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Any physical process can be represented as a quantum channel mapping an initial state to a final state. Hence it can be characterized from the point of view of communication theory, i.e., in terms of its ability to transfer information. Quantum information provides a theoretical framework and the proper mathematical tools to accomplish this. In this context the notion of codes and communication capacities have been introduced by generalizing them from the classical Shannon theory of information transmission and error correction. The underlying assumption of this approach is to consider the channel not as acting on a single system, but on sequences of systems, which, when properly initialized allow one to overcome the noisy effects induced by the physical process under consideration. While most of the work produced so far has been focused on the case in which a given channel transformation acts identically and independently on the various elements of the sequence (memoryless configuration in jargon), correlated error models appear to be a more realistic way to approach the problem. A slightly different, yet conceptually related, notion of correlated errors applies to a single quantum system which evolves continuously in time under the influence of an external disturbance which acts on it in a non-Markovian fashion. This leads to the study of memory effects in quantum channels: a fertile ground where interesting novel phenomena emerge at the intersection of quantum information theory and other branches of physics. A survey is taken of the field of quantum channels theory while also embracing these specific and complex settings

We determine the ultimate potential of quantum imaging for boosting the resolution of a far-field, diffractionlimited, linear imaging device within the paraxial approximation. First we show that the problem of estimating the separation between two point-like sources is equivalent to the estimation of the loss parameters of two lossy bosonic channels, i.e., the transmissivities of two beam splitters. Using this representation, we establish the ultimate precision bound for resolving two point-like sources in an arbitrary quantum state, with a simple formula for the specific case of two thermal sources. We find that the precision bound scales with the number of collected photons according to the standard quantum limit. Then we determine the sources whose separation can be estimated optimally, finding that quantum-correlated sources (entangled or discordant) can be super-resolved at the sub-Rayleigh scale. Our results set the upper bounds on any present or future imaging technology, from astronomical observation to microscopy, which is based on quantum detection as well as source engineering. Introduction. Quantum imaging aims at harnessing quantum features of light to obtain optical images of high resolution beyond the boundary of classical optics. Its range of potential applications is very broad, from telescopy to microscopy and medical diagnosis, and has motivated a substantial research activity [1][2][3][4][5][6][7][8][9][10]. Typically, quantum imaging is scrutinized to outperform classical imaging in two ways. First, to resolve details below the Rayleigh length (sub-Rayleigh imaging). Second, to improve the way the precision scales with the number of photons, by exploiting non-classical states of light. It is well known that a collective state of N quantum particles has an effective wavelength that is N times smaller than individual particles [11][12][13][14][15][16][17][18]. If N independent photons are measured one expects that the blurring of the image scales as 1/ √ N (known as standard quantum limit or shot-noise limit), while for N entangled photons one can sometimes achieve a 1/N scaling (known as the Heisenberg limit).

What is the ultimate performance for discriminating two arbitrary quantum channels acting on a finite-dimensional Hilbert space? Here we address this basic question by deriving a general and fundamental lower bound. More precisely, we investigate the symmetric discrimination of two arbitrary qudit channels by means of the most general protocols based on adaptive (feedbackassisted) quantum operations. In this general scenario, we first show how port-based teleportation can be used to simplify these adaptive protocols into a much simpler non-adaptive form, designing a new type of teleportation stretching. Then, we prove that the minimum error probability affecting the channel discrimination cannot beat a bound determined by the Choi matrices of the channels, establishing a general, yet computable formula for quantum hypothesis testing. As a consequence of this bound, we derive ultimate limits and no-go theorems for adaptive quantum illumination and single-photon quantum optical resolution. Finally, we show how the methodology can also be applied to other tasks, such as quantum metrology, quantum communication and secret key generation.

