Quantum information, though not precisely defined, is a fundamental concept of quantum information theory which predicts many fascinating phenomena and provides new physical resources. A basic problem is to recognize the features of quantum systems responsible for those phenomena. One of these important features is that non-commuting quantum states cannot be broadcast: two copies cannot be obtained out of a single copy, not even reproduced marginally on separate systems. We focus on the difference in information content between one copy and two copies, which is a basic manifestation of the gap between quantum and classical information. We show that if the chosen information measure is the Holevo quantity, the difference between the information content of one copy and two copies is zero if and only if the states can be broadcast. We propose a new approach in defining measures of quantumness of ensembles based on the difference in information content between the original ensemble and the ensemble of duplicated states. We comment on the permanence property of quantum states and the recently introduced superbroadcasting operation. We also provide an appendix where we discuss the status of quantum information in quantum physics, based on the so-called isomorphism principle.
Quantum information, though not precisely defined, is a fundamental concept of quantum information theory which predicts many fascinating phenomena and provides new physical resources. A basic problem is to recognize the features of quantum systems responsible for those phenomena. One of these important features is that non-commuting quantum states cannot be broadcast: two copies cannot be obtained out of a single copy, not even reproduced marginally on separate systems. We focus on the difference in information content between one copy and two copies, which is a basic manifestation of the gap between quantum and classical information. We show that if the chosen information measure is the Holevo quantity, the difference between the information content of one copy and two copies is zero if and only if the states can be broadcast. We propose a new approach in defining measures of quantumness of ensembles based on the difference in information content between the original ensemble and the ensemble of duplicated states. We comment on the permanence property of quantum states and the recently introduced superbroadcasting operation. We also provide an appendix where we discuss the status of quantum information in quantum physics, based on the so-called isomorphism principle.
Modern quantum physics now enables control of quantum systems at the level of individual trajectories, opening a new frontier that links quantum information theory, quantum many-body physics, and quantum thermodynamics, and uncovers novel non-equilibrium phenomena such as deep thermalization and measurement-induced entanglement. However, a central challenge remains: their characterization relies on measuring nonlinear properties of individual quantum states, a task tantamount to fine-grained cloning of a quantum ensemble. Here, the fundamental laws governing the cloning of quantum ensembles are investigated. First, a general no-cloning theorem for arbitrary ensembles is established from an information-theoretic perspective, even assuming multiple copies of the ensemble's purification. It is then shown that this barrier can be unexpectedly circumvented for physical ensembles generated by finite-time evolutions. Nevertheless, these tasks are proven to remain computationally intractable, even when the full circuit description of state preparation is known. This stands in sharp contrast to the conventional no-cloning theorem, which relies on the state being unknown. Together, these results establish new fundamental principles of quantum mechanics, reveal intrinsic trade-offs among sample complexity, computational complexity, and quantum measurements, and highlight the necessity of problem-specific strategies for probing measurement-induced quantum phenomena.
We prove a no-broadcasting theorem for the Holevo information of a noncommuting ensemble stating that no operation can generate a bipartite ensemble such that both copies have the same information as the original. We argue that upper bounds on the average information over both copies imply lower bounds on the quantum capacity required to send the ensemble without information loss. This is because a channel with zero quantum capacity has a unitary extension transferring at least as much information to its environment as it transfers to the output. For an ensemble being the time orbit of a pure state under a Hamiltonian evolution, we derive such a bound on the required quantum capacity in terms of properties of the input and output energy distribution. Moreover, we discuss relations between the broadcasting problem and entropy power inequalities. The broadcasting problem arises when a signal should be transmitted by a time-invariant device such that the outgoing signal has the same timing information as the incoming signal had. Based on previous results we argue that this establishes a link between quantum information theory and the theory of low power computing because the loss of timing information implies loss of free energy.
Abstract No-broadcasting theorem is one of the most fundamental results in quantum information theory; it guarantees that the simplest attacks on any quantum protocol, based on eavesdropping and copying of quantum information, are impossible. Due to the fundamental importance of the no-broadcasting theorem, it is essential to understand the exact boundaries of this limitation. We generalize the standard definition of broadcasting by restricting the set of states which we want to broadcast and restricting the sets of measurements which we use to test the broadcasting. We show that in some of the investigated cases broadcasting is equivalent to commutativity, while in other cases commutativity is not necessary.
We propose a new measure of relative incompatibility for a quantum system with respect to two non-commuting observables, and call it quantumness of relative incompatibility. In case of a classical state, order of observation is inconsequential, hence probability distribution of outcomes of any observable remains undisturbed. We define relative entropy of the two marginal probability distributions as a measure of quantumness in the state, which is revealed only in presence of two non-commuting observables. Like all other measures, we show that the proposed measure satisfies some basic axioms. Also, we find that this measure depicts complementarity with quantum coherence. The relation is more vivid when we choose one of the observables in such a way that its eigen basis matches with the basis in which the coherence is measured. Our result indicates that the quantumness in a single system is still an interesting question to explore and there can be an inherent feature of the state which manifests beyond the idea of quantum coherence.
We study the quantumness of bipartite correlations by proposing a quantity that combines a measure of total correlations-mutual information-with the notion of broadcast copies-i.e., generally nonfactorized copies-of bipartite states. By analyzing how our quantity increases with the number of broadcast copies, we are able to classify classical, separable, and entangled states. This motivates the definition of the broadcast regularization of mutual information, the asymptotic minimal mutual information per broadcast copy, which we show to have many properties of an entanglement measure.
The no-broadcasting theorem, first established by Barnum et al. [Phys. Rev. Lett. 76, 2818 (1996)], states that a set of quantum states can be broadcast if and only if it constitutes a commuting family. Quite recently, Piani et al. [Phys. Rev. Lett. 100, 090502 (2008)] showed, by using an ingenious and sophisticated method, that the correlations in a single bipartite state can be locally broadcast if and only if the state is effectively a classical one (i.e., the correlations therein are classical). In this Brief Report, under the condition of nondegenerate spectrum, we provide an alternative and significantly simpler proof of the latter result based on the original no-broadcasting theorem and the monotonicity of the quantum relative entropy. This derivation motivates us to conjecture the equivalence between these two elegant yet formally different no-broadcasting theorems and indicates a subtle and fundamental issue concerning spectral degeneracy which also lies at the heart of the conflict between the von Neumann projection postulate and the L\"uders ansatz for quantum measurements. This relation not only offers operational interpretations for commutativity and classicality but also illustrates the basic significance of noncommutativity in characterizing quantumness from the informational perspective.