We investigate an original family of quantum distinguishability problems, where the goal is to perfectly distinguish between *M* quantum states that become identical under a completely decohering map. Similarly, we study distinguishability of *M* quantum channels that cannot be distinguished when one is restricted to decohered input and output states. The studied problems arise naturally in the presence of a superselection rule, allow one to quantify the amount of information that can be encoded in phase degrees of freedom (coherences), and are related to time-energy uncertainty relation. We present a collection of results on both necessary and sufficient conditions for the existence of *M* perfectly distinguishable states (channels) that are classically indistinguishable.

We adopt the perspective of similarity equivalence, in gate set tomography called the gauge, to analyze various properties of quantum operations belonging to a semigroup, Φ=*e**^{Lt}*,and therefore given through the Lindblad operator. We first observe that the non unital part of the channel decouples from the time evolution. Focusing on unital operations we restrict our attention to the single-qubit case, showing that the semigroup embedded inside the tetrahedron of Pauli channels is bounded by the surface composed of product probability vectors and includes the identity map together with the maximally depolarizing channel. Consequently, every member of the Pauli semigroup is unitarily equivalent to a unistochastic map, describing a coupling with one-qubit environment initially in the maximally mixed state, determined by a unitary matrix of order four.

In this work we analyze properties of generic quantum channels in the case of large system size. We use the random matrix theory and free probability to show that the distance between two independent random channels tends to a constant value as the dimension of the system grows larger. As a measure of the distance we use the diamond norm. In the case of a flat Hilbert-Schmidt distribution on quantum channels, we obtain that the distance converges to 1/2*+2/π*. Furthermore, we show that for a random state *ρ* acting on a bipartite Hilbert space $H_A \otimes H_B$, sampled from the Hilbert-Schmidt distribution, the reduced states $Tr_A\rho$ and $Tr_B \rho$ are arbitrarily close to the maximally mixed state. This implies that, for large dimensions, the state *ρ* may be interpreted as a Jamio{\l}kowski state of a unital map.

Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel $\Phi$, i.e., we look for channels $\Phi^{\mathcal{C}}$ that induce the same classical transitions $T$, but are "more coherent". To quantify the coherence of a channel $\Phi$ we measure the coherence of the corresponding Jamio{\l}kowski state $J_{\Phi}$. We show that the classical transition matrix $T$ can be coherified to reversible unitary dynamics if and only if $T$ is unistochastic. Otherwise the Jamio{\l}kowski state $J_\Phi^{\mathcal{C}}$ of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To asses the extent to which an optimal process $\Phi^{\mathcal{C}}$ is indeterministic we find explicit bounds on the entropy and purity of $J_\Phi^{\mathcal{C}}$, and relate the latter to the unitarity of $\Phi^{\mathcal{C}}$. We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel $\Phi$ and reduces its rank (the minimal number of required Kraus operators) from $d^2$ to $d$.

}, doi = {10.1088/1367-2630/aaaff3}, url = {https://doi.org/10.1088/1367-2630/aaaff3}, author = {Kamil Korzekwa and Stanis{\l}aw Czach{\'o}rski and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {2606, title = {Gauge invariant information concerning quantum channels}, journal = {Quantum}, volume = {2}, year = {2018}, pages = {60}, abstract = {Motivated by the gate set tomography we study quantum channels from the perspective of information which is invariant with respect to the gauge realized through similarity of matrices representing channel superoperators. We thus use the complex spectrum of the superoperator to provide necessary conditions relevant for complete positivity of qubit channels and to express various metrics such as average gate fidelity.

}, doi = {10.22331/q-2018-04-11-60}, url = {https://doi.org/10.22331/q-2018-04-11-60}, author = {{\L}ukasz Rudnicki and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {2621, title = {Majorization uncertainty relations for mixed quantum states}, journal = {Journal of Physics A: Mathematical and Theoretical}, volume = {51}, year = {2018}, chapter = {175306}, abstract = {Majorization uncertainty relations are generalized for an arbitrary mixed quantum state *ρ* of a finite size *N*. In particular, a lower bound for the sum of two entropies characterizing probability distributions corresponding to measurements with respect to arbitrary two orthogonal bases is derived in terms of the spectrum of *ρ* and the entries of a unitary matrix *U* relating both bases. The obtained results can also be formulated for two measurements performed on a single subsystem of a bipartite system described by a pure state, and consequently expressed as uncertainty relation for the sum of conditional entropies.

