We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard et al., (2005). For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

We prove a new concentration result for non-catalytic decoupling by showing that, for suitably large$t$, applying a unitary chosen uniformly at random from an approximate$t$-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitary$t$-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.

This paper is eligible for the Jack Reil Wolf ISIT Student Paper Award.We prove new one shot achievability results for measurement compression of quantum instruments with side information at the receiver. Unlike previous one shot results for this problem, our one shot bounds are nearly optimal and do not need catalytic randomness. In fact, we state a more general problem called centralised multi link measurement compression with quantum side information and provide one shot achievability results for it. As a simple corollary, we obtain one shot measurement compression results for quantum instruments with side information that we mentioned earlier. All our one shot results lead to the standard results for this problem in the asymptotic iid setting. We prove our achievability bounds by first proving a novel sequential classical quantum multipartite covering lemma, which should be of independent interest.

In a breakthrough, Hastings showed that there exist quantum channels whose classical Holevo capacity is superadditive i.e. more classical information can be transmitted by quantum encoding strategies entangled across multiple channel uses as compared to unentangled quantum encoding strategies. Hastings’ proof used Haar random unitaries to exhibit superadditivity. In this paper we show that a unitary chosen uniformly at random from an approximate$n^{2/3}$-design gives rise to a quantum channel with superadditive classical Holevo capacity, where$n$is the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We follow the geometric functional analytic approach of Aubrun, Szarek and Werner in order to prove our result. More precisely we prove a sharp Dvoretzky-like theorem stating that, with high probability under the choice of a unitary from an approximate$t$-design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low’s technique for proving concentration of measure for an approximate$t$-design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work. Haar random unitaries require at least$\Omega (n^{2})$random bits in order to describe them with good precision. In contrast, there exist exact$n^{2/3}$-designs using only$O(n^{2/3} \log n)$random bits. Thus, our work can be viewed as a partial derandomisation of Hastings’ result, and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity. Finally we also show that for any$p > 1$, approximate unitary$n^{1.7}$-designs give rise to channels violating subadditivity of Rényi$p$-entropy. In addition to stratified analysis, the proof of this result uses a new technique of approximating a monotonic differentiable function defined on a closed bounded interval and its derivative by moderate degree polynomials which should be of independent interest.

In this work, we prove a novel one-shot ‘multi-sender’ decoupling theorem generalising Dupuis' seminal single sender decoupling theorem. We start off with a multipartite quantum state, say on$A_{1}A_{2}R$, where$A_{1}, A_{2}$are treated as the two ‘sender’ systems and$R$is the reference system. We apply independent Haar random unitaries in tensor product on$A_{1}$and$A_{2}$and then send the resulting systems through a quantum channel. We want the channel output$B$to be almost in tensor with the untouched reference$R$. Our main result shows that this is indeed the case if suitable entropic conditions are met. An immediate application of our main result is to obtain a one-shot simultaneous decoder for sending quantum information over a$k$-sender entanglement unassisted quantum multiple access channel (QMAC). The rate region achieved by this decoder is the natural one-shot quantum analogue of the pentagonal classical rate region. Assuming a simultaneous smoothing conjecture, this one-shot rate region approaches the optimal rate region of Yard et al. [20] in the asymptotic iid limit. Our work is the first one to obtain a non-trivial simultaneous decoder for the QMAC with limited entanglement assistance in both one-shot and asymptotic iid settings; previous works used unlimited entanglement assistance.

We prove the first non-trivial one-shot inner bounds for sending quantum information over an entanglement unassisted two-sender quantum multiple access channel (QMAC) and an unassisted two-sender two-receiver quantum interference channel (QIC). Previous works only studied the unassisted QMAC in the limit of many independent and identical uses of the channel also known as the asymptotic iid limit, and did not study the unassisted QIC at all. We employ two techniques, rate splitting and successive cancellation, in order to obtain our inner bound. Rate splitting was earlier used to obtain inner bounds, avoiding time sharing, for classical channels in the asymptotic iid setting. Our main technical contribution is to extend rate splitting from the classical asymptotic iid setting to the quantum one-shot setting. In the asymptotic iid limit our one-shot inner bound for QMAC approaches the rate region of Yard et al. [22]. For the QIC we get novel non-trivial rate regions in the asymptotic iid setting. All our results also extend to the case where limited entanglement assistance is provided, in both one-shot and asymptotic iid settings. The limited entanglement results for one-shot setting for both QMAC and QIC are new. For the QIC the limited entanglement results are new even in the asymptotic iid setting.

