Quantum theory can be regarded as a noncommutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the well-known isomorphism between completely positive maps and bipartite density operators is derived, which makes this connection much more explicit. This isomorphism is given an operational interpretation in terms of statistical correlations between ensemble preparation procedures and outcomes of measurements. Finally, the isomorphism is applied to elucidate the connection between no-cloning and no-broadcasting theorems and the monogamy of entanglement, and a simplified proof of the no-broadcasting theorem is obtained as a by-product.
Quantum theory can be regarded as a noncommutative generalization of classical probability. From this point of view, one expects quantum dynamics to be analogous to classical conditional probabilities. In this paper, a variant of the well-known isomorphism between completely positive maps and bipartite density operators is derived, which makes this connection much more explicit. This isomorphism is given an operational interpretation in terms of statistical correlations between ensemble preparation procedures and outcomes of measurements. Finally, the isomorphism is applied to elucidate the connection between no-cloning and no-broadcasting theorems and the monogamy of entanglement, and a simplified proof of the no-broadcasting theorem is obtained as a by-product.
The relations between dynamical maps and quantum states of bipartite systems are analyzed from the perspective of quantum conditional probability. In particular, we explore new interesting relations between completely positive maps, which correspond to quantum channels, and states of bipartite systems which correspond to correlations between the initial and final states. The new connection emerges in a natural way from the generalisation of the classical concept of conditional probability. We develop applications of these relations which prove to be very useful in both directions, either for the classification of positive maps which are not completely positive, the classification of non-decomposable dynamical maps or for the classification of positive partial transpose and entangled states.
Quantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: Channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.
Quantum theory can be viewed as a generalization of classical probability
theory, but the analogy as it has been developed so far is not complete.
Classical probability theory is independent of causal structure, whereas the
conventional quantum formalism requires causal structure to be fixed in
advance. In this paper, we develop the formalism of quantum conditional states,
which unifies the description of experiments involving two systems at a single
time with the description of those involving a single system at two times. The
analogies between quantum theory and classical probability theory are expressed
succinctly within the formalism and it unifies the mathematical description of
distinct concepts, such as ensemble preparation procedures, measurements, and
quantum dynamics. We introduce a quantum generalization of Bayes' theorem and
the associated notion of Bayesian conditioning. Conditioning a quantum state on
a classical variable is the correct rule for updating quantum states in light
of classical data, regardless of the causal relationship between the classical
variable and the quantum degrees of freedom, but it does not include the
projection postulate as a special case. We show that previous arguments that
projection is the quantum generalization of conditioning are based on
misleading analogies. Since our formalism is causally neutral, conditioning
provides a unification of the predictive and retrodictive formalisms for
prepare-and-measure experiments and leads to an elegant derivation of the set
of states that a system can be steered to by making measurements on a remote
system.
Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the update maps corresponding to a complete projective measurement, must be subjective, the situation is not so clear in other cases. Here, it is argued that all trace preserving completely positive maps, including unitary operators, should be regarded as subjective, in the same sense as a classical conditional probability distribution. The argument is based on a reworking of the Choi‐Jamiołkowski isomorphism in terms of “conditional” density operators and trace preserving completely positive maps, which mimics the relationship between conditional probabilities and stochastic maps in classical probability.
Abstract. Assuming that quantum states, including pure states, represent subjective degrees of belief rather than objective properties of systems, the question of what other elements of the quantum formalism must also be taken as subjective is addressed. In particular, we ask this of the dynamical aspects of the formalism, such as Hamiltonians and unitary operators. Whilst some operations, such as the update maps corresponding to a complete projective measurement, must be subjective, the situation is not so clear in other cases. Here, it is argued that all trace preserving completely positive maps, including unitary operators, should be regarded as subjective, in the same sense as a classical conditional probability distribution. The argument is based on a reworking of the Choi-Jamiołkowski isomorphism in terms of “conditional ” density operators and trace preserving completely positive maps, which mimics the relationship between conditional probabilities and stochastic maps in classical probability.