We investigate the impact of a local random unitary noise on multipartite quantum states of arbitrary dimension. We follow the dynamical approach, in which the single-particle unitaries are generated by local random Hamiltonians. Assuming short evolution time we derive an upper bound on the fidelity between an initial and the final state transformed by this type of noise. This result is based on averaging the Tamm-Mandelstam bound and holds for a wide class of distributions of random Hamiltonians fulfilling specific symmetry conditions. It is showed that the sensitivity of a given pure quantum state to the discussed type of noise depends only on the properties of a single-particle and bipartite reduced states.

}, doi = {10.1103/PhysRevA.95.032333}, url = {https://doi.org/10.1103/PhysRevA.95.032333}, author = {Marcin Markiewicz and Zbigniew Pucha{\l}a and Anna de Rosier and Wies{\l}aw Laskowski and Karol {\.Z}yczkowski} } @article {iitisid_0689, title = {Asymptotic entropic uncertainty relations}, journal = {J. Math. Phys. }, volume = {57}, year = {2016}, note = {arXiv:1412.7065v1}, chapter = {032204}, abstract = {Entropic uncertainty relations are analyzed for the case of N-dimensional Hilbert space and two orthogonal measurements performed in two generic bases, related by a Haar random unitary matrix U. We derive estimations for the average norms of truncations of U of a given size, which allow us to study state-independent lower bounds for the sum of two entropies describing the measurements outcomes. In particular, we show that the Maassen{\textendash}Uffink bound asymptotically behaves as lnN-lnlnN-ln2, while the strong entropic majorization relation yields a nearly optimal bound, lnN-const. Analogous results are also obtained for a more general case of several orthogonal measurements performed in generic bases.

}, doi = {10.1063/1.4944425}, url = {http://dx.doi.org/10.1063/1.4944425}, author = {Rados{\l}aw Adamczak and Rafa{\l} Lata{\l}a and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0734, title = {Distinguishability of generic quantum states}, journal = {Physical Review A}, volume = {93}, year = {2016}, note = {arXiv:1507.05123}, chapter = {062112}, abstract = {Properties of random mixed states of order N distributed uniformly with respect to the Hilbert-Schmidt measure are investigated. We show that for large N, due to the concentration of measure, the trace distance between two random states tends to a fixed number D\ =1/4+1/π, which yields the Helstrom bound on their distinguishability. To arrive at this result we apply free random calculus and derive the symmetrized Marchenko{\textendash}Pastur distribution, which is shown to describe numerical data for the model of two coupled quantum kicked tops. Asymptotic values for the fidelity, Bures and transmission distances between two random states are obtained. Analogous results for quantum relative entropy and Chernoff quantity provide other bounds on the distinguishablity of both states in a multiple measurement setup due to the quantum Sanov theorem.

}, doi = {10.1103/PhysRevA.93.062112}, url = {https://doi.org/10.1103/PhysRevA.93.062112}, author = {Zbigniew Pucha{\l}a and {\L}ukasz Pawela and Karol {\.Z}yczkowski} } @article {iitisid_0733, title = {Certainty relations, mutual entanglement and non-displacable manifolds}, journal = {Phys. Rev. A}, volume = {92}, year = {2015}, pages = {032109}, abstract = {{We derive explicit bounds for the average entropy characterizing measurements of a pure quantum state of size N in L orthogonal bases. Lower bounds lead to novel entropic uncertainty relations, while upper bounds allow us to formulate universal certainty relations. For L=2 the maximal average entropy saturates at logN as there exists a mutually coherent state, but certainty relations are shown to be nontrivial for L>=3 measurements. In the case of a prime power dimension