We provide the first inner bounds for sending private classical information over a quantum multiple access channel. We do so by using three powerful information theoretic techniques: rate splitting, quantum simultaneous decoding for multiple access channels, and a novel smoothed distributed covering lemma for classical quantum channels. Our inner bounds are given in the one shot setting and accordingly the three techniques used are all very recent ones specifically designed to work in this setting. The last technique is new to this work and is our main technical advancement. For the asymptotic iid setting, our one shot inner bounds lead to the natural quantum analogue of the best classical inner bounds for this problem. A full version of this paper is accessible at [5].

We consider the problem of communication over a classical-quantum broadcast channel with one sender and two receivers. Generalizing the classical inner bounds shown by Marton and the recent quantum asymptotic version shown by Savov and Wilde, we obtain one-shot inner bounds in the quantum setting. Our bounds are stated in terms of hypothesis testing and one-shot max divergences. These results give a full justification of the claims of Savov and Wilde in the classical-quantum asymptotic iid setting; the techniques also yield similar bounds in the information spectrum setting. We obtain these results using a different analysis of the random codebook argument; our method yields a classical one-shot Marton bound with a common message and a classical one-shot mutual covering lemma based on rejection sampling.

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups, including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in $\mathbb{Z}_p^n$, whenever $p$ is a fixed prime. For the induction step, we introduce the problem Translating Coset generalizing both Hidden Translation and Hidden Subgroup and prove a powerful self-reducibility result: Translating Coset in a finite solvable group $G$ is reducible to instances of Translating Coset in $G/N$ and $N$, for appropriate normal subgroups $N$ of $G$. Our self-reducibility framework, combined with Kuperberg's subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.

The existence of quantum uncertainty relations is the essential reason that some classically unrealizable cryptographic primitives become realizable when quantum communication is allowed. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking [DiVincenzo et al. 2004]. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n -bit message is encoded in a quantum system using a classical key of size much smaller than n . Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message, in which case the message is said to be “locked”. Furthermore, knowing the key, it is possible to recover, that is “unlock”, the message. In this article, we make the following contributions by exploiting a connection between uncertainty relations and low-distortion embeddings of Euclidean spaces into slightly larger spaces endowed with the ℓ 1 norm. We introduce the notion of a metric uncertainty relation and connect it to low-distortion embeddings of ℓ 2 into ℓ 1 . A metric uncertainty relation also implies an entropic uncertainty relation. We prove that random bases satisfy uncertainty relations with a stronger definition and better parameters than previously known. Our proof is also considerably simpler than earlier proofs. We then apply this result to show the existence of locking schemes with key size independent of the message length. Moreover, we give efficient constructions of bases satisfying metric uncertainty relations. The bases defining these metric uncertainty relations are computable by quantum circuits of almost linear size. This leads to the first explicit construction of a strong information locking scheme. These constructions are obtained by adapting an explicit norm embedding due to Indyk [2007] and an extractor construction of Guruswami et al. [2009]. We apply our metric uncertainty relations to exhibit communication protocols that perform equality testing of n -qubit states. We prove that this task can be performed by a single message protocol using O (log 2 n ) qubits and n bits of communication, where the computation of the sender is efficient.

We construct an encoding and decoding scheme achieving the Chong-Motani-Garg inner bound [1] for a two sender two receiver interference channel with classical input and quantum output. This automatically gives a similar inner bound for sending classical information through an interference channel with quantum inputs and outputs without entanglement assistance. Our result matches the best known inner bound for the interference channel in the classical setting. Achieving the Chong-Motani-Garg inner bound, which is known to be equivalent to the Han-Kobayashi inner bound [3], answers an open question raised recently by Fawzi et al. [4]. Our encoding strategy is the standard random encoding strategy. Our decoding strategy is a sequential strategy where a receiver loops through all candidate messages trying to project the received state onto a `typical' subspace for the candidate message under consideration, stopping if the projection succeeds for a message, which is then declared as the guess of the receiver for the sent message. On the way to our main result, we show that random encoding and sequential decoding strategies suffice to achieve rates up to the mutual information for a single sender single receiver channel, and the standard inner bound for a two sender single receiver multiple access channel, for channels with classical input and quantum output. Besides conceptual simplicity, a sequential decoding strategy is space efficient, and may have additional efficiency advantages in some settings. We prove our inner bounds using two new technical tools - a non-commutative union bound to analyse the decoding error probability, and a geometric notion of approximate interesection of two conditionally typical subspaces.

