A Singular Value Inequality for Heinz Means
We prove a matrix inequality for matrix monotone functions, and apply it to prove a singular value inequality for Heinz means recently conjectured by X. Zhan.
math.FA↗
arXiv subjects
Koenraad M. R. Audenaert
Publications and source records attributed to Koenraad M. R. Audenaert.
Entanglement on mixed stabiliser states--I: Normal Forms and Reduction Procedures
The stabiliser formalism allows the efficient description of a sizeable class of pure as well as mixed quantum states of N-qubit systems. That same formalism has important applications in the field of quantum error correcting codes, where mixed stabiliser states correspond to projectors on subspaces associated with stabiliser codes. Here, we derive efficient reduction procedures to obtain various useful normal forms for stabiliser states. We explicitly prove that these procedures will always converge to the correct result and that these procedures are efficient in that they only require a polynomial number of operations on the generators of the stabilisers. We obtain two single-party normal forms. The simplest, the row-reduced echelon form, is useful to calculate partial traces of stabiliser states. The second is the fully reduced form and allows for the efficient calculation of the overlap between two stabiliser states, as well as of the Uhlmann fidelity between them, and their Bures distance. We also find a reduction procedure of bipartite stabiliser states, where the operations involved are restricted to be local ones. Using this two-party normal form, we prove that every bipartite mixed stabiliser state is locally equivalent to a direct product of a number of maximally entangled states and a separable state. Using this normal form we can efficiently calculate every reasonable bipartite entanglement measure of mixed stabiliser states.
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Scaling of success probabilities for linear optics gates
By using the abstract linear-optical network derived in [S. Scheel and N. Lütkenhaus, New J. Phys. \textbf{6}, 51 (2004)] we show that for the lowest possible ancilla photon numbers the probability of success of realizing a (single-shot) generalized nonlinear sign shift gate on an ($N+1$)-dimensional signal state scales as $1/N^2$. We limit ourselves to single-shot gates without conditional feed-forward. We derive our results by using determinants of Vandermonde-type over a polynomial basis which is closely related to the well-known Jacobi polynomials.
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A Note on the p->q norms of Completely Positive Maps
King and Ruskai asked whether the $p\to q$ norm of a completely positive map $Φ$, acting between Schatten $p$ and $q$ classes of self-adjoint operators, $||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}$, is equal to the $p\to q$ norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.
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A Norm Compression Inequality for Block Partitioned Positive Semidefinite Matrices
Let $A$ be a positive semidefinite matrix, block partitioned as $$ A=\twomat{B}{C}{C^*}{D}, $$ where $B$ and $D$ are square blocks. We prove the following inequalities for the Schatten $q$-norm $||.||_q$, which are sharp when the blocks are of size at least $2\times2$: $$ ||A||_q^q \le (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 1\le q\le 2, $$ and $$ ||A||_q^q \ge (2^q-2) ||C||_q^q + ||B||_q^q+||D||_q^q, \quad 2\le q. $$ These bounds can be extended to symmetric partitionings into larger numbers of blocks, at the expense of no longer being sharp: $$ ||A||_q^q \le \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 1\le q\le 2, $$ and $$ ||A||_q^q \ge \sum_{i} ||A_{ii}||_q^q + (2^q-2) \sum_{i<j} ||A_{ij}||_q^q, \quad 2\le q. $$
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There, and Back Again: Quantum Theory and Global Optimisation
We consider a problem in quantum theory that can be formulated as an optimisation problem and present a global optimisation algorithm for solving it, the foundation of which relies in turn on a theorem from quantum theory. To wit, we consider the maximal output purity $ν_q$ of a quantum channel as measured by Schatten $q$-norms, for integer $q$. This quantity is of fundamental importance in the study of quantum channel capacities in quantum information theory. To calculate $ν_q$ one has to solve a non-convex optimisation problem that typically exhibits local optima. We show that this particular problem can be approximated to arbitrary precision by an eigenvalue problem over a larger matrix space, thereby circumventing the problem of local optima. The mathematical proof behind this algorithm relies on the Quantum de Finetti theorem, which is a theorem used in the study of the foundations of quantum theory. We expect that the approach presented here can be generalised and will turn out to be applicable to a larger class of global optimisation problems. We also present some preliminary numerical results, showing that, at least for small problem sizes, the present approach is practically realisable.
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On Strong Superadditivity of the Entanglement of Formation
We employ a basic formalism from convex analysis to show a simple relation between the entanglement of formation $E_F$ and the conjugate function $E^*$ of the entanglement function $E(ρ)=S(\trace_Aρ)$. We then consider the conjectured strong superadditivity of the entanglement of formation $E_F(ρ) \ge E_F(ρ_I)+E_F(ρ_{II})$, where $ρ_I$ and $ρ_{II}$ are the reductions of $ρ$ to the different Hilbert space copies, and prove that it is equivalent with subadditivity of $E^*$. As an application, we show that strong superadditivity would follow from multiplicativity of the maximal channel output purity for all non-trace-preserving quantum channels, when purity is measured by Schatten $p$-norms for $p$ tending to 1.
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