}, doi = {10.1103/PhysRevA.92.032109}, url = {https://doi.org/10.1103/PhysRevA.92.032109}, author = {Zbigniew Pucha{\l}a and {\L}. Rudnicki and K. Chabuda and M. Paraniak and Karol {\.Z}yczkowski} } @article {iitisid_0732, title = {Minimal Renyi-Ingarden-Urbanik entropy of multipartite quantum states}, journal = {Entropy}, volume = {17}, number = {7}, year = {2015}, pages = {5063{\textendash}5084}, abstract = {We study the entanglement of a pure state of a composite quantum system consisting of several subsystems with d levels each. It can be described by the R{\textquoteright}enyi-Ingarden-Urbanik entropy Sq of a decomposition of the state in a product basis, minimized over all local unitary transformations. In the case q=0 this quantity becomes a function of the rank of the tensor representing the state, while in the limit q{\textrightarrow}$\infty$ the entropy becomes related to the overlap with the closest separable state and the geometric measure of entanglement. For any bipartite system the entropy S1 coincides with the standard entanglement entropy. We analyze the distribution of the minimal entropy for random states of three and four-qubit systems. In the former case the distributions of 3-tangle is studied and some of its moments are evaluated, while in the latter case we analyze the distribution of the hyperdeterminant. The behavior of the maximum overlap of a three-qudit system with the closest separable state is also investigated in the asymptotic limit.

}, doi = {10.3390/e17075063}, url = {https://doi.org/10.3390/e17075063}, author = {M. Enr{\'\i}quez and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0686, title = {Real numerical shadow and generalized B-splines}, journal = {Linear Algebra and its Applications}, volume = {479}, year = {2015}, note = {arXiv:1409.4941}, pages = {12{\textendash}51}, abstract = {arXiv:1409.4941

}, doi = {10.1016/j.laa.2015.03.029}, url = {https://doi.org/10.1016/j.laa.2015.03.029}, author = {Charles F. Dunkl and Piotr Gawron and {\L}ukasz Pawela and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {2424, title = {Wybory samorz{\k a}dowe 2014 w poszukiwaniu anomalii statystycznych}, journal = {Electoral Studies}, volume = {30}, year = {2015}, pages = {534{\textendash}545}, author = {Piotr Gawron and {\L}ukasz Pawela and Zbigniew Pucha{\l}a and Szklarski, Jacek and Karol {\.Z}yczkowski} } @article {iitisid_0593, title = {Constructive entanglement test from triangle inequality}, journal = {J. Phys. A: Math. Theor.}, volume = {47}, number = {42}, year = {2014}, note = {arXiv:1211.2306}, pages = {424035}, abstract = {We derive a simple lower bound on the geometric measure of entanglement for mixed quantum states in the case of a general multipartite system. The main ingredient of the presented derivation is the triangle inequality applied to the root infidelity distance in the space of density matrices. The obtained bound leads to entanglement criteria with a straightforward interpretation. The proposed criteria provide an experimentally accessible, powerful entanglement test. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to {\textquoteright}50 years of Bell{\textquoteright}s theorem{\textquoteright}.

}, doi = {10.1088/1751-8113/47/42/424035}, url = {https://doi.org/10.1088/1751-8113/47/42/424035}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} } @article {iitisid_0682, title = {Diagonal unitary entangling gates and contradiagonal quantum states}, journal = {Phys. Rev. A}, volume = {90}, year = {2014}, note = {arXiv:1407.1169 doi:10.1103/PhysRevA.90.032303}, pages = {032303}, abstract = {Nonlocal properties of an ensemble of diagonal random unitary matrices of order N2 are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as logN, in contrast to the logN2 behavior characteristic to random unitary gates. Entangling power of a diagonal gate U is related to the von Neumann entropy of an auxiliary quantum state ρ=AA{\textdagger}/N2, where the square matrix A is obtained by reshaping the vector of diagonal elements of U of length N2 into a square matrix of order N. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices A, with all entries of the same modulus and random phases and the ensemble of quantum states ρ, such that all their diagonal entries are equal to 1/N. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the