Calculating the capacity of interference channels is a notorious open problem in classical information theory. Such channels have two senders and two receivers, and each sender would like to communicate with a partner receiver. The capacity of such channels is known exactly in the settings of “very strong” and “strong” interference, while the Han-Kobayashi coding strategy gives the best known achievable rate region in the general case. Here, we introduce and study the quantum interference channel, a natural generalization of the interference channel to the setting of quantum information theory. We restrict ourselves for the most part to channels with two classical inputs and two quantum outputs in order to simplify the presentation of our results (though generalizations of our results to channels with quantum inputs are straightforward). We are able to determine the exact classical capacity of this channel in the settings of “very strong” and “strong” interference, by exploiting Winter's successive decoding strategy and a novel two-sender quantum simultaneous decoder, respectively. We provide a proof that a Han-Kobayashi strategy is achievable with Holevo information rates, up to a conjecture regarding the existence of a three-sender quantum simultaneous decoder. This conjecture holds for a special class of quantum multiple-access channels with average output states that commute, and we discuss some other variations of the conjecture that hold. Finally, we detail a connection between the quantum interference channel and prior work on the capacity of bipartite unitary gates.

Quantum uncertainty relations are at the heart of many quantum cryptographic protocols performing classically impossible tasks. One operational manifestation of these uncertainty relations is a purely quantum effect referred to as information locking. A locking scheme can be viewed as a cryptographic protocol in which a uniformly random n-bit message is encoded in a quantum system using a classical key of size much smaller than n. Without the key, no measurement of this quantum state can extract more than a negligible amount of information about the message (the message is "locked"). Furthermore, knowing the key, it is possible to recover (or "unlock") the message.

It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore et al. [2005] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω( n log n ) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because in general it seems hard to implement a given highly entangled measurement. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL( n ,F p m ) and G n where G is finite and satisfies a suitable property.

We construct efficient or query efficient quantum property testers for two existential group properties which have exponential query complexity both for their decision problem in the quantum and for their testing problem in the classical model of computing. These are periodicity in groups and the common coset range property of two functions having identical ranges within each coset of some normal subgroup. Our periodicity tester is efficient in Abelian groups and generalizes, in several aspects, previous periodicity testers. This is achieved by introducing a technique refining the majority correction process widely used for proving robustness of algebraic properties. The periodicity tester in non-Abelian groups and the common coset range tester are query efficient.

We prove the following information-theoretic property about quantum states. Substate theorem: Let ρ and σ be quantum states in the same Hilbert space with relative entropy S (ρ ‖ σ) ≔ Tr ρ (log ρ - log σ) = c . Then for all ϵ > 0, there is a state ρ′ such that the trace distance ‖ρ′ - ρ‖ tr : Tr √(ρ′ - ρ) 2 ≤ ϵ, and ρ′/2 O ( c /ϵ 2 ) ≤ σ. It states that if the relative entropy of ρ and σ is small, then there is a state ρ′ close to ρ, i.e. with small trace distance ‖ρ′ - ρ‖ tr , that when scaled down by a factor 2 O ( c ) ‘sits inside’, or becomes a ‘substate’ of, σ. This result has several applications in quantum communication complexity and cryptography. Using the substate theorem, we derive a privacy trade-off for the set membership problem in the two-party quantum communication model. Here Alice is given a subset A ⊆ [ n ], Bob an input i ∈ [ n ], and they need to determine if i ∈ A . Privacy trade-off for set membership: In any two-party quantum communication protocol for the set membership problem, if Bob reveals only k bits of information about his input, then Alice must reveal at least n /2 O( k ) bits of information about her input. We also discuss relationships between various information theoretic quantities that arise naturally in the context of the substate theorem.

