Entanglement Cost Research Articles

Secure Key from Bound Entanglement

We characterize the set of shared quantum states which contain a cryptographically private key. This allows us to recast the theory of privacy as a paradigm closely related to that used in entanglement manipulation. It is shown that one can distill an arbitrarily secure key from bound entangled states. There are also states that have less distillable private keys than the entanglement cost of the state. In general, the amount of distillable key is bounded from above by the relative entropy of entanglement. Relationships between distillability and distinguishability are found for a class of states which have Bell states correlated to separable hiding states. We also describe a technique for finding states exhibiting irreversibility in entanglement distillation.

Uncertainty, Monogamy, and Locking of Quantum Correlations

Squashed entanglement and entanglement of purification are quantum-mechanical correlation measures and are defined as certain minimizations of entropic quantities. In this paper, we present the first nontrivial 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 accessible information, entanglement cost, and logarithmic negativity. In the case of squashed entanglement, the results are obtained using 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 regularized entanglement of purification is known to equal the entanglement needed to prepare many copies of a 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 d/spl times/d-dimensional system.

Classical information capacity of a class of quantum channels

We consider the additivity of the minimal output entropy and the classical information capacity of a class of quantum channels. For this class of channels, the norm of the output is maximized for the output being a normalized projection. We prove the additivity of the minimal output Renyi entropies with entropic parameters α ∊ [0, 2], generalizing an argument by Alicki and Fannes, and present a number of examples in detail. In order to relate these results to the classical information capacity, we introduce a weak form of covariance of a channel. We then identify various instances of weakly covariant channels for which we can infer the additivity of the classical information capacity. Both additivity results apply to the case of an arbitrary number of different channels. Finally, we relate the obtained results to instances of bi-partite quantum states for which the entanglement cost can be calculated.

Searching for highly entangled multi-qubit states

We present a simple numerical optimization procedure to search for highly entangled states of 2, 3, 4 and 5 qubits. We develop a computationally tractable entanglement measure based on the negative partial transpose criterion, which can be applied to quantum systems of an arbitrary number of qubits. The search algorithm attempts to optimize this entanglement cost function to find the maximal entanglement in a quantum system. We present highly entangled 4-qubit and 5-qubit states discovered by this search. We show that the 4-qubit state is not quite as entangled, according to two separate measures, as the conjectured maximally entangled Higuchi–Sudbery state. Using this measure, these states are more highly entangled than the 4-qubit and 5-qubit GHZ states. We also present a conjecture about the NPT measure, inspired by some of our numerical results, that the single-qubit reduced states of maximally entangled states are all totally mixed.

Dual entanglement measures based on no local cloning and no local deleting

Impossibility of cloning and deleting of unknown states are important restrictions on processing of information in the quantum world. On the other hand, a known quantum state can always be cloned or deleted. However if we restrict the class of allowed operations, there will arise restrictions on the ability of cloning and deleting machines. We have shown that cloning and deleting of known states is in general not possible by local operations. This impossibility hints at quantum correlation in the state. We propose dual measures of quantum correlation based on the dual restrictions of no local cloning and no local deleting. The measures are relative entropy distances of the desired states in a (generally impossible) perfect local cloning or local deleting process from the best approximate state that is actually obtained by imperfect local cloning or deleting machines. Just like the dual measures of entanglement cost and distillable entanglement, the proposed measures are based on important processes in quantum information. We discuss their properties. For the case of pure states, estimations of these two measures are also provided. Interestingly, the entanglement of cloning for a maximally entangled state of two two-level systems is not unity.

Ordering two-qubit states with concurrence and negativity

We study the ordering of two-qubit states with respect to the degree of bipartite entanglement using the Wootters concurrence -- a measure of the entanglement of formation, and the negativity -- a measure of the entanglement cost under the positive-partial-transpose-preserving operations. For two-qubit pure states, the negativity is the same as the concurrence. However, we demonstrate analytically on simple examples of various mixtures of Bell and separable states that the entanglement measures can impose different orderings on the states. We show which states, in general, give the maximally different predictions, (i) when one of the states has the concurrence greater but the negativity smaller than those for the other state, and (ii) when the states are entangled to the same degree according to one of the measures, but differently according to the other.

