We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation suggesting that the CQC relation is generally valid. Importantly, we also show that the CQC relation represents an improvement to an entropic uncertainty principle in the presence of a quantum memory, and that it can be used to verify an achievable secret key rate in the quantum one-time pad cryptographic protocol.
We postulate the existence of a universal uncertainty relation between the quantum and classical mutual informations between pairs of quantum systems. Specifically, we propose that the sum of the classical mutual information, determined by two mutually unbiased pairs of observables, never exceeds the quantum mutual information. We call this the complementary-quantum correlation (CQC) relation and prove its validity for pure states, for states with one maximally mixed subsystem, and for all states when one measurement is minimally disturbing. We provide results of a Monte Carlo simulation suggesting that the CQC relation is generally valid. Importantly, we also show that the CQC relation represents an improvement to an entropic uncertainty principle in the presence of a quantum memory, and that it can be used to verify an achievable secret key rate in the quantum one-time pad cryptographic protocol.
Abstract As a very fundamental principle in quantum physics, uncertainty principle has been studied intensively via various uncertainty inequalities. A natural and fundamental question is whether an equality exists for the uncertainty principle. Here we derive an entropic uncertainty equality relation for a bipartite system consisting of a quantum system and a coupled quantum memory, based on the information measure introduced by Brukner and Zeilinger (Phys. Rev. Lett. 83:3354, 1999). The equality indicates that the sum of measurement uncertainties over a complete set of mutually unbiased bases on a subsystem is equal to a total, fixed uncertainty determined by the initial bipartite state. For the special case where the system and the memory are the maximally entangled, all of the uncertainties related to each mutually unbiased base measurement are zero, which is substantially different from the uncertainty inequality relation. The results are meaningful for fundamental reasons and give rise to operational applications such as in quantum random number generation and quantum guessing games. Moreover, we experimentally verify the measurement uncertainty relation in the presence of quantum memory on a five-qubit spin system by directly measuring the corresponding quantum mechanical observables, rather than quantum state tomography in all the previous experiments of testing entropic uncertainty relations.
Abstract Quantum entanglement is regarded as one of the core concepts, which is used to describe the non-classical correlation between subsystems, and entropic uncertainty relation plays a vital role in quantum precision measurement. It is well known that entanglement of formation can be expressed by von Neumann entropy of subsystems for arbitrary pure states. An interesting question is naturally raised: is there any intrinsic correlation between the entropic uncertainty relation and quantum entanglement? Or if the relation can be applied to estimate the entanglement. In this work, we focus on exploring the complementary relation between quantum entanglement and the entropic uncertainty relation. The results show that there exists an inequality relation between both of them for an arbitrary two-qubit system, and specifically the larger uncertainty will induce the weaker entanglement of the probed system, and vice versa. Besides, we use randomly generated states as illustrations to verify our results. Therefore, we claim that our observations might offer and support the validity of using the entropy uncertainty relation to estimate quantum entanglement.
Entropic uncertainty relations provide an information-theoretic framework for quantifying the fundamental indeterminacy inherent in quantum mechanics. We propose more stringent quantum-memory-assisted entropic uncertainty relations for complete sets of mutually unbiased bases in multipartite scenarios. We present lower and upper bounds of the quantum uncertainties based on the complementarity of the observables, the purity of the measured state, the (conditional) von-Neumann entropies, the Holevo quantities and mutual information. The results are illustrated by several representative cases, showing that our bounds are tighter than and outperform previously existing bounds.
We conjecture uncertainty relations that restrict correlations between the results of measurements performed by two separate parties on a shared quantum state. The first uncertainty relation bounds the sum of two mutual informations when one party measures a single observable and the other party measures one of two observables. The uncertainty relation does not follow from the Maassen-Uffink uncertainty relation and is much stronger than the Hall uncertainty relation derived from the latter. The second uncertainty relation bounds the sum of two mutual informations when each party measures one of two observables. We provide numerical evidence for the validity of the conjectured uncertainty relations, and we prove them for large classes of states and observables.
The CQC conjecture by Schneeloch et al. (Physical Review A 90.6, 2014) asserts that the sum of classical mutual information between two parties obtained by measuring individual systems in two mutually unbiased bases cannot exceed their shared quantum mutual information. This remarkable conjecture, which deserves more work, still remains open. In this article, using a result by Coles and Piani (Physical Review A 89.2, 2014), we derive a sufficient condition for which the CQC conjecture holds. We show that this condition validates the conjecture for a wider class of states compared to the states that were mentioned in the original work. Furthermore, we extend this conjecture to a higher number of mutually unbiased bases and prime dimensions. We prove this extended CQC conjecture on isotropic states of arbitrary prime dimensions and simulate it extensively for dimensions 3 and 5 for random bipartite states, observing no contradiction.
The complementary quantum correlations (CQC) relation bounds the sum of two classical mutual informations, obtained from local mutually unbiased measurements, by the quantum mutual information of the premeasurement state. We refute it. Separable rank-two counterexamples exist in every local dimension pair m×n with m,n≥3, with closed-form excess at least 1/(8m2n2) nats, and in every qubit--qudit pair 2×n with n≥3 except n=3,5; every covered pair also admits full-rank separable counterexamples. We then analyse the two residual qubit--qudit dimensions. A dimension-free entropy envelope replaces the natural quadratic majorant and lowers the requirement for closing the equal-prior orthogonal two-ray family from a triangular-discrimination bound S≤8/3 to S≤3.8265583…; the associated gate matrix has trace exactly two, so its spectral test collapses to a single eigenvalue-free scalar; and the exact identity X=1−4Var(c) turns the prime-Fourier full-spark barrier into a variance bound. A 128-bit interval cover then closes that family at 2×3 with gap at least 0.012021 nats, and at 2×5 an exact saturator attaining S=(14+25)/5 refutes three competing routes. We also give a state-dependent corrected inequality that is universal, and show it is incomparable with CQC already at 2×2. Finally we quantify why the original searches had essentially no power to find these states: the violating set is a sliver against the low-rank boundary, the witness lies 7.3 standard deviations below the Hilbert--Schmidt mean, the bases must be aligned to about nine degrees (∼10−21 of frames), and extrapolating the sample minimum demands 1010 to 1018 samples against the 107 ever run.