Private states, quantum data hiding and the swapping of perfect secrecy

We derive a formal connection between quantum data hiding and quantum privacy, confirming the intuition behind the construction of bound entangled states from which secret bits can be extracted. We present three main results. First, we show how to simplify the class of private states and related states via reversible local operation and one-way communication. Second, we obtain a bound on the one-way distillable entanglement of private states in terms of restricted relative entropy measures, which is tight in many cases and shows that protocols for one-way distillation of key out of states with low distillable entanglement lead to the distillation of data hiding states. Third, we consider the problem of extending the distance of quantum key distribution with help of intermediate stations. In analogy to the quantum repeater, this paradigm has been called the quantum key repeater. We show that when extending private states with one-way communication, the resulting rate is bounded by the one-way distillable entanglement. In order to swap perfect secrecy it is thus essentially optimal to use entanglement swapping.Comment: v3 published version, some details of the main proofs have been moved to the appendix, 21 pages. v2 claims changed from LOCC to one-way LOCC in the process of correcting a mistake found in v1 (in proof of Lemma 3). v1: 15 pages, 9 figure

Asymptotic entanglement transformation between W and GHZ states

We investigate entanglement transformations with stochastic local operations and classical communication (SLOCC) in an asymptotic setting using the concepts of degeneration and border rank of tensors from algebraic complexity theory. Results well-known in that field imply that GHZ states can be transformed into W states at rate 1 for any number of parties. As a generalization, we find that the asymptotic conversion rate from GHZ states to Dicke states is bounded as the number of subsystems increase and the number of excitations is fixed. By generalizing constructions of Coppersmith and Winograd and by using monotones introduced by Strassen we also compute the conversion rate from W to GHZ states.Comment: 11 page

Uncertainty, Monogamy, and Locking of Quantum Correlations

Squashed entanglement and entanglement of purification are quantum mechanical correlation measures and defined as certain minimisations of entropic quantities. We present the first non-trivial calculations of both quantities. Our results lead to the conclusion that both measures can drop by an arbitrary amount when only a single qubit of a local system is lost. This property is known as "locking" and has previously been observed for other correlation measures, such as the accessible information, entanglement cost and the logarithmic negativity. In the case of squashed entanglement, the results are obtained with the help of an inequality that can be understood as a quantum channel analogue of well-known entropic uncertainty relations. This inequality may prove a useful tool in quantum information theory. The regularised entanglement of purification is known to equal the entanglement needed to prepare a many copies of quantum state by local operations and a sublinear amount of communication. Here, monogamy of quantum entanglement (i.e., the impossibility of a system being maximally entangled with two others at the same time) leads to an exact calculation for all quantum states that are supported either on the symmetric or on the antisymmetric subspace of a dxd-dimensional system.Comment: 7 pages revtex4, no figures. v2 has improved presentation and a couple of references adde

Post-selection technique for quantum channels with applications to quantum cryptography

We propose a general method for studying properties of quantum channels acting on an n-partite system, whose action is invariant under permutations of the subsystems. Our main result is that, in order to prove that a certain property holds for any arbitrary input, it is sufficient to consider the special case where the input is a particular de Finetti-type state, i.e., a state which consists of n identical and independent copies of an (unknown) state on a single subsystem. A similar statement holds for more general channels which are covariant with respect to the action of an arbitrary finite or locally compact group. Our technique can be applied to the analysis of information-theoretic problems. For example, in quantum cryptography, we get a simple proof for the fact that security of a discrete-variable quantum key distribution protocol against collective attacks implies security of the protocol against the most general attacks. The resulting security bounds are tighter than previously known bounds obtained by proofs relying on the exponential de Finetti theorem [Renner, Nature Physics 3,645(2007)].Comment: 3.5 page

Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication

We study nondeterministic multiparty quantum communication with a quantum generalization of broadcasts. We show that, with number-in-hand classical inputs, the communication complexity of a Boolean function in this communication model equals the logarithm of the support rank of the corresponding tensor, whereas the approximation complexity in this model equals the logarithm of the border support rank. This characterisation allows us to prove a log-rank conjecture posed by Villagra et al. for nondeterministic multiparty quantum communication with message-passing. The support rank characterization of the communication model connects quantum communication complexity intimately to the theory of asymptotic entanglement transformation and algebraic complexity theory. In this context, we introduce the graphwise equality problem. For a cycle graph, the complexity of this communication problem is closely related to the complexity of the computational problem of multiplying matrices, or more precisely, it equals the logarithm of the asymptotic support rank of the iterated matrix multiplication tensor. We employ Strassen's laser method to show that asymptotically there exist nontrivial protocols for every odd-player cyclic equality problem. We exhibit an efficient protocol for the 5-player problem for small inputs, and we show how Young flattenings yield nontrivial complexity lower bounds

Entanglement distillation from Greenberger-Horne-Zeilinger shares

We study the problem of converting a product of Greenberger-Horne-Zeilinger (GHZ) states shared by subsets of several parties in an arbitrary way into GHZ states shared by every party. Our result is that if SLOCC transformations are allowed, then the best asymptotic rate is the minimum of bipartite log-ranks of the initial state. This generalizes a result by Strassen on the asymptotic subrank of the matrix multiplication tensor.Comment: 8 pages, v2: minor correction

Smooth Entropy Bounds on One-Shot Quantum State Redistribution

In quantum state redistribution as introduced in [Luo and Devetak (2009)] and [Devetak and Yard (2008)], there are four systems of interest: the $A$ system held by Alice, the $B$ system held by Bob, the $C$ system that is to be transmitted from Alice to Bob, and the $R$ system that holds a purification of the state in the $ABC$ registers. We give upper and lower bounds on the amount of quantum communication and entanglement required to perform the task of quantum state redistribution in a one-shot setting. Our bounds are in terms of the smooth conditional min- and max-entropy, and the smooth max-information. The protocol for the upper bound has a clear structure, building on the work [Oppenheim (2008)]: it decomposes the quantum state redistribution task into two simpler quantum state merging tasks by introducing a coherent relay. In the independent and identical (iid) asymptotic limit our bounds for the quantum communication cost converge to the quantum conditional mutual information $I(C:R|B)$, and our bounds for the total cost converge to the conditional entropy $H(C|B)$. This yields an alternative proof of optimality of these rates for quantum state redistribution in the iid asymptotic limit. In particular, we obtain a strong converse for quantum state redistribution, which even holds when allowing for feedback.Comment: v3: 29 pages, 1 figure, extended strong converse discussio

Asymptotic tensor rank of graph tensors: beyond matrix multiplication

We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on $k$ vertices. For $k\geq4$, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication ($k=3$), which is approximately 0.79. We raise the question whether for some $k$ the exponent per edge can be below $2/3$, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalise to higher order tensors a result by Strassen on the asymptotic subrank of tight tensors and a result by Coppersmith and Winograd on the asymptotic rank of matrix multiplication. Our results have applications in entanglement theory and communication complexity

A Generic Security Proof for Quantum Key Distribution

Quantum key distribution allows two parties, traditionally known as Alice and Bob, to establish a secure random cryptographic key if, firstly, they have access to a quantum communication channel, and secondly, they can exchange classical public messages which can be monitored but not altered by an eavesdropper, Eve. Quantum key distribution provides perfect security because, unlike its classical counterpart, it relies on the laws of physics rather than on ensuring that successful eavesdropping would require excessive computational effort. However, security proofs of quantum key distribution are not trivial and are usually restricted in their applicability to specific protocols. In contrast, we present a general and conceptually simple proof which can be applied to a number of different protocols. It relies on the fact that a cryptographic procedure called privacy amplification is equally secure when an adversary's memory for data storage is quantum rather than classical.Comment: Analysis of B92 protocol adde