results for au:Morgan_C in:quant-ph

- Feb 29 2016 quant-ph arXiv:1602.08362v2Establishing the strong converse theorem for a communication channel confirms that the capacity of that channel, that is, the maximum achievable rate of reliable information communication, is the ultimate limit of communication over that channel. Indeed, the strong converse theorem for a channel states that coding at a rate above the capacity of the channel results in the convergence of the error to its maximum value 1 and that there is no trade-off between communication rate and decoding error. Here we prove that the strong converse theorem holds for the product-state capacity of quantum channels with ergodic Markovian correlated memory.
- Dec 17 2015 quant-ph arXiv:1512.05324v3Several information measures have recently been defined which capture the notion of "recoverability." In particular, the fidelity of recovery quantifies how well one can recover a system $A$ of a tripartite quantum state, defined on systems $ABC$, by acting on system $C$ alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper, we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform his actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal Type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the Type I error is constrained to be no larger than a constant.
- Sep 18 2015 quant-ph arXiv:1509.05155v5We investigate decoupling, one of the most important primitives in quantum Shannon theory, by replacing the uniformly distributed random unitaries commonly used to achieve the protocol, with repeated applications of random unitaries diagonal in the Pauli-$Z$ and -$X$ bases. This strategy was recently shown to achieve an approximate unitary $2$-design after a number of repetitions of the process, which implies that the strategy gradually achieves decoupling. Here, we prove that even fewer repetitions of the process achieve decoupling at the same rate as that with the uniform ones, showing that rather imprecise approximations of unitary $2$-designs are sufficient for decoupling. We also briefly discuss efficient implementations of them and implications of our decoupling theorem to coherent state merging and relative thermalisation.
- Feb 27 2015 quant-ph arXiv:1502.07514v4Unitary $2$-designs are random unitaries simulating up to the second order statistical moments of the uniformly distributed random unitaries, often referred to as Haar random unitaries. They are used in a wide variety of theoretical and practical quantum information protocols, and also have been used to model the dynamics in complex quantum many-body systems. Here, we show that unitary $2$-designs can be approximately implemented by alternately repeating random unitaries diagonal in the Pauli-$Z$ basis and that in the Pauli-$X$ basis. We also provide a converse about the number of repetitions needed to achieve unitary $2$-designs. These results imply that the process after $\ell$ repetitions achieves a $\Theta(d^{-\ell})$-approximate unitary $2$-design. Based on the construction, we further provide quantum circuits that efficiently implement approximate unitary $2$-designs. Although a more efficient implementation of unitary $2$-designs is known, our quantum circuit has its own merit that it is divided into a constant number of commuting parts, which enables us to apply all commuting gates simultaneously and leads to a possible reduction of an actual execution time. We finally interpret the result in terms of the dynamics generated by time-dependent Hamiltonians and provide for the first time a random disordered time-dependent Hamiltonian that generates a unitary $2$-design after switching interactions only a few times.
- We present a new achievable rate region for the two-user binary-input classical-quantum broadcast channel. The result is a generalization of the classical Marton-Gelfand-Pinsker region and is provably larger than the best previously known rate region for classical-quantum broadcast channels. The proof of achievability is based on the recently introduced polar coding scheme and its generalization to quantum network information theory.
- Polar coding is a method for communication over noisy classical channels which is provably capacity-achieving and has an efficient encoding and decoding. Recently, this method has been generalized to the realm of quantum information processing, for tasks such as classical communication, private classical communication, and quantum communication. In the present work, we apply the polar coding method to network quantum information theory, by making use of recent advances for related classical tasks. In particular, we consider problems such as the compound multiple access channel and the quantum interference channel. The main result of our work is that it is possible to achieve the best known inner bounds on the achievable rate regions for these tasks, without requiring a so-called quantum simultaneous decoder. Thus, our work paves the way for developing network quantum information theory further without requiring a quantum simultaneous decoder.
- Proving achievability of protocols in quantum Shannon theory usually does not consider the efficiency at which the goal of the protocol can be achieved. Nevertheless it is known that protocols such as coherent state merging are efficiently achievable at optimal rate. We aim to investigate this fact further in a general one-shot setting, by considering certain classes of decoupling theorems and give exact rates for these classes. Moreover we compare results of general decoupling theorems using Haar distributed unitaries with those using smaller sets of operators, in particular $\epsilon$-approximate 2-designs. We also observe the behavior of our rates in special cases such as $\epsilon$ approaching zero and the asymptotic limit.
- We exhibit a possible road towards a strong converse for the quantum capacity of degradable channels. In particular, we show that all degradable channels obey what we call a "pretty strong" converse: When the code rate increases above the quantum capacity, the fidelity makes a discontinuous jump from 1 to at most 0.707, asymptotically. A similar result can be shown for the private (classical) capacity. Furthermore, we can show that if the strong converse holds for symmetric channels (which have quantum capacity zero), then degradable channels obey the strong converse: The above-mentioned asymptotic jump of the fidelity at the quantum capacity is then from 1 down to 0.
- Aug 23 2011 quant-ph arXiv:1108.4282v2The strong capacity of a particular channel can be interpreted as a sharp limit on the amount of information which can be transmitted reliably over that channel. To evaluate the strong capacity of a particular channel one must prove both the direct part of the channel coding theorem and the strong converse for the channel. Here we consider the strong converse theorem for the periodic quantum channel and show some rather surprising results. We first show that the strong converse does not hold in general for this channel and therefore the channel does not have a strong capacity. Instead, we find that there is a scale of capacities corresponding to error probabilities between integer multiples of the inverse of the periodicity of the channel. A similar scale also exists for the random channel.
- Jul 20 2011 quant-ph arXiv:1107.3741v1We obtain a maximizer for the quantum mutual information for classical information sent over the quantum qubit amplitude damping channel. This is achieved by limiting the ensemble of input states to antipodal states, in the calculation of the product-state capacity for the channel, the resulting maximizing ensemble consisting of just two non-orthogonal states. We also consider the product-state capacity of a convex combination of two memoryless channels and demonstrate in particular that it is in general not given by the minimum of the capacities of the respective memoryless channels.
- Jul 19 2010 quant-ph arXiv:1007.2723v1In this thesis we analyse the type of states and ensembles which achieve the capacity for certain quantum channels carrying classical information. We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprised of states taken from ensembles of non-entangled states and sending them over copies of the quantum channel. Using the "single-letter" formula proved independently by Holevo and by Schumacher and Westmoreland we obtain the product-state capacity of the qubit quantum amplitude-damping channel, which is determined by a transcendental equation in a single real variable and can be solved numerically. We demonstrate that the product-state capacity of this channel can be achieved using a minimal ensemble of non-orthogonal pure states. Next we consider the classical capacity of two quantum channels with memory, namely a periodic channel with quantum depolarising channel branches and a convex combination of quantum channels. We prove that the classical capacity for each of the classical memory channels mentioned above is, in fact, equal to the respective product-state capacities. For those channels this means that the classical capacity is achieved without the use of entangled input-states. Next we introduce the channel coding theorem for memoryless quantum channels, providing a known proof by Winter for the strong converse of the theorem. We then consider the strong converse to the channel coding theorem for a periodic quantum channel.
- Feb 18 2009 quant-ph arXiv:0902.2834v2We investigate the classical capacity of two quantum channels with memory: a periodic channel with depolarizing channel branches, and a convex combination of depolarizing channels. We prove that the capacity is additive in both cases. As a result, the channel capacity is achieved without the use of entangled input states. In the case of a convex combination of depolarizing channels the proof provided can be extended to other quantum channels whose classical capacity has been proved to be additive in the memoryless case.