}, doi = {10.1103/PhysRevA.90.032303}, url = {https://doi.org/10.1103/PhysRevA.90.032303}, author = {A. Lakshminarayan and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0634, title = {Numerical range for random matrices}, journal = {Journal of Mathematical Analysis and Applications}, volume = {418}, number = {1}, year = {2014}, note = {arXiv:1309.6203}, pages = {516}, author = {Beno{\^\i}t Collins and Piotr Gawron and Alexander E. Litvak and Karol {\.Z}yczkowski} } @article {iitisid_0656, title = {Strong Majorization Entropic Uncertainty Relations}, journal = {Phys. Rev. A}, volume = {89}, number = {5}, year = {2014}, note = {arXiv:1402.0129}, pages = {052115}, abstract = {We analyze entropic uncertainty relations in a finite dimensional Hilbert space and derive several strong bounds for the sum of two entropies obtained in projective measurements with respect to any two orthogonal bases. We improve the recent bounds by Coles and Piani [ArXiv:1307.4265], which are known to be stronger than the well known result of Maassen and Uffink. Furthermore, we find a novel bound based on majorization techniques, which also happens to be stronger than the recent results involving largest singular values of submatrices of the unitary matrix connecting both bases. The firsts set of new bounds give better results for unitary matrices close to the Fourier matrix, while the second one works better in the opposite sectors. Some results derived admit generalization to arbitrary mixed states and the bounds are increased by the von Neumann entropy of the measured state

