Restricted Numerical Range and some its applications

Restricted Numerical Range and some its applications Karol Życzkowski in collaboration with P. Gawron, J. Miszczak, Z. Pucha la (Gliwice), L. Skowronek (Kraków), M.D. Choi (Toronto), C. Dunkl (Virginia) and J. Holbrook (Guelph) Jagiellonian University, Cracow, & Academy of Sciences, Warsaw see preprint arxiv: 0905.3646 10-th WONRA, Cracow, June 28-29, 2010 KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 1 / 33

Numerical Range (Field of Values) Definition For any operator A acting on H N one defines its NUMERICAL RANGE as a subset of the complex plane defined by: Λ(A) = { x A x : x H N, x x = 1}. (1) In physics: Rayleigh quotient, R(A) := x A x / x x Hermitian case For any hermitian operator A = A with spectrum λ 1 λ 2 λ N its numerical range forms an interval: the set of all possible expectation values of the observable A among arbitrary pure states, Λ(A) = [λ 1, λ N ]. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 2 / 33

Numerical range and its properties Compactness Λ(A) is a compact subset of C. Convexity: Hausdorf-Toeplitz theorem - Λ(A) is a convex subset of C. Example Numerical range of a random (not hermitian) matrix of order N = 6 red dots represent its eigenvalues 2 1 2 1 1 2 e 1 2 KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 3 / 33

Numerical range for a normal matrix Convex hull of the spectrum & Numerical range Lemma. If A is normal i.e. A A = AA, then Λ(A) = Co(σ(A)). Proof. The eigen decomposition reads A φ 1 = z i φ 1 and φ i φ j = δ ij. Take convex combination λ of arbitrary two eigenvalues, λ = az 1 + (1 a)z 2 and the superposition ψ := a φ 1 + 1 a φ 2. Then ψ A ψ = az 1 + (1 a)z 2 = λ hence λ Λ(A). Example Numerical range of a random normal matrix for N = 6 0.4 0.2 0.5 0.5 1.0 0.2 0.4 0.6 KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 4 / 33

Generalization of NUMERICAL RANGE Restricted Numerical Range defined by restricting the set of states to a given subset Ω X Ω of the set Ω of all normalized states, Λ X (A) = { ψ A ψ : ψ Ω X }, Physically motivated Examples of restricted numerical range: a) Real States, Ω R = { ψ R N } leads to Numerical range restricted to Real states Notions for composite Hilbert space, e.g. H MN = H M H N. b) Product States, Ω P = { ψ 1 ψ 2 } leads to Product (local) Numerical range c) Maximally Entangled states, Ω E = {U 1 U 2 ψ + } with ψ + = 1 N N i=1 i i and local unitaries U 1, U 2 U(N) leads to Entangled Numerical range d) Separable (mixed) states, (belonging to the convex hull of all product states) leads to Separable Numerical range These restrictions can be combined, so one can work e.g. with Real, entangled Numerical range KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 5 / 33

Cognate notions Restricted Numerical Radius defined by restricting the set of states to a given subset Ω X Ω of the set Ω of all normalized states. r X (A) = max{ z : z Λ X (A)} As before, physical applications motivate us to study several examples of restricted numerical radius a) Real States, Ω R = { ψ R N } leads to Numerical radius restricted to Real states Notions for composite Hilbert space, e.g. H MN = H M H N. b) Product States, Ω P = { ψ 1 ψ 2 } leads to Product (local) Numerical radius c) Maximally Entangled states, Ω E = {U 1 U 2 ψ + } with ψ + = 1 N N i=1 i i and local unitaries U 1, U 2 U(N) leads to Numerical Radius with respect to Entangled States d) Separable (mixed) states, (belonging to the convex hull of all product states) leads to Separable Numerical radius KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 6 / 33

Key message: Numerical range (and its generalizations) is a a) nice algebraic tool b) and usefull also in theoretical physics! Consider your research on generalized numerical ranges and their properties! KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 7 / 33

Product (local) numerical range (PNR) Definition Product numerical range of an operator A acting on H N H M Λ (A) = { ψ φ A ψ φ : ψ C N, φ C M}, Product numerical range forms a subset of numerical range Product numerical range of a random normal matrix Product numerical range of a random matrix KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 8 / 33

