A simple operational interpretation of the fidelity

This note presents a corollary to Uhlmann's theorem which provides a simple operational interpretation for the fidelity of mixed states.Comment: 1 pag

Better bound on the exponent of the radius of the multipartite separable ball

We show that for an m-qubit quantum system, there is a ball of radius asymptotically approaching kappa 2^{-gamma m} in Frobenius norm, centered at the identity matrix, of separable (unentangled) positive semidefinite matrices, for an exponent gamma = (1/2)((ln 3/ln 2) - 1), roughly .29248125. This is much smaller in magnitude than the best previously known exponent, from our earlier work, of 1/2. For normalized m-qubit states, we get a separable ball of radius sqrt(3^(m+1)/(3^m+3)) * 2^{-(1 + \gamma)m}, i.e. sqrt{3^{m+1}/(3^m+3)}\times 6^{-m/2} (note that \kappa = \sqrt{3}), compared to the previous 2 * 2^{-3m/2}. This implies that with parameters realistic for current experiments, NMR with standard pseudopure-state preparation techniques can access only unentangled states if 36 qubits or fewer are used (compared to 23 qubits via our earlier results). We also obtain an improved exponent for m-partite systems of fixed local dimension d_0, although approaching our earlier exponent as d_0 approaches infinity.Comment: 30 pp doublespaced, latex/revtex, v2 added discussion of Szarek's upper bound, and reference to work of Vidal, v3 fixed some errors (no effect on results), v4 involves major changes leading to an improved constant, same exponent, and adds references to and discussion of Szarek's work showing that exponent is essentially optimal for qubit case, and Hildebrand's alternative derivation for qubit case. To appear in PR

Introduction to Quantum Error Correction

In this introduction we motivate and explain the ``decoding'' and ``subsystems'' view of quantum error correction. We explain how quantum noise in QIP can be described and classified, and summarize the requirements that need to be satisfied for fault tolerance. Considering the capabilities of currently available quantum technology, the requirements appear daunting. But the idea of ``subsystems'' shows that these requirements can be met in many different, and often unexpected ways.Comment: 44 pages, to appear in LA Science. Hyperlinked PDF at http://www.c3.lanl.gov/~knill/qip/ecprhtml/ecprpdf.pdf, HTML at http://www.c3.lanl.gov/~knill/qip/ecprhtm

Compressibility of Mixed-State Signals

We present a formula that determines the optimal number of qubits per message that allows asymptotically faithful compression of the quantum information carried by an ensemble of mixed states. The set of mixed states determines a decomposition of the Hilbert space into the redundant part and the irreducible part. After removing the redundancy, the optimal compression rate is shown to be given by the von Neumann entropy of the reduced ensemble.Comment: 7 pages, no figur

Generalization of entanglement to convex operational theories: Entanglement relative to a subspace of observables

We define what it means for a state in a convex cone of states on a space of observables to be generalized-entangled relative to a subspace of the observables, in a general ordered linear spaces framework for operational theories. This extends the notion of ordinary entanglement in quantum information theory to a much more general framework. Some important special cases are described, in which the distinguished observables are subspaces of the observables of a quantum system, leading to results like the identification of generalized unentangled states with Lie-group-theoretic coherent states when the special observables form an irreducibly represented Lie algebra. Some open problems, including that of generalizing the semigroup of local operations with classical communication to the convex cones setting, are discussed.Comment: 19 pages, to appear in proceedings of Quantum Structures VII, Int. J. Theor. Phy

Three-dimensionality of space and the quantum bit: an information-theoretic approach

