasing interest in f(R) theories of gravit

Recycling of quantum information: Multiple observations of quantum systems

Given a finite number of copies of an unknown qubit state that have already been measured optimally, can one still extract any information about the original unknown state? We give a positive answer to this question and quantify the information obtainable by a given observer as a function of the number of copies in the ensemble, and of the number of independent observers that, one after the other, have independently measured the same ensemble of qubits before him. The optimality of the protocol is proven and extensions to other states and encodings are also studied. According to the general lore, the state after a measurement has no information about the state before the measurement. Our results manifestly show that this statement has to be taken with a grain of salt, specially in situations where the quantum states encode confidential information.Comment: 4 page

Secrecy content of two-qubit states

We analyze the set of two-qubit states from which a secret key can be extracted by single-copy measurements plus classical processing of the outcomes. We introduce a key distillation protocol and give the corresponding necessary and sufficient condition for positive key extraction. Our results imply that the critical error rate derived by Chau, Phys. Rev. A {\bf 66}, 060302 (2002), for a secure key distribution using the six-state scheme is tight. Remarkably, an optimal eavesdropping attack against this protocol does not require any coherent quantum operation.Comment: 5 pages, RevTe

Beating noise with abstention in state estimation

We address the problem of estimating pure qubit states with non-ideal (noisy) measurements in the multiple-copy scenario, where the data consists of a number N of identically prepared qubits. We show that the average fidelity of the estimates can increase significantly if the estimation protocol allows for inconclusive answers, or abstentions. We present the optimal such protocol and compute its fidelity for a given probability of abstention. The improvement over standard estimation, without abstention, can be viewed as an effective noise reduction. These and other results are exemplified for small values of N. For asymptotically large N, we derive analytical expressions of the fidelity and the probability of abstention, and show that for a fixed fidelity gain the latter decreases with N at an exponential rate given by a Kulback-Leibler (relative) entropy. As a byproduct, we obtain an asymptotic expression in terms of this very entropy of the probability that a system of N qubits, all prepared in the same state, has a given total angular momentum. We also discuss an extreme situation where noise increases with N and where estimation with abstention provides a most significant improvement as compared to the standard approach

On the geometry of four qubit invariants

The geometry of four-qubit entanglement is investigated. We replace some of the polynomial invariants for four-qubits introduced recently by new ones of direct geometrical meaning. It is shown that these invariants describe four points, six lines and four planes in complex projective space ${\bf CP}^3$. For the generic entanglement class of stochastic local operations and classical communication they take a very simple form related to the elementary symmetric polynomials in four complex variables. Moreover, their magnitudes are entanglement monotones that fit nicely into the geometric set of $n$-qubit ones related to Grassmannians of $l$-planes found recently. We also show that in terms of these invariants the hyperdeterminant of order 24 in the four-qubit amplitudes takes a more instructive form than the previously published expressions available in the literature. Finally in order to understand two, three and four-qubit entanglement in geometric terms we propose a unified setting based on ${\bf CP}^3$ furnished with a fixed quadric.Comment: 19 page

Estimation of pure qubits on circles

Gisin and Popescu [PRL, 83, 432 (1999)] have shown that more information about their direction can be obtained from a pair of anti-parallel spins compared to a pair of parallel spins, where the first member of the pair (which we call the pointer member) can point equally along any direction in the Bloch sphere. They argued that this was due to the difference in dimensionality spanned by these two alphabets of states. Here we consider similar alphabets, but with the first spin restricted to a fixed small circle of the Bloch sphere. In this case, the dimensionality spanned by the anti-parallel versus parallel alphabet is now equal. However, the anti-parallel alphabet is found to still contain more information in general. We generalize this to having N parallel spins and M anti-parallel spins. When the pointer member is restricted to a small circle these alphabets again span spaces of equal dimension, yet in general, more directional information can be found for sets with smaller |N-M| for any fixed total number of spins. We find that the optimal POVMs for extracting directional information in these cases can always be expressed in terms of the Fourier basis. Our results show that dimensionality alone cannot explain the greater information content in anti-parallel combinations of spins compared to parallel combinations. In addition, we describe an LOCC protocol which extract optimal directional information when the pointer member is restricted to a small circle and a pair of parallel spins are supplied.Comment: 23 pages, 8 figure

Multiple copy 2-state discrimination with individual measurements

We address the problem of non-orthogonal two-state discrimination when multiple copies of the unknown state are available. We give the optimal strategy when only fixed individual measurements are allowed and show that its error probability saturates the collective (lower) bound asymptotically. We also give the optimal strategy when adaptivity of individual von Neumann measurements is allowed (which requires classical communication), and show that the corresponding error probability is exactly equal to the collective one for any number of copies. We show that this strategy can be regarded as Bayesian updating.Comment: 5 pages, RevTe