Useful lemma on dimensionality of subspaces Lemma Consider a composed complex Hilbert space H n = H k H m. Then 1 Any subspace S d H n of dimension d = (k 1)(m 1) + 1 contains at least one product state, 2 There exists a subspace of dimension d 1 = (k 1)(m 1) which does not contain any product states. can be proved by dimension counting... Wallach 2002; Parthasarathy 2004; Cubitt, Montanaro and Winter, 2008. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 9 / 33

Product numerical range for Hermitian operators Definition By minimal (maximal) local values we call: λ min loc (A) = min (Λ (A)), λ max loc (A) = max (Λ (A)). Convexity for hermitian operators For any Hermitian A its local numerical range is convex and forms an interval of the real axis, Λ (A) = [λ min loc (A), λmax loc (A)]. Proposition for 2 2 problem Local numerical range includes the central segment of the spectrum, Λ 2 (A) = [λ 2, λ 3 ] Λ (A) Λ(A). KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 10 / 33

2 2 example: two qubit density matrix ρ ρ(α) := U α DU α, with the spectrum D = diag{0, 1/6, 2/6, 3/6}, and a non local unitary matrix U α of eigenvectors U α = exp(iα σ x σ x ). Figure: Spectrum and product numerical range (grey) for the family of states ρ = ρ(α) KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 11 / 33

Example: one qubit map Ψ One qubit map Ψ : ρ Ψ(ρ) can be represented by the dynamical matrix, (Choi matrix) D Ψ = (Ψ 1) ψ + ψ + where ψ + = 1 2 ( 0, 0 + 1, 1 ), D Ψ = 1 2 a 0 0 1 ā 2 b 0 1 0 b 2 c 1 0 0 c 2 where a, b C and c = xa for some x R. Checking positivity of polynomials ( Skowronek, Życzkowski, J. Phys. A, (2009) we find [ 1 Λ (D Ψ ) = 2 M, 1 ] 2 + M, where M = 1 4, ( ) a + c + a c 2 + b 2. Jamio lkowski theorem: a map Ψ is positive iff D Ψ is block positive, x, y D Ψ x, y 0 λ loc min (D Ψ) 0 Conlusion: The map Ψ(a, b, c) is positive if M(a, b, c) 1/2. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 12 / 33

Product numerical range for Hermitian opertors 2 m problem Product numerical range includes the central segment of the spectrum, Λ m (A) = [λ m, λ m+1 ] Λ (A). 3 3 problem Product numerical range includes the central eigenvalue, λ 5 Λ (A). Moreover, this statement is optimal since there exist A = A acting on H 9 such that λ 4 / Λ (A). General N M problem Product numerical ranges contains barycenter of the spectrum, TrA/NM Λ (A). (this property holds also for an arbitrary non-hermitian operator A) KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 13 / 33

Product numerical range for non-hermitian operators PNR forms a connected set in the complex plane. This is true since product numerical range is a continuous image of a connected set. Product numerical range needs not to be convex! Two qubit example ( 1 0 X = 0 0 ) ( 1 0 0 0 Local numerical range Λ (X) is not convex, ) ( 0 0 + i 0 1 ) ( 0 0 0 1 Λ (X) = { x + yi : 0 x, 0 y, x + y 1 }. ). 1.0 0.8 loc Xs,t Xs,t 0.6 Im 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 Re KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 14 / 33

PNR for a tensor product of operators Minkowski sum of two sets in complex plane Z 1 Z 2 := {z : z = z 1 + z 2, z 1 Z 1, z 2 Z 2 }. Minkowski product of two sets in complex plane Z 1 Z 2 := {z : z = z 1 z 2, z 1 Z 1, z 2 Z 2 }. product rule Product numerical range of the Kronecker product (tensor product) of arbitrary operators is equal to the Minkowski product of the numerical ranges of both factors, Λ (A B) = Λ(A) Λ(B), This statement can be generalised for multipartite system. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 15 / 33