It is sometimes pointed out as a curiosity that the state space of quantum two-level systems, i.e. the qubit, and actual physical space are both three-dimensional and Euclidean. In this paper, we suggest an information-theoretic analysis of this relationship, by proving a particular mathematical result: suppose that physics takes place in d spatial dimensions, and that some events happen probabilistically (not assuming quantum theory in any way). Furthermore, suppose there are systems that carry "minimal amounts of direction information", interacting via some continuous reversible time evolution. We prove that this uniquely determines spatial dimension d=3 and quantum theory on two qubits (including entanglement and unitary time evolution), and that it allows observers to infer local spatial geometry from probability measurements.Comment: 13 + 22 pages, 9 figures. v4: some clarifications, in particular in Section V / Appendix C (added Example 39

Efficient solvability of Hamiltonians and limits on the power of some quantum computational models

We consider quantum computational models defined via a Lie-algebraic theory. In these models, specified initial states are acted on by Lie-algebraic quantum gates and the expectation values of Lie algebra elements are measured at the end. We show that these models can be efficiently simulated on a classical computer in time polynomial in the dimension of the algebra, regardless of the dimension of the Hilbert space where the algebra acts. Similar results hold for the computation of the expectation value of operators implemented by a gate-sequence. We introduce a Lie-algebraic notion of generalized mean-field Hamiltonians and show that they are efficiently ("exactly") solvable by means of a Jacobi-like diagonalization method. Our results generalize earlier ones on fermionic linear optics computation and provide insight into the source of the power of the conventional model of quantum computation.Comment: 6 pages; no figure

Indeterminate-length quantum coding

The quantum analogues of classical variable-length codes are indeterminate-length quantum codes, in which codewords may exist in superpositions of different lengths. This paper explores some of their properties. The length observable for such codes is governed by a quantum version of the Kraft-McMillan inequality. Indeterminate-length quantum codes also provide an alternate approach to quantum data compression.Comment: 32 page

Lower bound for the quantum capacity of a discrete memoryless quantum channel

We generalize the random coding argument of stabilizer codes and derive a lower bound on the quantum capacity of an arbitrary discrete memoryless quantum channel. For the depolarizing channel, our lower bound coincides with that obtained by Bennett et al. We also slightly improve the quantum Gilbert-Varshamov bound for general stabilizer codes, and establish an analogue of the quantum Gilbert-Varshamov bound for linear stabilizer codes. Our proof is restricted to the binary quantum channels, but its extension of to l-adic channels is straightforward.Comment: 16 pages, REVTeX4. To appear in J. Math. Phys. A critical error in fidelity calculation was corrected by using Hamada's result (quant-ph/0112103). In the third version, we simplified formula and derivation of the lower bound by proving p(Gamma)+q(Gamma)=1. In the second version, we added an analogue of the quantum Gilbert-Varshamov bound for linear stabilizer code

Trading quantum for classical resources in quantum data compression

We study the visible compression of a source E of pure quantum signal states, or, more formally, the minimal resources per signal required to represent arbitrarily long strings of signals with arbitrarily high fidelity, when the compressor is given the identity of the input state sequence as classical information. According to the quantum source coding theorem, the optimal quantum rate is the von Neumann entropy S(E) qubits per signal. We develop a refinement of this theorem in order to analyze the situation in which the states are coded into classical and quantum bits that are quantified separately. This leads to a trade--off curve Q(R), where Q(R) qubits per signal is the optimal quantum rate for a given classical rate of R bits per signal. Our main result is an explicit characterization of this trade--off function by a simple formula in terms of only single signal, perfect fidelity encodings of the source. We give a thorough discussion of many further mathematical properties of our formula, including an analysis of its behavior for group covariant sources and a generalization to sources with continuously parameterized states. We also show that our result leads to a number of corollaries characterizing the trade--off between information gain and state disturbance for quantum sources. In addition, we indicate how our techniques also provide a solution to the so--called remote state preparation problem. Finally, we develop a probability--free version of our main result which may be interpreted as an answer to the question: ``How many classical bits does a qubit cost?'' This theorem provides a type of dual to Holevo's theorem, insofar as the latter characterizes the cost of coding classical bits into qubits.Comment: 51 pages, 7 figure