We demonstrate that there exists a universal, near-optimal recovery map-the transpose channel-for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement.

Uncertainty relations are often considered to be a measure of incompatibility of noncommuting observables. However, such a consideration is not valid in general, motivating the need for an alternate measure that applies to any set of noncommuting observables. We present an operational approach to quantifying incompatibility without invoking uncertainty relations. Our measure aims to capture the incompatibility of noncommuting observables as manifest in the nonorthogonality of their eigenstates. We prove that this measure has all the desired properties. It is zero when the observables commute, strictly greater than zero when they do not, and is maximum when they are mutually unbiased. We also obtain tight upper bounds on this measure for any N noncommuting observables and compute it exactly when the observables are mutually unbiased.In quantum theory, any observable or a set of commuting observables can in principle be measured with any desired precision. This is because commuting observables have a complete set of simultaneous eigenkets, and therefore, measurement of one does not disturb the measurement result obtained for the other. This no longer holds when the observables do not commute. Noncommuting observables do not have a complete set of common eigenkets, and therefore it is impossible to specify definite values simultaneously. This is the essence of the celebrated uncertainty principle [1,2, 5]. Uncertainty relations [1-4, 6-16, 20] express the uncertainty principle in a quantitative way by providing a lower bound on the "uncertainty" in the result of a simultaneous measurement of noncommuting observables.Observables are defined to be compatible when they commute, and incompatible when they do not. The uncertainty principle, therefore, is a manifestation of the incompatibility of noncommuting observables. Despite the conceptual importance of incompatible observables and applications of such observables in quantum state determination [34][35][36] and quantum cryptography [26,27,[29][30][31], there does not seem to be a good general measure of their incompatibility, although entropic uncertainty relations have often been considered for this purpose (see, for example, [15,[17][18][19]).To see in what sense uncertainty relations quantify incompatibility of noncommuting observables, consider, for example, the entropic uncertainty relation due to Maassen and Uffink [4]. For any quantum state ρ ∈ H with dim H = d, and measurement of any two observables A and B with eigenvectors {|a i } and {|b i }, respectively, it was shown that [4]where c = max | a|b |: |a ∈ {|a i } , |b ∈ {|b i }, andB} is the Shannon entropy (all logarithms are taken to base 2). Observe that the right-hand side of the above inequality is independent of ρ. The incompatibility of the observables A and B can be measured by either the sum of the entropies [left hand side of (1)] minimized over all ρ (if it is not known whether equality is achieved) or the lower bound when equality is achieved for some state. We then say that a set of observable...

It is demonstrated here that local dynamics have the ability to strongly modify the entangling power of unitary quantum gates acting on a composite system. The scenario is common to numerous physical systems, in which the time evolution involves local operators and nonlocal interactions. To distinguish between distinct classes of gates with zero entangling power we introduce a complementary quantity called gate-typicality and study its properties. Analyzing multiple applications of any entangling operator interlaced with random local gates, we prove that both investigated quantities approach their asymptotic values in a simple exponential form. This rapid convergence to equilibrium, valid for subsystems of arbitrary size, is illustrated by studying multiple actions of diagonal unitary gates and controlled unitary gates.Introduction: The uniquely nonclassical phenomenon of entanglement is a well-known resource for quantum information [1]. It is increasingly used to characterize complex states, from many-body ground states [2], to infinite temperature quantum phase transitions such as the ergodic to localized phase in strongly interacting many-body systems [3]. Simple coupled quantum chaotic models have been studied [4,5] and experimentally realized [6,7] to demonstrate the large entanglement growth wherein the subsystems are nearly maximally mixed. In general the dynamics of entanglement in a non-equilibrium context is responsible for thermalization [8]. More recently in a different setting, black-holes are conjectured to scramble quantum information in a time that is logarithmic in the entropy via entanglement [9].While much work has centered on properties of states, quantum operators have also been studied as a physical resource for creating entanglement [10][11][12][13][14][15]. Studying directly the operators, such as propagators in time, frees us from the arbitrariness of initial states or the choice of eigenvectors. The entangling power [10,16] while referring to an inherent property of operators on a bipartite composite system is also related to how much state entanglement can be created, on the average, using one application of the unitary on product states. Investigations on quantum transport in light harvesting complexes [17], quantum chaos [18] and thermalization [19]

We provide a construction of sets of $d/2+1$ mutually unbiased bases (MUBs) in dimensions $d=4,8$ using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. Moreover, specific examples of sets of MUBs obtained using our construction are shown to be {\it strongly unextendible}; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions $d=2^{n} (n>3) $ as well.} {Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a $H_{2}$ entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in $d=4$.