Two qubit example Disks in complex plane ( ) c 2r Numerical range of X = forms a disk of radius r centered at c. 0 c Consider then a family of tensor products ( 1 2r1 Y (r 1, r 2 ) := X 1 X 2 = 0 1 ) ( 1 2r2 0 1 Product numercial range of Y is equal to the Minkowski product of two discs, ). One gets: a) cardioid for (r 1, r 2 ) = (1, 1), b) limaçon of Pascal for (0.7, 1), and c) Cartesian oval for (0.5, 1.2) in consistency with Farouki and Pottmann 02. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 16 / 33

Separable numerical range Product numerical range: Λ (A) := {TrσA : σ = ψ φ ψ φ } Separable numerical range of A Λ sep (A) := { Tr σa : σ is separable, σ = i p i ρ A i ρ B i }, Lemma. Λ sep (A) = conv[λ (A)] 0.5 0.5 Im 0.0 Im 0.0 0.5 0.5 1.0 1.0 0.5 0.0 0.5 0 1.0 1.0 0.5 0.0 0.5 0 Numerical range (light gray), separable numerical range (dark gray) and product numerical range (black) for two exemplary unitary matrices of size N = 4 with spectrum (1, i, 1, i). KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 17 / 33

Multipartite operators The notion of product numerical range can be extended for spaces with the multipartite structure described in a Hilbert space formed by an m fold tensor product, H m = H k1 H km, with N = K 1... K m. Then product numerical range of an operator A acting on H m reads Λ (A) = { ψ A ψ : where ψ = φ 1 φ 2 φ m } (i.e. the set of expectation values among product states) KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 18 / 33

Example (three qubits: PNR not simply connected, Schülte-Herbrueggen, Dirr, Helmke and Glaser 2008) Consider a unitary operator acting on H 2 H 2 H 2 diagonal in the computational basis { 0, 0, 0,..., 1, 1, 1, A 2 = diag(1, i, i, i, i, i, i, 1). Figure: Product numerical range of A 2 with genus g = 1. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 19 / 33

PNR for a tensor power of one qubit operator Example (Multi qubit system) Consider a diagonal unitary matrix U of size two, [ ] 1 0 U := 0 e iφ Figure: Product numerical range of U n for n = 1, 2, 3, 4. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 20 / 33

Figure: Product numerical range of U n for n = 5, 6, 7, 8. Note that for n = 7 genus of Λ (U n ) equals 2. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 21 / 33

Figure: Product numerical range of U 7 contains two holes. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 22 / 33

Multipartite case: Genus 2 example Product numerical range for a four-partite operator Let A be a unitary operator of order 16, diagonal in the the product basis A = diag ( ) e iπ 4, i, i, e 3iπ 4, 1, e 3iπ 4, e iπ 4, 1, 1, e iπ 4, e 3iπ 4, 1, e 3iπ 4, i, i, e iπ 4 0.5 Im 0.0 0.5 1.0 1.0 0.5 0.0 0.5 0 Product Numerical Range Λ (A) analyzed numerically by taking a sample of random states from H2 4 Observe non uniform density of points in the set Λ (A). What is the probability distribution P A (z)? KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 23 / 33

Numerical range & quantum states: Set of pure states of a finite size N: Let Ω N = CP N 1 denotes the set of all pure states in H N. Example: for N = 2 one obtains Bloch sphere, Ω 2 = CP 1. Probability distributions on complex plane supported by Λ(A) - Fix an arbitrary operator A of size N, - Generate random pure states ψ of size N (with respect to unitary invariant, Fubini Study measure) and analyze their contribution ψ A ψ to the numerical range on the complex plane... Probability distribution: (a numerical shadow ) P A (z) := is supported on Λ(A). How it looks like? Ω N dψ δ ( z ψ A ψ ) KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 24 / 33

Normal case: a shadow of the classical simplex... Normal matrices of size N = 2 with spectrum {λ 1, λ 2 } Distribution P A (z) covers uniformly the interval [λ 1, λ 2 ] on the complex plane, Examples for diagonal matrices A of size two and three 1.0 Density plot of numerical range of operator [1, 0; 0, i] 1.0 Density plot of numerical range of operator diag(1, i, i) 0.5 0.5 0.0 0.0 0.5 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 1.0 0.5 0.0 0.5 1.0 Normal matrices of order N = 3 with spectrum {λ 1, λ 2, λ 3 } Distribution P A (z) covers uniformly the triangle (λ 1, λ 2, λ 3 ) on the complex plane. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 25 / 33

