arXiv subjects

Graeme Mitchison

Publications and source records attributed to Graeme Mitchison.

At least 19 recordsLinked to original sources

arXiv2014

Symmetric polynomials in information theory: entropy and subentropy

Entropy and other fundamental quantities of information theory are customarily expressed and manipulated as functions of probabilities. Here we study the entropy H and subentropy Q as functions of the elementary symmetric polynomials in the probabilities, and reveal a series of remarkable properties. Derivatives of all orders are shown to satisfy a complete monotonicity property. H and Q themselves become multivariate Bernstein functions and we derive the density functions of their Levy-Khintchine representations. We also show that H and Q are Pick functions in each symmetric polynomial variable separately. Furthermore we see that H and the intrinsically quantum informational quantity Q become surprisingly closely related in functional form, suggesting a special significance for the symmetric polynomials in quantum information theory. Using the symmetric polynomials we also derive a series of further properties of H and Q.

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arXiv2013

Entropy, subentropy and the elementary symmetric functions

We use complex contour integral techniques to study the entropy H and subentropy Q as functions of the elementary symmetric polynomials, revealing a series of striking properties. In particular for these variables, derivatives of -Q are equal to derivatives of H of one higher order and the first derivatives of H and Q are seen to be completely monotone functions. It then follows that exp (-H) and exp(-Q) are Laplace transforms of infinitely divisible probability distributions.

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arXiv2013

On exact quantum query complexity

We present several families of total boolean functions which have exact quantum query complexity which is a constant multiple (between 1/2 and 2/3) of their classical query complexity, and show that optimal quantum algorithms for these functions cannot be obtained by simply computing parities of pairs of bits. We also characterise the model of nonadaptive exact quantum query complexity in terms of coding theory and completely characterise the query complexity of symmetric boolean functions in this context. These results were originally inspired by numerically solving the semidefinite programs characterising quantum query complexity for small problem sizes. We include numerical results giving the optimal success probabilities achievable by quantum algorithms computing all boolean functions on up to 4 bits, and all symmetric boolean functions on up to 6 bits.

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arXiv2012

The hierarchical structure of local unitary invariants

Local unitary invariants allow one to test whether multipartite states are equivalent up to local basis changes. Equivalently, they specify the geometry of the "orbit space" obtained by factoring out local unitary action from the state space. This space is of interest because of its intimate relationship to entanglement. Unfortunately, the dimension of the orbit space grows exponentially with the number of subsystems, and the number of invariants needed to characterise orbits grows at least as fast. This makes the study of entanglement via local unitary invariants seem very daunting. I point out here that there is a simplifying principle: Invariants fall into families related by the tracing-out of subsystems, and these families grow exponentially with the number of subsystems. In particular, in the case of pure qubit systems, there is a family whose size is about half the dimension of orbit space. These invariants are closely related to cumulants and to multipartite separability. Members of the family have been repeatedly discovered in the literature, but the fact that they are related to cumulants and constitute a family has apparently not been observed.

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arXiv2009

Cumulants and the moment algebra: tools for analysing weak measurements

Recently it has been shown that cumulants significantly simplify the analysis of multipartite weak measurements. Here we consider the mathematical structure that underlies this, and find that it can be formulated in terms of what we call the moment algebra. Apart from resulting in simpler proofs, the flexibility of this structure allows generalizations of the original results to a number of weak measurement scenarios, including one where the weakly interacting pointers reach thermal equilibrium with the probed system.

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arXiv2008

One-and-a-half quantum de Finetti theorems

We prove a new kind of quantum de Finetti theorem for representations of the unitary group U(d). Consider a pure state that lies in the irreducible representation U_{mu+nu} for Young diagrams mu and nu. U_{mu+nu} is contained in the tensor product of U_mu and U_nu; let xi be the state obtained by tracing out U_nu. We show that xi is close to a convex combination of states Uv, where U is in U(d) and v is the highest weight vector in U_mu. When U_{mu+nu} is the symmetric representation, this yields the conventional quantum de Finetti theorem for symmetric states, and our method of proof gives near-optimal bounds for the approximation of xi by a convex combination of product states. For the class of symmetric Werner states, we give a second de Finetti-style theorem (our 'half' theorem); the de Finetti-approximation in this case takes a particularly simple form, involving only product states with a fixed spectrum. Our proof uses purely group theoretic methods, and makes a link with the shifted Schur functions. It also provides some useful examples, and gives some insight into the structure of the set of convex combinations of product states.

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arXiv2007

Weak measurement takes a simple form for cumulants

A weak measurement on a system is made by coupling a pointer weakly to the system and then measuring the position of the pointer. If the initial wavefunction for the pointer is real, the mean displacement of the pointer is proportional to the so-called weak value of the observable being measured. This gives an intuitively direct way of understanding weak measurement. However, if the initial pointer wavefunction takes complex values, the relationship between pointer displacement and weak value is not quite so simple, as pointed out recently by R. Jozsa. This is even more striking in the case of sequential weak measurements. These are carried out by coupling several pointers at different stages of evolution of the system, and the relationship between the products of the measured pointer positions and the sequential weak values can become extremely complicated for an arbitrary initial pointer wavefunction. Surprisingly, all this complication vanishes when one calculates the cumulants of pointer positions. These are directly proportional to the cumulants of sequential weak values. This suggests that cumulants have a fundamental physical significance for weak measurement.

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arXiv2007

Sequential weak measurement

The notion of weak measurement provides a formalism for extracting information from a quantum system in the limit of vanishing disturbance to its state. Here we extend this formalism to the measurement of sequences of observables. When these observables do not commute, we may obtain information about joint properties of a quantum system that would be forbidden in the usual strong measurement scenario. As an application, we provide a physically compelling characterisation of the notion of counterfactual quantum computation.

