It is known that general convertibility of bipartite entangled states is not possible to arbitrary error without some classical communication. While some tradeoffs between communication cost and conversion error have been proven, these bounds can be very loose. In particular, there are many cases in which tolerable error might be achievable using zero-communication protocols. In this work we address these cases by deriving the optimal fidelity of pure quantum state conversions under local unitaries as well as local operations and shared randomness. We also use these results to explore catalytic conversions between pure states using zero communication.
It is known that general convertibility of bipartite entangled states is not possible to arbitrary error without some classical communication. While some tradeoffs between communication cost and conversion error have been proven, these bounds can be very loose. In particular, there are many cases in which tolerable error might be achievable using zero-communication protocols. In this work we address these cases by deriving the optimal fidelity of pure quantum state conversions under local unitaries as well as local operations and shared randomness. We also use these results to explore catalytic conversions between pure states using zero communication.
We consider the problem of deciding if some multiparty entangled pure state can be converted, with a nonzero success probability, into a given bipartite pure state shared between two specified parties through local quantum operations and classical communication. We show that this question is equivalent to the well-known computational problem of deciding if a multivariate polynomial is identically zero. Efficient randomized algorithms developed to study the latter can thus be applied to our question. As a result, a given transformation is possible if and only if it is generically attainable by a simple randomized protocol.
Entanglement assistance is known to reduce the quantum communication complexity of evaluating functions with distributed inputs. But does the type of entanglement matter, or are EPR pairs always sufficient? This is a natural question because in several other settings maximally entangled states are known to be less useful as a resource than some partially entangled state. These include non-local games, tasks with quantum communication between players and referee, and simulating bipartite unitaries or communication channels. By contrast, we prove that the bounded-error entanglement-assisted quantum communication complexity of a function cannot be improved by more than a constant factor by replacing maximally entangled states with arbitrary entangled states. In particular, we show that every quantum communication protocol using Q qubits of communication and arbitrary shared entanglement can be ε-approximated by a protocol using O(Q/ε+log(1/ε)/ε) qubits of communication and only EPR pairs as shared entanglement. Our second result concerns an old question in quantum information theory: How much quantum communication is required to approximately convert one pure bipartite entangled state into another? We show that the communication cost of converting between two bipartite quantum states is upper bounded, up to a constant multiplicative factor, by a natural and efficiently computable quantity which we call the ℓ∞-Earth Mover's Distance (EMD) between those two states. Furthermore, we prove a complementary lower bound on the cost of state conversion by the ε-smoothed ℓ∞-EMD, which is a natural smoothing of the ℓ∞-EMD that we will define via a connection with optimal transport theory.
We prove continuity of distillable entanglement, the optimal asymptotic rate at which copies of a bipartite state can be converted into maximally entangled pairs by local operations and classical communication (LOCC) with vanishing error. We establish an explicit asymptotic continuity bound that holds for arbitrary bipartite states and depends logarithmically on the system dimension. This bound has the same structure as known continuity bounds for conditional entropy, entanglement cost, and relative entropy of entanglement. Our proof constructs an explicit LOCC protocol that converts either of two arbitrary bipartite states into the other at an ebit cost that vanishes as their trace distance tends to zero.
Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by local operations and classical communication (LOCC) when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies that an error-free reversible conversion is asymptotically possible even between states whose copies can never be locally unitarily equivalent with any finite numbers of copies, although such a conversion is impossible in the finite setting. In fact, we show such an example. Moreover, we establish how to overcome the irreversibility of LOCC conversion in two ways. As for the first method, we evaluate how many copies of the initial state are to be lost to overcome the irreversibility of LOCC conversion. The second method is to add a supplementary state appropriately, which also works for local unitary conversion unlike the first method. Especially, for the qubit system, any nonmaximally pure entangled state can be a universal resource for the asymptotic reversibility when copies of the state are sufficiently many. More interestingly, our analysis implies that far-from-maximally entangled states can be better than nearly maximally entangled states as this type of resource. This fact brings new insight to the resource theory of state conversion.
Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.
The phenomenon of quantum entanglement is a fundamental feature of quantum mechanics which, as a counterintuitive and inherently ”quantum” phenomenon (with no classical analogue) has been the subject of much study, especially in quantum information theory. One fruitful approach to the description of entanglement has been in its operational description - that is, in the consideration of what can be achieved using entangled states under certain restrictions, typically the regime of local operations and classical communications.\n\nWe present results here related to the operational characterisation of entanglement in the resource model, in both bipartite and multipartite cases. First, we consider the conversion between pure bipartite entangled states in terms of an often-ignored resource - the classical communication cost. Using prior results for more specific conversions, we derive lower bounds on this cost (and the related quantity of the conversion inefficiency) for general bipartite pure states.\n\nWe also consider pure-state conversions of multipartite entanglement, in particular the class of protocols in which multipartite states are converted to states shared between fewer parties. We have found a previously-unconsidered variety of such conversions, in which the target state of the conversion is a state shared between a random subset of the parties. We find that when such post-selection of parties in the protocol is permitted allows for a wider variety of achievable target states; certain states which can not be reliably obtained between predetermined parties (even some where the probability of doing so is arbitrarily small) can be obtained between random parties. We consider a variety of states in which this phenomenon occurs, as well as bounds on such protocols can achieve.\n\nFinally we consider a practical use of entanglement as a resource, in an experimental implementation of a multipartite QKD protocol. This is based on the tripartite GHZ entangled state, but can be implemented using only bipartite entanglement. We adapt existing QKD results for both the bipartite and multipartite case to derive a secure key rate for this implementation, taking into account the ways in which it differs from the idealised theoretical case.