Let$U_{d}$be a unitary operator representing an arbitrary$d$-dimensional unitary quantum operation. This work presents optimal quantum circuits for transforming a number$k$of calls of$U_{d}$into its complex conjugate$\overline {U_{d}}$. Our circuits admit a parallel implementation and are proven to be optimal for any$k$and$d$with an average fidelity of$\left \langle{ {F}}\right \rangle =\frac {k+1}{d(d-k)}$. Optimality is shown for average fidelity, robustness to noise, and other standard figures of merit. This extends previous works which considered the scenario of a single call ($k=1$) of the operation$U_{d}$, and the special case of$k=d-1$calls. We then show that our results encompass optimal transformations from$k$calls of$U_{d}$to$f(U_{d})$for any arbitrary homomorphism$f$from the group of$d$-dimensional unitary operators to itself, since complex conjugation is the only non-trivial automorphism on the group of unitary operators. Finally, we apply our optimal complex conjugation implementation to design a probabilistic circuit for reversing arbitrary quantum evolutions.

Epsilon-nets and approximate unitary$t$-designs are natural notions that capture properties of unitary operations relevant for numerous applications in quantum information and quantum computing. In this work we study quantitative connections between these two notions. Specifically, we prove that, for$d$dimensional Hilbert space, unitaries constituting$\delta $-approximate$t$-expanders form$\epsilon $-nets for$t\simeq \frac {d^{5/2}}{ \epsilon }$and$\delta \simeq \left ({\frac { \epsilon ^{3/2}}{d}}\right)^{d^{2}}$. We also show that for arbitrary$t$,$\epsilon $-nets can be used to construct$\delta $-approximate unitary$t$-designs for$\delta \simeq \epsilon t$, where the notion of approximation is based on the diamond norm. Finally, we prove that the degree of an exact unitary$t$design necessary to obtain an$\epsilon $-net must grow at least as fast as$\frac {1}{ \epsilon }$(for fixed dimension) and not slower than$d^{2}$(for fixed$\epsilon $). This shows near optimality of our result connecting$t$-designs and$\epsilon $-nets. We apply our findings in the context of quantum computing. First, we show that that approximate t-designs can be generated by shallow random circuits formed from a set of universal two-qudit gates in the parallel and sequential local architectures considered in (Brandão et al., 2016). Importantly, our gate sets need not to be symmetric (i.e., contains gates together with their inverses) or consist of gates with algebraic entries. Second, we consider compilation of quantum gates and prove a non-constructive Solovay-Kitaev theorem for general universal gate sets. Our main technical contribution is a new construction of efficient polynomial approximations to the Dirac delta in the space of quantum channels, which can be of independent interest.

In this manuscript we analyse generalised port-based teleportation (PBT) schemes, allowing for transmitting more than one unknown quantum state (or a composite quantum state) in one go, where the state ends up in several ports at Bob’s side. We investigate the efficiency of our scheme discussing both deterministic and probabilistic case, where parties share maximally entangled states. It turns out that the new scheme gives better performance than various variants of the optimal PBT protocol used for the same task. All the results are presented in group-theoretic manner depending on such quantities like dimensions and multiplicities of irreducible representations in the Schur-Weyl duality. The presented analysis was possible by considering the algebra of permutation operators acting on$n$systems distorted by the action of partial transposition acting on more than one subsystem. Considering its action on the$n-$fold tensor product of the Hilbert space with finite dimension, we present construction of the respective irreducible matrix representations, which are in fact matrix irreducible representations of the Walled Brauer Algebra. I turns out that the introduced formalism, and symmetries beneath it, appears in many aspects of theoretical physics and mathematics - theory of anti ferromagnetism, aspects of gravity theory or in the problem of designing quantum circuits for special task like for example inverting an unknown unitary.