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arXiv2007

A most compendious and facile quantum de Finetti theorem

In its most basic form, the finite quantum de Finetti theorem states that the reduced k-partite density operator of an n-partite symmetric state can be approximated by a convex combination of k-fold product states. Variations of this result include Renner's "exponential" approximation by "almost-product" states, a theorem which deals with certain triples of representations of the unitary group, and D'Cruz et al.'s result for infinite-dimensional systems. We show how these theorems follow from a single, general de Finetti theorem for representations of symmetry groups, each instance corresponding to a particular choice of symmetry group and representation of that group. This gives some insight into the nature of the set of approximating states, and leads to some new results, including an exponential theorem for infinite-dimensional systems.

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arXiv2007

A dual de Finetti theorem

The quantum de Finetti theorem says that, given a symmetric state, the state obtained by tracing out some of its subsystems approximates a convex sum of power states. The more subsystems are traced out, the better this approximation becomes. Schur-Weyl duality suggests that there ought to be a dual result that applies to a unitarily invariant state rather than a symmetric state. Instead of tracing out a number of subsystems, one traces out part of every subsystem. The theorem then asserts that the resulting state approximates the fully mixed state, and the larger the dimension of the traced-out part of each subsystem, the better this approximation becomes. This paper gives a number of propositions together with their dual versions, to show how far the duality holds.

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arXiv2007

The limits of counterfactual computation

We show that the protocol recently proposed by Hosten et al. does not allow all possible results of a computation to be obtained counterfactually, as was claimed. It only gives a counterfactual outcome for one of the computer outputs. However, we confirm the observation that the protocol gives some protection against decoherence. In some situations, though, it may be more effective simply to run the computer several times.

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arXiv2006

The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group

Determining the relationship between composite systems and their subsystems is a fundamental problem in quantum physics. In this paper we consider the spectra of a bipartite quantum state and its two marginal states. To each spectrum we can associate a representation of the symmetric group defined by a Young diagram whose normalised row lengths approximate the spectrum. We show that, for allowed spectra, the representation of the composite system is contained in the tensor product of the representations of the two subsystems. This gives a new physical meaning to representations of the symmetric group. It also introduces a new way of using the machinery of group theory in quantum informational problems, which we illustrate by two simple examples.

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arXiv2005

On Nonzero Kronecker Coefficients and their Consequences for Spectra

A triple of spectra (r^A, r^B, r^{AB}) is said to be admissible if there is a density operator rho^{AB} with (Spec rho^A, Spec rho^B, Spec rho^{AB})=(r^A, r^B, r^{AB}). How can we characterise such triples? It turns out that the admissible spectral triples correspond to Young diagrams (mu, nu, lambda) with nonzero Kronecker coefficient [M. Christandl and G. Mitchison, to appear in Comm. Math. Phys., quant-ph/0409016; A. Klyachko, quant-ph/0409113]. This means that the irreducible representation V_lambda is contained in the tensor product of V_mu and V_nu. Here, we show that such triples form a finitely generated semigroup, thereby resolving a conjecture of Klyachko. As a consequence we are able to obtain stronger results than in [M. Ch. and G. M. op. cit.] and give a complete information-theoretic proof of the correspondence between triples of spectra and representations. Finally, we show that spectral triples form a convex polytope.

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arXiv2004

Sparse Graph Codes for Quantum Error-Correction

We present sparse graph codes appropriate for use in quantum error-correction. Quantum error-correcting codes based on sparse graphs are of interest for three reasons. First, the best codes currently known for classical channels are based on sparse graphs. Second, sparse graph codes keep the number of quantum interactions associated with the quantum error correction process small: a constant number per quantum bit, independent of the blocklength. Third, sparse graph codes often offer great flexibility with respect to blocklength and rate. We believe some of the codes we present are unsurpassed by previously published quantum error-correcting codes.

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arXiv2003

Towards a geometrical interpretation of quantum information compression

Let S be the von Neumann entropy of a finite ensemble E of pure quantum states. We show that S may be naturally viewed as a function of a set of geometrical volumes in Hilbert space defined by the states and that S is monotonically increasing in each of these variables. Since S is the Schumacher compression limit of E, this monotonicity property suggests a geometrical interpretation of the quantum redundancy involved in the compression process. It provides clarification of previous work in which it was shown that S may be increased while increasing the overlap of each pair of states in the ensemble. As a byproduct, our mathematical techniques also provide a new interpretation of the subentropy of E.

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arXiv2001

How many photons are needed to distinguish two transparencies?

We give a bound on the minimum number of photons that must be absorbed by any quantum protocol to distinguish between two transparencies. We show how a quantum Zeno method in which the angle of rotation is varied at each iteration can attain this bound in certain situations.

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arXiv2001

Minimal Absorption Measurements

We show that it is not possible to discriminate two close transparencies without a certain number of photons being absorbed. We extend this to the discrimination of patterns of transparency (images).

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arXiv2000

"Interaction-free" discrimination between semi-transparent objects

Absorption-free (also known as ``interaction-free'') measurement aims to detect the presence of an opaque object using a test particle without that particle being absorbed by the object. Here we consider semi-transparent objects which have an amplitude $α$ of transmitting a particle while leaving the state of the object unchanged and an amplitude $β$ of absorbing the particle. The task is to devise a protocol that can decide which of two known transmission amplitudes is present while ensuring that no particle interacts with the object. We show that the probabilities of being able to achieve this are limited by an inequality. This inequality implies that absorption free distinction between complete transparency and any partial transparency is always possible with probabilities approaching 1, but that two partial transparencies can only be distinguished with probabilities less than 1.

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