}, doi = {10.1103/PhysRevA.89.052115}, url = {https://doi.org/10.1103/PhysRevA.89.052115}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0562, title = {Entropic trade-off relations for quantum operations}, journal = {Phys. Rev. A}, volume = {87}, number = {3}, year = {2013}, note = {arXiv:1206.2536 doi: 10.1103/PhysRevA.87.032308}, pages = {032308}, abstract = {Spectral properties of an arbitrary matrix can be characterized by the entropy of its rescaled singular values. Any quantum operation can be described by the associated dynamical matrix or by the corresponding superoperator. The entropy of the dynamical matrix describes the degree of decoherence introduced by the map, while the entropy of the superoperator characterizes the a priori knowledge of the receiver of the outcome of a quantum channel Φ. We prove that for any map acting on an N-dimensional quantum system the sum of both entropies is not smaller than lnN. For any bistochastic map this lower bound reads 2lnN. We investigate also the corresponding R{\'e}nyi entropies, providing an upper bound for their sum, and analyze the entanglement of the bi-partite quantum state associated with the channel.}, author = {Wojciech Roga and Zbigniew Pucha{\l}a and {\L}. Rudnicki and Karol {\.Z}yczkowski} } @article {iitisid_0617, title = {Majorization entropic uncertainty relations}, journal = {J. Phys. A: Math. Theor.}, volume = {46}, year = {2013}, note = {arXiv:1304.7755}, pages = {272002}, abstract = {Entropic uncertainty relations in a finite dimensional Hilbert space are investigated. Making use of the majorization technique we derive explicit lower bounds for the sum of Renyi entropies describing probability distributions associated with a given pure state expanded in eigenbases of two observables. The bounds obtained are expressed in terms of the largest singular values of submatrices of the unitary rotation matrix. For a generic unitary matrix of size N = 5 the bound obtained is stronger than the one of Maassen and Uffink (MU) with probability larger than 98\%, and this ratio increases with N. We show also that the bounds investigated are invariant for unitary matrices equivalent up to dephasing and permutation and derive a classical analogue of the MU uncertainty relation formulated for stochastic transition matrices.}, author = {Zbigniew Pucha{\l}a and {\L}. Rudnicki and Karol {\.Z}yczkowski} } @article {iitisid_0589, title = {Collectibility for Mixed Quantum States}, journal = {Phys. Rev. A}, volume = {86}, number = {6}, year = {2012}, note = {arXiv:1211.0573 doi:10.1103/PhysRevA.86.062329}, pages = {062329}, abstract = {Bounds analogous to entropic uncertainty relations allow one to design practical tests to detect quantum entanglement by a collective measurement performed on several copies of the state analyzed. This approach, initially worked out for pure states only [ Phys. Rev. Lett. 107 150502 (2011)], is extended here for mixed quantum states. We define collectibility for any mixed states of a multipartite system. Deriving bounds for collectibility for positive partially transposed states of given purity provides insight into the structure of entangled quantum states. In the case of two qubits the application of complementary measurements and coincidence based detections leads to a test of entanglement of pseudopure states.}, author = {{\L}. Rudnicki and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and Karol {\.Z}yczkowski} } @article {iitisid_0536, title = {Restricted numerical shadow and geometry of quantum entanglement}, journal = {J. Phys. A: Math. Theor.}, volume = {45}, year = {2012}, note = {arXiv:1201.2524}, pages = {415309}, abstract = {The restricted numerical range W\_R(A) of an operator A acting on a D-dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset R of the of set of pure quantum states of dimension D. One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of A {\textendash} a normalized probability distribution on the complex plane supported in W\_R(A). Its value at point z in C is equal to the probability that the expectation value is equal to z, where |psi> represents a random quantum state in subset R distributed according to the natural measure on this set, induced by the unitarily invariant Fubini{\textendash}Study measure. Studying restricted shadows of operators of a fixed size D=N\_A N\_B we analyse the geometry of sets of separable and maximally entangled states of the N\_A x N\_B composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on D=2^3 dimensional Hilbert space allows us to investigate the structure of the orbits of GHZ and W quantum states of a three{\textendash}qubit system.}, author = {Zbigniew Pucha{\l}a and J.A. Miszczak and Piotr Gawron and Charles F. Dunkl and J.A. Holbrook and Karol {\.Z}yczkowski} } @article {iitisid_0509, title = {Numerical shadow and geometry of quantum states}, journal = {J. Phys. A: Math. Theor.}, volume = {44}, number = {33}, year = {2011}, note = {arXiv:1104.2760 IF=1.641(2010);}, pages = {335301}, abstract = {The totality of normalised density matrices of order N forms a convex set Q\_N in R^(N^2-1). Working with the flat geometry induced by the Hilbert-Schmidt distance we consider images of orthogonal projections of Q\_N onto a two-plane and show that they are similar to the numerical ranges of matrices of order N. For a matrix A of a order N one defines its numerical shadow as a probability distribution supported on its numerical range W(A), induced by the unitarily invariant Fubini-Study measure on the complex projective manifold CP^(N-1). We define generalized, mixed-states shadows of A and demonstrate their usefulness to analyse the structure of the set of quantum states and unitary dynamics therein.