Decoherence of two maximally entangled qubits in a lossy nonlinear cavity

Decoherence effect on quantum entanglement of two optical qubits in a lossy cavity interacting with a nonlinear medium (Kerr nonlinearity) is analyzed. The qubits are assumed to be initially in the maximally entangled states (Bell or Bell-like states) or the maximally entangled mixed states, on the example of Werner and Werner-like states. Two kinds of measures of the entanglement are considered: the concurrence to describe a decay of the entanglement of formation of the qubits, and the negativity to determine a decay of the entanglement cost under positive-partial-transpose-preserving operations. It is observed that the Kerr nonlinearity, in the discussed decoherence model, does not affect the entanglement of the qubits initially in the Bell or Werner states, although the evolution of the qubits can depend on this nonlinearity explicitly. However, it is shown that for the initial Bell-like state and the corresponding Werner-like state, the loss of the entanglement can be periodically reduced by inserting the Kerr nonlinearity in the lossy cavity. Moreover, the relativity of the entanglement measures is demonstrated, to our knowledge for the first time, as a result of a physical process.

Irreversibility of entanglement distillation for a class of symmetric states

We investigate the irreversibility of entanglement distillation for a symmetric d-1 parameter family of mixed bipartite quantum states acting on Hilbert spaces of arbitrary dimension d x d. We prove that in this family the entanglement cost is generically strictly larger than the distillable entanglement, such that the set of states for which the distillation process is asymptotically reversible is of measure zero. This remains true even if the distillation process is catalytically assisted by pure state entanglement and every operation is allowed, which preserves the positivity of the partial transpose. It is shown, that reversibility occurs only in cases where the state is quasi-pure in the sense that all its pure state entanglement can be revealed by a simple operation on a single copy. The reversible cases are shown to be completely characterized by minimal uncertainty vectors for entropic uncertainty relations.

Entanglement cost of antisymmetric states and additivity of capacity of some quantum channels

We study the entanglement cost of the states in the contragredient space, which consists of $(d-1)$ $d$-dimensional systems. The cost is always $\log_2 (d-1)$ ebits when the state is divided into bipartite $\C^d \otimes (\C^d)^{d-2}$. Combined with the arguments in \cite{Matsumoto02}, additivity of channel capacity of some quantum channels is also shown.

Remarks on Additivity of the Holevo Channel Capacity and of the Entanglement of Formation

The purpose of these notes is to discuss the relation between the additivity questions regarding the quantities (Holevo) capacity of a quantum channel T and entanglement of formation of a given bipartite state. In particular, using the Stinespring dilation theorem, we give a formula for the channel capacity involving entanglement of formation. This can be used to show that additivity of the latter for some states can be inferred from the additivity of capacity for certain channels. We demonstrate this connection for a family of group--covariant channels, allowing us to calculate the entanglement cost for many states, including some where a strictly smaller upper bound on the distillable entanglement is known. Group symmetry is used for more sophisticated analysis, giving formulas valid for a class of channels. This is presented in a general framework, extending recent findings of Vidal, Dur and Cirac (e-print quant-ph/0112131). We speculate on a general relation of superadditivity of the entanglement of formation, which would imply both the general additivity of this function under tensor products and of the Holevo capacity (with or without linear cost constraints).

Entanglement cost of generalised measurements

Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According...

Entanglement cost of three-level antisymmetric states

We show that the entanglement cost of the three-dimensional antisymmetric states is one ebit.

Entanglement cost under positive-partial-transpose-preserving operations.

