Some Bipartite States do not Arise from Channels

It is well-known that the action of a quantum channel on a state can be represented, using an auxiliary space, as the partial trace of an associated bipartite state. Recently, it was observed that for the bipartite state associated with the optimal average input of the channel, the entanglement of formation is simply the entropy of the reduced density matrix minus the Holevo capacity. It is natural to ask if every bipartite state can be associated with some channel in this way. We show that the answer is negative.Comment: 7 pages; minor typos corrected. To appear in special issue of the IBM Journal of Research and Development for Charles Bennett's 60th birthda

Invariance of quantum correlations under local channel for a bipartite quantum state

We show that the quantum discord and the measurement induced non-locality (MiN) in a bipartite quantum state is invariant under the action of a local quantum channel if and only if the channel is invertible. In particular, these quantities are invariant under a local unitary channel.Comment: 4 pages, no figures, proof of theorm 2 modifie

Capacity Theorems for Quantum Multiple Access Channels: Classical-Quantum and Quantum-Quantum Capacity Regions

We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multi-letter characterizations of two different two-dimensional capacity regions. The first region is comprised of the rates at which it is possible for one sender to send classical information, while the other sends quantum information. The second region consists of the rates at which each sender can send quantum information. For each region, we give an example of a channel for which the corresponding region has a single-letter description. One of our examples relies on a new result proved here, perhaps of independent interest, stating that the coherent information over any degradable channel is concave in the input density operator. We conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information.Comment: 38 pages, 1 figure. Fixed typos, added new example. Submitted to IEEE Tranactions on Information Theor

Mixed State Entanglement and Quantum Error Correction

Entanglement purification protocols (EPP) and quantum error-correcting codes (QECC) provide two ways of protecting quantum states from interaction with the environment. In an EPP, perfectly entangled pure states are extracted, with some yield D, from a mixed state M shared by two parties; with a QECC, an arbi- trary quantum state $|\xi\rangle$ can be transmitted at some rate Q through a noisy channel $\chi$ without degradation. We prove that an EPP involving one- way classical communication and acting on mixed state $\hat{M}(\chi)$ (obtained by sharing halves of EPR pairs through a channel $\chi$) yields a QECC on $\chi$ with rate $Q=D$, and vice versa. We compare the amount of entanglement E(M) required to prepare a mixed state M by local actions with the amounts $D_1(M)$ and $D_2(M)$ that can be locally distilled from it by EPPs using one- and two-way classical communication respectively, and give an exact expression for $E(M)$ when $M$ is Bell-diagonal. While EPPs require classical communica- tion, QECCs do not, and we prove Q is not increased by adding one-way classical communication. However, both D and Q can be increased by adding two-way com- munication. We show that certain noisy quantum channels, for example a 50% depolarizing channel, can be used for reliable transmission of quantum states if two-way communication is available, but cannot be used if only one-way com- munication is available. We exhibit a family of codes based on universal hash- ing able toachieve an asymptotic $Q$ (or $D$) of 1-S for simple noise models, where S is the error entropy. We also obtain a specific, simple 5-bit single- error-correcting quantum block code. We prove that {\em iff} a QECC results in high fidelity for the case of no error the QECC can be recast into a form where the encoder is the matrix inverse of the decoder.Comment: Resubmission with various corrections and expansions. See also http://vesta.physics.ucla.edu/~smolin/ for related papers and information. 82 pages latex including 19 postscript figures included using psfig macro

Markovian embedding of non-Markovian quantum collisional models

A wide class of non-Markovian completely positive master equations can be formulated on the basis of quantum collisional models. In this phenomenological approach the dynamics of an open quantum system is modeled through an ensemble of stochastic realizations that consist in the application at random times of a (collisional) completely positive transformation over the system state. In this paper, we demonstrate that these kinds of models can be embedded in bipartite Markovian Lindblad dynamics consisting of the system of interest and an auxiliary one. In contrast with phenomenological formulations, here the stochastic ensemble dynamics an the inter-event time interval statistics are obtained from a quantum measurement theory after assuming that the auxiliary system is continuously monitored in time. Models where the system inter-collisional dynamics is non-Markovian [B. Vacchini, Phys. Rev. A 87, 030101(R) (2013)] are also obtained from the present approach. The formalism is exemplified through bipartite dynamics that leads to non-Markovian system effects such as an environment-to-system back flow of information.Comment: 12 pages, 2 figure