}, issn = {1751-8113}, author = {Charles F. Dunkl and Piotr Gawron and J.A. Holbrook and J.A. Miszczak and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0504, title = {Numerical shadows: measures and densities on the numerical range}, journal = {Linear Algebra Appl.}, volume = {434}, year = {2011}, note = {arXiv:1010.4189 IF=1.005(2010);}, pages = {2042{\textendash}2080}, abstract = {For any operator M acting on an N-dimensional Hilbert space H\_N we introduce its numerical shadow, which is a probability measure on the complex plane supported by the numerical range of M. The shadow of M at point z is defined as the probability that the inner product (Mu,u) is equal to z, where u stands for a random complex vector from H\_N, satisfying ||u||=1. In the case of N=2 the numerical shadow of a non-normal operator can be interpreted as a shadow of a hollow sphere projected on a plane. A similar interpretation is provided also for higher dimensions. For a hermitian M its numerical shadow forms a probability distribution on the real axis which is shown to be a one dimensional B-spline. In the case of a normal M the numerical shadow corresponds to a shadow of a transparent solid simplex in R^{N-1} onto the complex plane. Numerical shadow is found explicitly for Jordan matrices J\_N, direct sums of matrices and in all cases where the shadow is rotation invariant. Results concerning the moments of shadow measures play an important role. A general technique to study numerical shadow via the Cartesian decomposition is described, and a link of the numerical shadow of an operator to its higher-rank numerical range is emphasized.}, issn = {0024-3795}, author = {Charles F. Dunkl and Piotr Gawron and J.A. Holbrook and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} } @article {iitisid_0473, title = {Product numerical range in a space with tensor product structure}, journal = {Linear Algebra Appl.}, volume = {434}, number = {1}, year = {2011}, note = {arXiv:1008.3482 IF=1.005(2010);}, pages = {327{\textendash}342}, abstract = {We study operators acting on a tensor product Hilbert space and investigate their product numerical range, product numerical radius and separable numerical range. Concrete bounds for the product numerical range for Hermitian operators are derived. Product numerical range of a non-Hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum. While the latter set is convex, the product range needs not to be convex nor simply connected. The product numerical range of a tensor product is equal to the Minkowski product of numerical ranges of individual factors.}, issn = {0024-3795}, author = {Zbigniew Pucha{\l}a and Piotr Gawron and J.A. Miszczak and {\L}. Skowronek and Man-Duen Choi and Karol {\.Z}yczkowski} } @article {iitisid_0476, title = {Restricted numerical range: A versatile tool in the theory of quantum information}, journal = {J. Math. Phys.}, volume = {51}, number = {10}, year = {2010}, note = {arXiv:0905.3646 IF=1.291(2010);}, pages = {102204}, issn = {00222488}, author = {Piotr Gawron and Zbigniew Pucha{\l}a and J.A. Miszczak and {\L}. Skowronek and Karol {\.Z}yczkowski} } @article {iitisid_0421, title = {Sub- and super-fidelity as bounds for quantum fidelity}, journal = {Quantum Information \& Computation}, volume = {9}, number = {1\&2}, year = {2009}, note = {arXiv:0805.2037 IF=2.980(209); IF5=2.402(2010);}, month = {1}, pages = {0103{\textendash}0130}, abstract = {We derive several bounds on fidelity between quantum states. In particular we show that fidelity is bounded from above by a simple to compute quantity we call super{\textendash}fidelity. It is analogous to another quantity called sub{\textendash}fidelity. For any two states of a two{\textendash}dimensional quantum system ($N=2$) all three quantities coincide. We demonstrate that sub{\textendash} and super{\textendash}fidelity are concave functions. We also show that super{\textendash}fidelity is super{\textendash}multiplicative while sub{\textendash}fidelity is sub{\textendash}multiplicative and design feasible schemes to measure these quantities in an experiment. Super{\textendash}fidelity can be used to define a distance between quantum states. With respect to this metric the set of quantum states forms a part of a $N^2-1$ dimensional hypersphere.}, issn = {1533-7146}, author = {J.A. Miszczak and Zbigniew Pucha{\l}a and Pawe{\l} Horodecki and A. Uhlmann and Karol {\.Z}yczkowski} } @article {iitisid_0402, title = {Quantum state discrimination: A geometric approach}, journal = {Phys. Rev. A}, volume = {77}, number = {4}, year = {2008}, note = {arXiv:0711.4286 IF=2.908(2008);}, pages = {042111}, abstract = {We analyze the problem of finding sets of quantum states that can be deterministically discriminated. From a geometric point of view, this problem is equivalent to that of embedding a simplex of points whose distances are maximal with respect to the Bures distance (or trace distance). We derive upper and lower bounds for the trace distance and for the fidelity between two quantum states, which imply bounds for the Bures distance between the unitary orbits of both states. We thus show that, when analyzing minimal and maximal distances between states of fixed spectra, it is sufficient to consider diagonal states only. Hence when optimal discrimination is considered, given freedom up to unitary orbits, it is sufficient to consider diagonal states. This is illustrated geometrically in terms of Weyl chambers.}, issn = {1094-1622}, author = {D. Markham and J.A. Miszczak and Zbigniew Pucha{\l}a and Karol {\.Z}yczkowski} }