We study the entanglement cost under quantum operations preserving the positivity of the partial transpose (PPT operations). We demonstrate that this cost is directly related to the logarithmic negativity, thereby providing the operational interpretation for this entanglement measure. As examples we discuss general Werner states and arbitrary bipartite Gaussian states. Then we prove that for the antisymmetric Werner state PPT cost and PPT entanglement of distillation coincide. This is the first example of a truly mixed state for which entanglement manipulation is asymptotically reversible, which points towards a unique entanglement measure under PPT operations.

Entanglement cost of generalised measurements

Bipartite entanglement is one of the fundamental quantifiable resources of quantum information theory. We propose a new application of this resource to the theory of quantum measurements. According to Naimark's theorem any rank 1 generalised measurement (POVM) M may be represented as a von Neumann measurement in an extended (tensor product) space of the system plus ancilla. By considering a suitable average of the entanglements of these measurement directions and minimising over all Naimark extensions, we define a notion of entanglement cost E_{\min}(M) of M. We give a constructive means of characterising all Naimark extensions of a given POVM. We identify various classes of POVMs with zero and non-zero cost and explicitly characterise all POVMs in 2 dimensions having zero cost. We prove a constant upper bound on the entanglement cost of any POVM in any dimension. Hence the asymptotic entanglement cost (i.e. the large n limit of the cost of n applications of M, divided by n) is zero for all POVMs. The trine measurement is defined by three rank 1 elements, with directions symmetrically placed around a great circle on the Bloch sphere. We give an analytic expression for its entanglement cost. Defining a normalised cost of any $d$-dimensional POVM by E_{\min} (M)/\log_2 d, we show (using a combination of analytic and numerical techniques) that the trine measurement is more costly than any other POVM with d>2, or with d=2 and ancilla dimension 2. This strongly suggests that the trine measurement is the most costly of all POVMs.

The uniqueness theorem for entanglement measures

We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyze properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entangle ment.

The entanglement of purification

We introduce a measure of both quantum as well as classical correlations in a quantum state, the entanglement of purification. We show that the (regularized) entanglement of purification is equal to the entanglement cost of creating a state ρ asymptotically from maximally entangled states, with negligible communication. We prove that the classical mutual information and the quantum mutual information divided by two are lower bounds for the regularized entanglement of purification. We present numerical results of the entanglement of purification for Werner states in H2⊗H2.

Entanglement cost of bipartite mixed states.

We compute the entanglement cost of several families of bipartite mixed states, including arbitrary mixtures of two Bell states. This is achieved by developing a technique that allows us to ascertain the additivity of the entanglement of formation for any state supported on specific subspaces. As a side result, the proof of the irreversibility in asymptotic local manipulations of entanglement is extended to two-qubit systems.

Remote control of restricted sets of operations: Teleportation of angles

We study the remote implementation of a unitary transformation on a qubit. We show the existence of non-trivial protocols (i.e., using less resources than bidirectional state teleportation) which allow the perfect remote implementation of certain continuous sets of quantum operations. We prove that, up to a local change of basis, only two subsets exist that can be implemented remotely with a non-trivial protocol: Arbitrary rotations around a fixed direction $\vec{n}$ and rotations by a fixed angle around an arbitrary direction lying in a plane orthogonal to $\vec{n}$. The overall classical information and distributed entanglement cost required for the remote implementation depends on whether it is a priori known to which of the two teleportable subsets the transformation belongs to. If it is so, the optimal protocol consumes one e-bit of entanglement and one c-bit in each direction. If the subset is not known, two e-bits of entanglement need to be consumed while the classical channel becomes asymmetric, two c-bits are conveyed from Alice to Bob but only one from Bob to Alice.

The asymptotic entanglement cost of preparing a quantum state

We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state ρ is equal to limn→∞ Ef (ρ ⊗n )/n where Ef is the entanglement of formation.

Irreversibility in asymptotic manipulations of entanglement.

We show that the process of entanglement distillation is irreversible by showing that the entanglement cost of a bound entangled state is finite. Such irreversibility remains even if extra pure entanglement is loaned to assist the distillation process.