The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement, already present in a number of settings (see Shimony [1] and Barnum and Linden [2]), is explored for bipartite and multipartite pure and mixed states. The measure is determined analytically for arbitrary twoqubit mixed states and for generalized Werner and isotropic states, and is also applied to certain multipartite mixed states. In particular, a detailed analysis is given for arbitrary mixtures of threequbit GHZ, W and inverted-W states. Along the way, we point out connections of the geometric measure of entanglement with entanglement witnesses and with the Hartree approximation method.

Dense coding is arguably the protocol that launched the field of quantum communication 1 . Today, however, more than a decade after its initial experimental realization 2 , the channel capacity remains fundamentally limited as conceived for photons using linear elements. Bob can only send to Alice three of four potential messages owing to the impossibility of carrying out the deterministic discrimination of all four Bell states with linear optics 3,4 , reducing the attainable channel capacity from 2 to log 2 3 ≈ 1.585 bits. However, entanglement in an extra degree of freedom enables the complete and deterministic discrimination of all Bell states 5-7 . Using pairs of photons simultaneously entangled in spin and orbital angular momentum 8,9 , we demonstrate the quantum advantage of the ancillary entanglement. In particular, we describe a dense-coding experiment with the largest reported channel capacity and, to our knowledge, the first to break the conventional linear-optics threshold. Our encoding is suited for quantum communication without alignment 10 and satellite communication. The first realization of quantum dense coding was optical, using pairs of photons entangled in polarization 2 . Dense coding has since been realized in various physical systems and broadened theoretically to include high-dimension quantum states with multiparties 11 , and even coding of quantum states 12 . The protocol extension to continuous variables 13,14 has also been experimentally explored optically, using superimposed squeezed beams 15 . Other physical approaches include a simulation in nuclear magnetic resonance with temporal averaging 16 , and an implementation with atomic qubits on demand without postselection 17 . However, photons remain the optimal carriers of information given their resilience to decoherence and ease of creation and transportation.Quantum dense coding was conceived 1 such that Bob could communicate 2 bits of classical information to Alice with the transmission of a single qubit, as follows. Initially, each party holds one spin-1/2 particle of a maximally entangled pair, such as one of the four Bell states. Bob then encodes his 2-bit message by applying one of four unitary operations on his particle, which he then transmits to Alice. Finally, Alice decodes the 2-bit message by discriminating the Bell state of the pair.Alice's decoding step, deterministically resolving the four Bell states, is known as Bell-state analysis (BSA). Although in principle attainable with nonlinear interactions, such BSA with photons is very difficult to achieve with present technology, yielding extremely low efficiencies and low discrimination fidelities 18 . Therefore, current fundamental studies and technological developments demand the use of linear optics. However, for quantum communication, standard BSA with linear optics is fundamentally impossible 3,4 . At best, only two Bell states can be discriminated; for quantum communication, the other two are considered together for a three-message encoding. Consequently, the maximu...

Complete and precise characterization of a quantum dynamical process can be achieved via the method of quantum process tomography. Using a source of correlated photons, we have implemented several methods, each investigating a wide range of processes, e.g., unitary, decohering, and polarizing. One of these methods, ancilla-assisted process tomography (AAPT), makes use of an additional "ancilla system," and we have theoretically determined the conditions when AAPT is possible. Surprisingly, entanglement is not required. We present data obtained using both separable and entangled input states. The use of entanglement yields superior results, however.

Maximally entangled mixed states are those states that, for a given mixedness, achieve the greatest possible entanglement. For two-qubit systems and for various combinations of entanglement and mixedness measures, the form of the corresponding maximally entangled mixed states is determined primarily analytically. As measures of entanglement, we consider entanglement of formation, relative entropy of entanglement, and negativity; as measures of mixedness, we consider linear and von Neumann entropies. We show that the forms of the maximally entangled mixed states can vary with the combination of (entanglement and mixedness) measures chosen. Moreover, for certain combinations, the forms of the maximally entangled mixed states can change discontinuously at a specific value of the entropy.