We develop a framework which unifies seemingly different extension (or "joinability") problems for bipartite quantum states and channels. This includes well known extension problems such as optimal quantum cloning and quantum marginal problems as special instances. Central to our generalization is a variant of the Choi-Jamiolkowski isomorphism between bipartite states and dynamical maps which we term the "homocorrelation map": while the former emphasizes the preservation of the positivity constraint, the latter is designed to preserve statistical correlations, allowing direct contact with entanglement. In particular, we define and analyze state-joining, channel-joining, and local-positive joining problems in three-party settings exhibiting collective UxUxU symmetry, obtaining exact analytical characterizations in low dimension. Suggestively, we find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree, while quantum channels are limited in the degree to which their measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces an upper bound on the extent of positive correlation across a bipartite system at a given time, as well as on the extent of negative correlation between the state of a same system across two instants of time. We argue that these general statistical bounds inform the quantum joinability limitations, and show that they are in fact sufficient for the three-party UxUxU-invariant setting.
On state versus channel quantum extension problems: exact results for U⊗U⊗U symmetry
We develop a framework which unifies seemingly different extension (or "joinability") problems for bipartite quantum states and channels. This includes well known extension problems such as optimal quantum cloning and quantum marginal problems as special instances. Central to our generalization is a variant of the Choi-Jamiolkowski isomorphism between bipartite states and dynamical maps which we term the "homocorrelation map": while the former emphasizes the preservation of the positivity constraint, the latter is designed to preserve statistical correlations, allowing direct contact with entanglement. In particular, we define and analyze state-joining, channel-joining, and local-positive joining problems in three-party settings exhibiting collective UxUxU symmetry, obtaining exact analytical characterizations in low dimension. Suggestively, we find that bipartite quantum states are limited in the degree to which their measurement outcomes may agree, while quantum channels are limited in the degree to which their measurement outcomes may disagree. Loosely speaking, quantum mechanics enforces an upper bound on the extent of positive correlation across a bipartite system at a given time, as well as on the extent of negative correlation between the state of a same system across two instants of time. We argue that these general statistical bounds inform the quantum joinability limitations, and show that they are in fact sufficient for the three-party UxUxU-invariant setting.
We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global N-partite quantum state. Intuitively, we say that the multipartite state ``joins'' the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible N-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in d dimensions and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N-representability problem to the case where subsystems are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-n sharability for arbitrary d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.
We investigate some basic scenarios in which a given set of bipartite quantum states may consistently arise as the set of reduced states of a global N-partite quantum state. Intuitively, we say that the multipartite state joins the underlying correlations. Determining whether, for a given set of states and a given joining structure, a compatible N-partite quantum state exists is known as the quantum marginal problem. We restrict to bipartite reduced states that belong to the paradigmatic classes of Werner and isotropic states in d dimensions, and focus on two specific versions of the quantum marginal problem which we find to be tractable. The first is Alice-Bob, Alice-Charlie joining, with both pairs being in a Werner or isotropic state. The second is m-n sharability of a Werner state across N subsystems, which may be seen as a variant of the N-representability problem to the case where subsystems are partitioned into two groupings of m and n parties, respectively. By exploiting the symmetry properties that each class of states enjoys, we determine necessary and sufficient conditions for three-party joinability and 1-n sharability for arbitrary d. Our results explicitly show that although entanglement is required for sharing limitations to emerge, correlations beyond entanglement generally suffice to restrict joinability, and not all unentangled states necessarily obey the same limitations. The relationship between joinability and quantum cloning as well as implications for the joinability of arbitrary bipartite states are discussed.
We introduce and study the notion of steerability for channels. This generalizes the notion of steerability of bipartite quantum states. We discuss a key conceptual difference between the case of states and the case of channels: while state steering deals with the notion of “hidden” states, steerability in the channel case is better understood in terms of coherence of channel extensions, rather than in terms of “hidden” channels. This distinction vanishes in the case of states. We further argue how the proposed notion of lack of coherence of channel extensions coincides with the notion of channel extensions realized via local operations and classical communication. We also discuss how the Choi–Jamiołkowski isomorphism allows the direct application of many results about states to the case of channels. We introduce measures for the steerability of channel extensions.
