We consider the existence in arbitrary finite dimensions d of a POVM comprised of d 2 rank-one operators all of whose operator inner products are equal. Such a set is called a "symmetric, informationally complete" POVM (SIC-POVM) and is equivalent to a set of d 2 equiangular lines in C d . SIC-POVMs are relevant for quantum state tomography, quantum cryptography, and foundational issues in quantum mechanics. We construct SIC-POVMs in dimensions two, three, and four. We further conjecture that a particular kind of group-covariant SIC-POVM exists in arbitrary dimensions, providing numerical results up to dimension 45 to bolster this claim.

The uncertainty principle, originally formulated by Heisenberg 1 , clearly illustrates the difference between classical and quantum mechanics. The principle bounds the uncertainties about the outcomes of two incompatible measurements, such as position and momentum, on a particle. It implies that one cannot predict the outcomes for both possible choices of measurement to arbitrary precision, even if information about the preparation of the particle is available in a classical memory. However, if the particle is prepared entangled with a quantum memory, a device that might be available in the not-too-distant future 2 , it is possible to predict the outcomes for both measurement choices precisely. Here, we extend the uncertainty principle to incorporate this case, providing a lower bound on the uncertainties, which depends on the amount of entanglement between the particle and the quantum memory. We detail the application of our result to witnessing entanglement and to quantum key distribution.Uncertainty relations constrain the potential knowledge one can have about the physical properties of a system. Although classical theory does not limit the knowledge we can simultaneously have about arbitrary properties of a particle, such a limit does exist in quantum theory. Even with a complete description of its state, it is impossible to predict the outcomes of all possible measurements on the particle. This lack of knowledge, or uncertainty, was quantified by Heisenberg 1 using the standard deviation (which we denote by R for an observable R). If the measurement on a given particle is chosen from a set of two possible observables, R and S, the resulting bound on the uncertainty can be expressed in terms of the commutator 3 :In an information-theoretic context, it is more natural to quantify uncertainty in terms of entropy rather than the standard deviation. Entropic uncertainty relations for position and momentum were derived in ref. 4 and later a relation was developed that holds for any pair of observables 5 . An improvement of this relation was subsequently conjectured 6 and then proved 7 . The improved relation iswhere H (R) denotes the Shannon entropy of the probability distribution of the outcomes when R is measured. The term 1/c quantifies the complementarity of the observables. For non-degenerate observables, c := max j,k | ψ j |φ k | 2 , where |ψ j and |φ k are the eigenvectors of R and S, respectively. One way to think about uncertainty relations is through the following game (the uncertainty game) between two players, Alice and Bob. Before the game commences, Alice and Bob agree on two measurements, R and S. The game proceeds as follows. Bob prepares a particle in a quantum state of his choosing and sends it to Alice. Alice then carries out one of the two measurements and announces her choice to Bob. Bob's task is to minimize his uncertainty about Alice's measurement outcome. This is illustrated in Fig. 1.Equation (1) bounds Bob's uncertainty in the case that he has no quantum memory-all information Bob hold...

The ideas of thermodynamics have proved fruitful in the setting of quantum information theory, in particular the notion that when the allowed transformations of a system are restricted, certain states of the system become useful resources with which one can prepare previously inaccessible states. The theory of entanglement is perhaps the best-known and most well-understood resource theory in this sense. Here, we return to the basic questions of thermodynamics using the formalism of resource theories developed in quantum information theory and show that the free energy of thermodynamics emerges naturally from the resource theory of energy-preserving transformations. Specifically, the free energy quantifies the amount of useful work which can be extracted from asymptotically many copies of a quantum system when using only reversible energy-preserving transformations and a thermal bath at fixed temperature. The free energy also quantifies the rate at which resource states can be reversibly interconverted asymptotically, provided that a sublinear amount of coherent superposition over energy levels is available, a situation analogous to the sublinear amount of classical communication required for entanglement dilution. Quantum resource theories are specified by a restriction on the quantum operations (state preparations, measurements, and transformations) that can be implemented by one or more parties. This singles out a set of states which can be prepared under the restricted operations. If the parties facing the restriction acquire a quantum state outside the restricted set of states, then they can use this state to implement measurements and transformations that are outside the class of allowed operations, consuming the state in the process. Thus, such states are useful resources.A few prominent examples serve to illustrate the idea: if two or more parties are restricted to communicating classically and implementing local quantum operations, then entangled states become a resource [1]; if a party is restricted to quantum operations that have a particular symmetry, then states that break this symmetry become a resource [2][3][4]; if a party is restricted to preparing states that are completely mixed and performing unitary operations, then any state that is not completely mixed, i.e., any state that has some purity, becomes a resource [5].In this Letter, we develop the quantum resource theory of states that are T athermal, i.e., not thermal at temperature T. This provides a useful new formulation of equilibrium and nonequilibrium thermodynamics for finite-dimensional quantum systems, and allows us to apply new mathematical tools to the subject. The restricted class of operations which defines our resource theory includes only those that can be achieved through energy-conserving unitaries and the preparation of any ancillary system in a thermal state at temperature T, as first studied by Janzing et al. [6] in the context of Landauer's principle. Here, the ancillary systems can have an arbitrary Hilbert space and an a...

We conjecture a new entropic uncertainty principle governing the entropy of complementary observations made on a system given side information in the form of quantum states, generalizing the entropic uncertainty relation of Maassen and Uffink [Phys. Rev. Lett. 60, 1103 (1988)]. We prove a special case for certain conjugate observables by adapting a similar result found by Christandl and Winter pertaining to quantum channels [IEEE Trans. Inf. Theory 51, 3159 (2005)], and discuss possible applications of this result to the decoupling of quantum systems and for security analysis in quantum cryptography.