Most methods of optimal control cannot obtain accurate time-optimal protocols. The quantum brachistochrone equation is an exception, and has the potential to provide accurate time-optimal protocols for a wide range of quantum control problems. So far, this potential has not been realized, however, due to the inadequacy of conventional numerical methods to solve it. Here we show that the quantum brachistochrone problem can be recast as that of finding geodesic paths in the space of unitary operators. We expect this brachistochrone-geodesic connection to have broad applications, as it opens up minimaltime control to the tools of geometry. As one such application, we use it to obtain a fast numerical method to solve the brachistochrone problem, and apply this method to two examples demonstrating its power. The ability to realize a prescribed evolution for a given physical quantum device is important in a range of applications. A powerful approach to this task is to vary the Hamiltonian of the device with time [1][2][3]. A prescription for a time-dependent Hamiltonian realizing a desired evolution is called a "control protocol," and a protocol that achieves this task in the minimal time is called "timeoptimal" [4,5]. Because the ever-present noise from the environment degrades quantum states over time, generating the fastest possible evolution is important in information processing [6][7][8], metrology [9][10][11], cooling [12,13], and experiments that probe quantum behavior [14][15][16][17]. Finding accurate time-optimal protocols is difficult because it is a two-objective optimization problem: one must minimize the error in the resulting evolution while simultaneously minimizing the time taken by the protocol (hereafter, the "protocol time"). Finding approximate protocols, on the other hand, is relatively easy: one can minimize a weighted sum of the two objectives [1], or search for protocols at a range of fixed times to locate a likely minimum time. But there is presently no practical way to refine these further. Analytical methods that use the Pontryagin maximum principle or the geometry of the unitary group are useful only for specific kinds of problems and constraints [5,[18][19][20][21][22][23][24][25][26]. In view of this, the quantum brachistochrone equation (QBE) was a significant development [27][28][29]. It has the potential to provide accurate time-optimal protocols under two physically relevant constraints: (i) the system has a finite energy bandwidth (the norm of the Hamiltonian is bounded); and (ii) the Hamiltonian is restricted to a subspace of Hermitian operators. Nevertheless, an obstacle remains that has prevented the QBE from becoming a practical tool: the QBE transforms the optimization problem into that of solving an ordinary differential equation (ODE) with boundary values, but there exists no numerical method that can solve such a boundary-value problem (BVP) efficiently in high dimensions. The available methods, namely "simple shooting," "multiple shooting," finite difference, and finite-el...

The readout of a classical memory can be modelled as a problem of quantum channel discrimination, where a decoder retrieves information by distinguishing the different quantum channels encoded in each cell of the memory [S. Pirandola, Phys. Rev. Lett. 106, 090504 (2011)]. In the case of optical memories, such as CDs and DVDs, this discrimination involves lossy bosonic channels and can be remarkably boosted by the use of nonclassical light (quantum reading). Here we generalize these concepts by extending the model of memory from single-cell to multi-cell encoding. In general, information is stored in a block of cells by using a channel-codeword, i.e., a sequence of channels chosen according to a classical code. Correspondingly, the readout of data is realized by a process of "parallel" channel discrimination, where the entire block of cells is probed simultaneously and decoded via an optimal collective measurement. In the limit of an infinite block we define the quantum reading capacity of the memory, quantifying the maximum number of readable bits per cell. This notion of capacity is nontrivial when we suitably constrain the physical resources of the decoder. For optical memories (encoding bosonic channels), such a constraint is energetic and corresponds to fixing the mean total number of photons per cell. In this case, we are able to prove a separation between the quantum reading capacity and the maximum information rate achievable by classical transmitters, i.e., arbitrary classical mixtures of coherent states. In fact, we can easily construct nonclassical transmitters that are able to outperform any classical transmitter, thus showing that the advantages of quantum reading persist in the optimal multi-cell scenario.

The locking effect is a phenomenon that is unique to quantum information theory and represents one of the strongest separations between the classical and quantum theories of information. The Fawzi-HaydenSen locking protocol harnesses this effect in a cryptographic context, whereby one party can encode n bits into n qubits while using only a constant-size secret key. The encoded message is then secure against any measurement that an eavesdropper could perform in an attempt to recover the message, but the protocol does not necessarily meet the composability requirements needed in quantum key distribution applications. In any case, the locking effect represents an extreme violation of Shannon's classical theorem, which states that information-theoretic security holds in the classical case if and only if the secret key is the same size as the message. Given this intriguing phenomenon, it is of practical interest to study the effect in the presence of noise, which can occur in the systems of both the legitimate receiver and the eavesdropper. This paper formally defines the locking capacity of a quantum channel as the maximum amount of locked information that can be reliably transmitted to a legitimate receiver by exploiting many independent uses of a quantum channel and an amount of secret key sublinear in the number of channel uses. We provide general operational bounds on the locking capacity in terms of other well-known capacities from quantum Shannon theory. We also study the important case of bosonic channels, finding limitations on these channels' locking capacity when coherent-state encodings are employed and particular locking protocols for these channels that might be physically implementable.

This corrects the article DOI: 10.1103/PhysRevLett.118.100502.

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