We show that measuring any two low rank quantum states in a random orthonormal basis gives, with high probability, two probability distributions having total variation distance at least a universal constant times the Frobenius distance between the two states. This implies that for any finite ensemble of quantum states there is a single POVM that distinguishes between every pair of states from the ensemble by at least a constant times their Frobenius distance; in fact, with high probability a random POVM, under a suitable definition of randomness, suffices. There are examples of ensembles with constant pairwise trace distance where a single POVM cannot distinguish pairs of states by much better than their Frobenius distance, including the important ensemble of coset states of hidden subgroups of the symmetric group (Moore at al., 2005). We next consider the random Fourier method for the hidden subgroup problem (HSP) which consists of Fourier sampling the coset state of the hidden subgroup using random orthonormal bases for the group representations. In cases where every representation of the group has polynomially bounded rank when averaged over the hidden subgroup, the random Fourier method gives a POVM for the HSP operating on one coset state at a time and using totally a polynomial number of coset states. In particular, we get such POVMs whenever the group and the hidden subgroup form a Gel'fand pair, e.g., Abelian, dihedral and Heisenberg groups. This gives a positive counterpart to earlier negative results about random Fourier sampling when the above rank is exponentially large (Grigni et al., 2004), which happens for example in the HSP in the symmetric group. The drawback of random POVMs is that they are not efficient to implement, since measuring in a random basis takes exponential time as can be seen by a counting argument. This leads us to the open question of efficiently implementable pseudorandom measurement bases

It has been known for some time that graph isomorphism reduces to the hidden subgroup problem (HSP). What is more, most exponential speedups in quantum computation are obtained by solving instances of the HSP. A common feature of the resulting algorithms is the use of quantum coset states, which encode the hidden subgroup. An open question has been how hard it is to use these states to solve graph isomorphism. It was recently shown by Moore, Russell, and Schulman [30] that only an exponentially small amount of information is available from one, or a pair of coset states. A potential source of power to exploit are entangled quantum measurements that act jointly on many states at once. We show that entangled quantum measurements on at least Ω(n log n) coset states are necessary to get useful information for the case of graph isomorphism, matching an information theoretic upper bound. This may be viewed as a negative result because highly entangled measurements seem hard to implement in general. Our main theorem is very general and also rules out using joint measurements on few coset states for some other groups, such as GL(n,Fpm) and Gn where G is finite and satisfies a suitable property.

Consider a two-party quantum communication protocol for computing some function f : {0, 1}/sup n/ /spl times/ {0, 1}/sup n/ /spl rarr/ Z. We show that the first message of P can be compressed to 0(k) classical bits using prior entanglement if it carries at most k bits of information about the sender's input. This implies a general direct sum result for one-round and simultaneous quantum protocols. It also implies a new round elimination lemma in quantum communication, which allows us to extend recent classical lower bounds on the cell probe complexity of some data structure problems, e.g. approximate nearest neighbor searching on the Hamming cube {0, 1}/sup n/, to the quantum setting. We then show an optimal tradeoff between the privacy losses of Alice and Bob in computing f in terms of the one-round quantum communication complexity of f with prior entanglement. This tradeoff is independent of the number of rounds of communication. The above message compression and privacy tradeoff results use a lot of qubits of prior entanglement, leading one to wonder how much prior entanglement is really required by a quantum protocol. We show that Newman's [1991] technique of reducing the number of public coins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of black-box reduction of prior entanglement that subsumes Newman's technique. Intuitively, a black-box reduction does not change the unitary transforms of Alice and Bob; it only decreases the amount of entanglement of the prior entangled state. We prove that such a black-box reduction is impossible for quantum protocols by exhibiting a particular one-round quantum protocol for the equality function where the black-box technique fails to reduce the amount of prior entanglement by more than a constant factor.

We consider a fundamental problem in data structures, static predecessor searching: Given a subset S of size n from the universe [m], store S so that queries of the form "What is the predecessor of x in S?" can be answered efficiently. We study this problem in the cell probe model introduced by Yao [1981]. Recently, Beame and Fich [2002] obtained optimal bounds on the number of probes needed by any deterministic query scheme if the associated storage scheme uses only n/sup O(1)/ cells of word size (log m)/sup O(1)/ bits. We give a new lower bound proof for this problem that matches the bounds of Beame and Fich. Our lower bound proof has the following advantages: it works for randomised query schemes too, while Beame and Fich's proof works for deterministic query schemes only. In addition, it is simpler than Beame and Fich's proof. We prove our lower bound using the round elimination approach of Miltersen, Nisan, Safra and Wigderson [1998]. Using tools from information theory, we prove a strong round elimination lemma for communication complexity that enables us to obtain a tight lower bound for the predecessor problem. We also use our round elimination lemma to obtain a rounds versus communication tradeoff for the 'greater-than' problem, improving on the tradeoff in [1998]. We believe that our round elimination lemma is of independent interest and should have other applications.

