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

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

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

Verschraenkung versus Stosszahlansatz: Disappearance of the Thermodynamic Arrow in a High-Correlation Environment

The crucial role of ambient correlations in determining thermodynamic behavior is established. A class of entangled states of two macroscopic systems is constructed such that each component is in a state of thermal equilibrium at a given temperature, and when the two are allowed to interact heat can flow from the colder to the hotter system. A dilute gas model exhibiting this behavior is presented. This reversal of the thermodynamic arrow is a consequence of the entanglement between the two systems, a condition that is opposite to molecular chaos and shown to be unlikely in a low-entropy environment. By contrast, the second law is established by proving Clausius' inequality in a low-entropy environment. These general results strongly support the expectation, first expressed by Boltzmann and subsequently elaborated by others, that the second law is an emergent phenomenon that requires a low-entropy cosmological environment, one that can effectively function as an ideal information sink.Comment: 4 pages, REVTeX

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

Necessary Condition for the Quantum Adiabatic Approximation

A gapped quantum system that is adiabatically perturbed remains approximately in its eigenstate after the evolution. We prove that, for constant gap, general quantum processes that approximately prepare the final eigenstate require a minimum time proportional to the ratio of the length of the eigenstate path to the gap. Thus, no rigorous adiabatic condition can yield a smaller cost. We also give a necessary condition for the adiabatic approximation that depends on local properties of the path, which is appropriate when the gap varies.Comment: 5 pages, 1 figur

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

Quantum communication using a bounded-size quantum reference frame

Typical quantum communication schemes are such that to achieve perfect decoding the receiver must share a reference frame with the sender. Indeed, if the receiver only possesses a bounded-size quantum token of the sender's reference frame, then the decoding is imperfect, and we can describe this effect as a noisy quantum channel. We seek here to characterize the performance of such schemes, or equivalently, to determine the effective decoherence induced by having a bounded-size reference frame. We assume that the token is prepared in a special state that has particularly nice group-theoretic properties and that is near-optimal for transmitting information about the sender's frame. We present a decoding operation, which can be proven to be near-optimal in this case, and we demonstrate that there are two distinct ways of implementing it (corresponding to two distinct Kraus decompositions). In one, the receiver measures the orientation of the reference frame token and reorients the system appropriately. In the other, the receiver extracts the encoded information from the virtual subsystems that describe the relational degrees of freedom of the system and token. Finally, we provide explicit characterizations of these decoding schemes when the system is a single qubit and for three standard kinds of reference frame: a phase reference, a Cartesian frame (representing an orthogonal triad of spatial directions), and a reference direction (representing a single spatial direction).Comment: 17 pages, 1 figure, comments welcome; v2 published versio

A Lower Bound for Quantum Phase Estimation

We obtain a query lower bound for quantum algorithms solving the phase estimation problem. Our analysis generalizes existing lower bound approaches to the case where the oracle Q is given by controlled powers Q^p of Q, as it is for example in Shor's order finding algorithm. In this setting we will prove a log (1/epsilon) lower bound for the number of applications of Q^p1, Q^p2, ... This bound is tight due to a matching upper bound. We obtain the lower bound using a new technique based on frequency analysis.Comment: 7 pages, 1 figur

Discrete phase space based on finite fields

The original Wigner function provides a way of representing in phase space the quantum states of systems with continuous degrees of freedom. Wigner functions have also been developed for discrete quantum systems, one popular version being defined on a 2N x 2N discrete phase space for a system with N orthogonal states. Here we investigate an alternative class of discrete Wigner functions, in which the field of real numbers that labels the axes of continuous phase space is replaced by a finite field having N elements. There exists such a field if and only if N is a power of a prime; so our formulation can be applied directly only to systems for which the state-space dimension takes such a value. Though this condition may seem limiting, we note that any quantum computer based on qubits meets the condition and can thus be accommodated within our scheme. The geometry of our N x N phase space also leads naturally to a method of constructing a complete set of N+1 mutually unbiased bases for the state space.Comment: 60 pages; minor corrections and additional references in v2 and v3; improved historical introduction in v4; references to quantum error correction in v5; v6 corrects the value quoted for the number of similarity classes for N=