The rapid advance of quantum information technology requires precise control and manipulation of quantum systems and of their properties; in particular, it is essential to certify that quantum processors truly work quantum mechanically, in order to validate experiments and their results. The problem of certification of quantum states and devices is a demanding one, thus many attempts have been put forward to find ways of efficiently test their basic functionalities, such as their entanglement properties. The main problems addressed in this thesis concern indeed finding optimal strategies for the estimation and characterization of quantum states and channels, with a special focus on entanglement correlations. In our first work we considered the certification of entanglement across a given partition of a multi-qubit system, when only partial information about the corresponding quantum state is available; we tackled the problem of finding the best measurement strategy introducing the statistics of lengths of measurement sequences, choosing multi-qubit Pauli-measurements as observables. Using a sample of random unknown states, we were able to identify the (on average) shortest, i.e. most efficient, measurement sequence to detect entanglement. The investigation was carried out with an algorithm based on the truncated moment sequences (tms) problem, which provides necessary and sufficient conditions for entanglement or separability of a quantum state of a finite dimensional system. This study results in a very efficient strategy especially for symmetric states, for which only a tiny fraction (10−6) of randomly chosen entangled states are undetected for systems with 6 qubits or more. We continued in our second work with the problem of separability of quantum channels to answer the question whether a given quantum device is able to create entanglement or not. The Choi matrix representation, provided by the Choi–Jamiołkowski isomorphism, casts the problem in terms of systems and ancillas, giving different classes of separability depending on the cut considered between them. Once again, the tms framework turned out to be well suited for this study, giving a unifying approach for the different separability problems; a solution is found in terms of the tms associated to the coordinates of the Choi state, in a fixed basis, and semidefinite programming is used to get a separability certificate. We explored examples of families of 2-qubit and single-qutrit channels, for which our algorithm can give an answer in cases where other criteria remain inconclusive. In a further work we shifted our attention to a different direction, highlighting a connection between the fields of quantum information, Bose-Einstein condensation (BEC) and analogue gravity, thus showing the relevance of quantum information notions in a broader context. We studied the entanglement properties of a BEC analogue of a black hole, which is described by a three-mode Gaussian state; we investigated bipartite and tripartite entanglement measures based on the covariance matrix description, both at zero and finite temperature, providing the best experimental configuration for entanglement detection. In a final stage of this thesis, we started looking at different measurement strategies for testing quantum devices whose functionality can be described by a single parameter; we used a Bayesian approach to estimate this parameter, exploiting the information collected through measurements to update a conditional probability distribution, without the need of estimating expected values of observables. The problem of finding the optimal experimental design can be in this case translated in the optimization of a utility function, which usually guides the update process adaptively at each step; we tried to outperform this approach looking ahead not only at the next step, but at a few ones, in order to obtain an answer of correct or wrong functioning as soon as possible. Moreover, we chose the input state for the quantum channel considered minimizing the error probability given by the Chernoff distance between the outputs of an ideal and a faulty channel; finally, we considered two different final decision criteria and compared their efficiency.
The well-known duality relating entangled states and noisy quantum channels is expressed in terms of a channel ket, a pure state on a suitable tripartite system, which functions as a pre-probability allowing the calculation of statistical correlations between, for example, the entrance and exit of a channel, once a framework has been chosen so as to allow a consistent set of probabilities. In each framework the standard notions of ordinary (classical) information theory apply, and it makes sense to ask whether information of a particular sort about one system is or is not present in another system. Quantum effects arise when a single pre-probability is used to compute statistical correlations in different incompatible frameworks, and various constraints on the presence and absence of different kinds of information are expressed in a set of all-or-nothing theorems which generalize or give a precise meaning to the concept of ``no-cloning.'' These theorems are used to discuss the location of information in quantum channels modeled using a mixed-state environment, the classical-quantum channels introduced by Holevo, and the location of information in the physical carriers of a quantum code. It is proposed that both channel and entanglement problems be classified in terms of pure states (functioning as pre-probabilities) on systems of $p\ensuremath{\geqslant}2$ parts, with mixed bipartite entanglement and simple noisy channels belonging to the category p=3, a five-qubit code to the category p=6, etc., then by the dimensions of the Hilbert spaces of the component parts, along with other criteria yet to be determined.
We derive simple and unified closed-form expressions for projections with respect to fidelity (equivalently, the Bures and purified distances) onto several sets of interest. These include projections of bipartite positive semidefinite (PSD) matrices onto the set of PSD matrices with a given marginal, and projections of ensembles of PSD matrices onto the set of PSD decompositions of a given matrix, with important special cases corresponding to projections onto the set of quantum channels (via the Choi isomorphism) and onto the set of measurements. We introduce prior-channel decompositions of completely positive (CP) maps, which uniquely decompose any CP map into a prior PSD matrix and a quantum channel. This decomposition generalizes the Choi-Jamiolkowski isomorphism by establishing a bijective correspondence between arbitrary bipartite PSD matrices and channel-state pairs, and we show that it arises naturally from the fidelity projections developed here. As applications, we show that the pretty good measurement - associated with a weighted ensemble - is the fidelity projection of the ensemble onto the set of measurements, and that the Petz recovery map - associated with a reference state and forward channel - is the projection of a CP map (constructed from the channel-state pair) onto the set of reverse quantum channels, thereby recasting the well-known identification of the Petz map with quantum Bayes' rule in information-geometric terms. Our results also provide an information-geometric underpinning of the Leifer-Spekkens quantum state over time formalism [Leifer and Spekkens, Phys. Rev. A 88, 052130 (2013)].