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this paper, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with$0 \leq \varepsilon < (2^{(1/12)} - 1)/(2(2^{(1/12)} + 1)) \approx 0.0144$). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source, for the parameter range$0 \leq \varepsilon <0.0132$, against a class of attacks given as a mixture of product box sequences, made of extremal no-signaling boxes, with additional symmetry conditions. Composable security proof against this class of attacks is provided.

In this paper, existence of bound entangled states with nonpositive partial transpose (NPT) is considered. As one knows, existence of such states would in particular imply nonadditivity of distillable entanglement. Moreover, it would rule out a simple mathematical description of the set of distillable states. The particular state, known to be 1-copy nondistillable and supposed to be bound entangled, is considered. The problem of its two-copy distillability, which boils down to show that maximal overlap of some projectorQwith Schmidt rank two states does not exceed 1/2 (called thehalf-property), is studied. First, it is shown that the maximum overlap can be attained on vectors that are not of the simple product form with respect to cut between two copies. Then, the problem in attacked twofold way: (a) the half-property is proved for some wide classes of Schmidt rank two states; (b) the overlap forallSchmidt rank two states is bounded from above bycA⊗I+I⊗BwithA,Btraceless 4 × 4 matrices, and TrAfA+ TrBfB=1/4.

In this paper, we develop a formalism for distilling a classical key from a quantum state in a systematic way, expanding on our previous work on a secure key from bound entanglement (Horodecki, 2005). More detailed proofs, discussion, and examples are provided of the main results. Namely, we demonstrate that all quantum cryptographic protocols can be recast in a way which looks like entanglement theory, with the only change being that instead of distilling Einstein–Podolsky–Rosen (EPR) pairs, the parties distill private states. The form of these general private states are given, and we show that there are a number of useful ways of expressing them. Some of the private states can be approximated by certain states, which are bound entangled. Thus, distillable entanglement is not a requirement for a private key. We find that such bound entangled states are useful for a cryptographic primitive we call a controlled private quantum channel (PQC). We also find a general class of states, which have negative partial transpose (are NPT), but which appear to be bound entangled. The relative entropy distance is shown to be an upper bound on the rate of a key. This allows us to compute theexactvalue of a distillable key for a certain class of private states.

New measures of multipartite entanglement are constructed based on two definitions of multipartite information and different methods of optimizing over extensions of the states. One is a generalization of the squashed entanglement where one takes the mutual information of parties conditioned on the state's extension and takes the infimum over such extensions. Additivity of the multipartite squashed entanglement is proved for both versions of the multipartite information which turn out to be related. The second one is based on taking classical extensions. This scheme is generalized, which enables to construct measures of entanglement based on the mixed convex roof of a quantity, which in contrast to the standard convex roof method involves optimization over all decompositions of a density matrix rather than just the decompositions into pure states. As one of the possible applications of these results we prove that any multipartite monotone is an upper bound on the amount of multipartite distillable key. The findings are finally related to analogous results in classical key agreement.

In this paper, we prove unconditional security for a quantum key distribution (QKD) protocol based on distilling pbits (twisted ebits) from an arbitrary untrusted state that is claimed to contain distillable key. Our main result is that we can verify security using only public communication-via parameter estimation of the given untrusted state. The technique applies even to bound-entangled states, thus extending QKD to the regime where the available quantum channel has zero quantum capacity. We also show how to convert our purification-based QKD schemes to prepare/measure schemes.

In this paper, we provide a class of bound entangled states that have positive distillable secure key rate. The smallest state of this kind is 4 otimes 4, which shows that peculiar security contained in bound entangled states does not need high-dimensional systems. We show that for these states a positive key rate can be obtained byone-wayDevetak-Winter (DW) protocol. Subsequently, the volume of bound entangled key-distillable states formotimesnHilbert space withm,n> 4 is shown to be nonzero. We provide a scheme of verification of cryptographic quality of experimentally prepared state in terms of local observables. Proposed set of seven collective settings is proven to be optimal in number of settings.