We show lower bounds in the multi-party quantum communication complexity model. In this model, there are t parties where the ith party has input X/sub i/ /spl sube/ [n]. These parties communicate with each other by transmitting qubits to determine with high probability the value of some function F of their combined input (X/sub 1/,...,X/sub t/). We consider the class of Boolean valued functions whose value depends only on X/sub 1/ /spl cap/.../spl cap/ X/sub t/; that is, for each F in this class there is an f/sub F/ : 2/sup [n]/ /spl rarr/ {0,1}, such that F(X/sub 1/,...,X/sub t/) = f/sub F/(X/sub 1/ /spl cap/.../spl cap/ X/sub t/). We show that the t-party k-round communication complexity of F is /spl Omega/(s/sub m/(f/sub F/)/(k/sup 2/)), where s/sub m/(f/sub F/) stands for the monotone sensitivity of f/sub F/' and is defined by s/sub m/(f/sub F/) = /sup /spl utri// max/sub S/spl sube//[n] |{i : f/sub F/(S /spl cup/ {i}) /spl ne/ f/sub F/(S)}|. For two-party quantum communication protocols for the set disjointness problem, this implies that the two parties must exchange /spl Omega/(n/k/sup 2/) qubits. An upper bound of O(n/k) can be derived from the O(/spl radic/n) upper bound due to S. Aaronson and A. Ambainis (2003). For k = 1, our lower bound matches the /spl Omega/(n) lower bound observed by H. Buhrman and R. de Wolf (2001) (based on a result of A. Nayak (1999)), and for 2 /spl les/ k /spl Lt/ n/sup 1/4 /, improves the lower bound of /spl Omega/(/spl radic/n) shown by A. Razborov (2002). For protocols with no restrictions on the number of rounds, we can conclude that the two parties must exchange /spl Omega/(n/sup 1/3/) qubits. This, however, falls short of the optimal /spl Omega/ (/spl radic/n) lower bound shown by A. Razborov (2002). Our result is obtained by adapting to the quantum setting the elegant information-theoretic arguments of Z. Bar-Yossef et al. (2002). Using this method we can show similar lower bounds for the L/sub /spl infin// function considered in Z. Bar-Yossef et al. (2002).

We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of non-abelian groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z pn, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful self-reducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G.

We prove a fundamental theorem about the relative entropy of quantum states, which roughly states that if the relative entropy, S(/spl rho//spl par//spl sigma/)/spl Delta/=Tr /spl rho/(log /spl rho/-log /spl sigma/), of two quantum states /spl rho/ and /spl sigma/ is at most c, then /spl rho//2/sup O(c)/ 'sits inside' /spl sigma/. Using this 'substate' theorem, we give tight lower bounds for the privacy loss of bounded error quantum communication protocols for the index function problem. We also use the 'substate' theorem to give tight lower bounds for the k-round bounded error quantum communication complexity of the pointer chasing problem, when the wrong player starts, and all the log n bits of the kth pointer are desired.

Studies the quantum complexity of the static set membership problem: given a subset S (|S|/spl les/n) of a universe of size m(/spl Gt/n), store it as a table, T:(0,1)/sup r//spl rarr/(0,1), of bits so that queries of the form 'is x in S?' can be answered. The goal is to use a small table and yet answer queries using a few bit probes. This problem was considered by H. Buhrman et al. (2000), who showed lower and upper bounds for this problem in the classical deterministic and randomised models. In this paper, we formulate this problem in the "quantum bit-probe model". We assume that access to the table T is provided by means of a black-box (oracle) unitary transform O/sub T/ that takes the basis state (y,b) to the basis state |y,b/spl oplus/T(y)>. The query algorithm is allowed to apply O/sub T/ on any superposition of basis states. We show tradeoff results between the space (defined as 2/sup r/) and the number of probes (oracle calls) in this model. Our results show that the lower bounds shown by Buhrman et al. for the classical model also hold (with minor differences) in the quantum bit-probe model. These bounds almost match the classical upper bounds. Our lower bounds are proved using linear algebraic arguments.