Normal case: a shadow of the classical simplex... Normal matrices of size N = 4 with spectrum {λ 1, λ 2, λ 3, λ 4 } Distribution P A (z) covers the quadrangle (λ 1, λ 2, λ 3, λ 4 ) with the density determined by projection of the regular tetrahedron (3 simplex) on the complex plane 1.0 Density plot of numerical range of operator diag(1, exp(iπ/3), exp(i2π/3), 1) 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 Examples for diagonal matrices A of size four. Observe shadow of the edges of the tetrahedron = the set of all classical mixed states for N = 4. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 26 / 33

Normal case: a shadow of the 4-D simplex... Normal matrices of size N = 5 with spectrum (λ 1, λ 2, λ 3, λ 4, λ 5 ) Distribution P A (z) covers the pentagon {exp(2πj/5)}, j = 0,..., 4 with the density determined by projection of the 4-D simplex on the complex plane KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 27 / 33

General case for N = 2 Not normal matrices of size N = 2 Distribution P A (z) covers entire (eliptical) disk on the complex plane with non-uniform (!) density determined by projection of (empty!) Bloch sphere, S 2 = CP 1 on the complex plane. 1.0 Density plot of numerical range of operator [0, 1; 0, 1/2] 2.0 Density plot of numerical range of operator [0, 1; 0, 1] 1.5 0.5 1.0 0.5 0.0 0.0 0.5 0.5 1.0 1.5 1.0 1.0 0.5 0.0 0.5 1.0 2.0 2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0 Examples for non-normal matrices A of size two. Note the shadow of the projective space CP 1 = S 2 = set of quantum pure states for N = 2. KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 28 / 33

How complex / real projective space looks like? Not normal matrices of size N = 3 Distribution PA (z) covers the numerical range Λ(A) on the complex plane with a non-uniform density Density plot of numerical range of operator O5 subject to complex states Density plot of numerical range of operator O5 subject to real states 0.5 0.5 im 1.0 im 1.0 0.0 0.0 0.5 0.5 1.0 0.5 re 0.0 1.0 0.5 0.5 shadow of the projective space ΩN = CP (left) and N = 3 non-normal matrix A = [0, 1, 1; 0, i, 1; 0, 0, 1] 2 KZ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory 0.0 re 0.5 RP (right) for a 2 June 28-29, 2010 29 / 33

more about the set of quantum states for larger N Geometry of Quantum States an Introduction to Quantum Entanglement I. Bengtsson and K. Życzkowski Cambridge University Press, 2006 (hardcover) Cambridge, 2008 (improved version in paperback) KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 30 / 33

Numerical range & quantum entanglement: Shadow with respect to a restricted set Ω R of quantum states Let Ω Sep (Ω Ent ) denotes the set of all separable (entangled) pure states in H N H N. For an arbitrary operator A of size N 2 one defines its numerical shadow with respect to separable (entangled) states P S A(z) := Ω Sep dψ δ ( z ψ A ψ ) The distribution P S A (z) is supported on Λ (A). How does it look? KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 31 / 33

Concluding Remarks Standard Numerical Range (NR) is a useful algebraic tool... We advocate to study its generalisation - the Restricted NR: a) product numerical range (PNR) useful for composite systems. b) separable numerical range (SNR) PNR needs not to be convex or simply connected. Some bounds for product numerical range are obtained, but we do not know how to find it for an arbitrary operator! For a product operator the PNR is given by the Minkowski product of numerical ranges of all the factors. PNR is a versatile tool: it can be used to analyze positivity of quantum maps, quantum entanglement, local fidelity, local distinguishability and other problems in quantum theory of composite systems. We introduce a concept of numerical shadow of an operator. If A is normal one obtains images of the set of classical mixed states, and the set of quantum pure states plays the role for the generic, non-normal A. Numerical shadow restricted to separable/entangled states could be useful for investigations of the geometry of quantum entanglement! KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 32 / 33

Thanks for your interest in our tool! If you want to know more about this product please ask your local dealer (or consult our preprint arxiv 0905.3646...) KŻ (IF UJ/CFT PAN ) Numerical Range & Quantum Theory June 28-29, 2010 33 / 33