Dynamics of Quantum Entanglement & Separable Numerical ...

Dynamics of Quantum Entanglement 

& Separable Numerical Shadow 

Karol ˙Zyczkowski 

in collaboration with 

P. Gawron, J. Miszczak, Z. Pucha̷la (Gliwice), 

C. Dunkl (Virginia) and J. Holbrook (Guelph) 

IF UJ, Cracow 2012 

Monday, January 23 

K ˙Z (ChInka) Quantum entanglement & Numerical shadow 23.01.2012 1 / 32

Composed systems & entangled states 

bi-partite systems: H = HA ⊗ HB 

separable pure states: |ψ〉 = |φA〉 ⊗ |φB〉 

entangled pure states: all states not of the above product form. 

Two–qubit system: N = 2 × 2 = 4 

Maximally entangled Bell state |ϕ + 〉 := 1 

 

√ |00〉 + |11〉 

2 

Entanglement measures 

For any pure state |ψ〉 ∈ HA ⊗ HB define its partial trace σ = TrB|ψ〉〈ψ|. 

Definition: Entanglement entropy of |ψ〉 is equal to von Neuman entropy 

of the partial trace 

E(|ψ〉) := −Tr σ ln σ 

The more mixed partial trace, the more entangled initial pure state... 


Entanglement of mixed quantm states 

Mixed states 

separable mixed states: ρsep = 

j pjρ A j ⊗ ρB j (∗∗) 

entangled mixed states: all states not of the above product form. 

How to find, whether a given density matrix ρ 

can be written in the form (**) and is separable ? 

The separability problem is solved only for the simplest cases 

for 2 × 2 and 2 × 3 problems... 

Entanglement of formation for a mixed state ρ 

 

E(ρ) := min qiE(|ψj〉 

E 

where the minimum is taken over all ensembles 

E = {qj, |ψj〉} : 

j qj|ψj〉〈ψj| = ρ. 

j 


Two–qubit mixed states 

The maximal ball inscribed into M (4) of radius r4 = 1/ √ 12 centred at 

ρ∗ = /4 is separable ! 

thetrahedron of eigenvalues 

K. ˙Z, P.Horodecki, M.Lewenstein, A.Sanpera, 1998 


Two–qubit mixed states 

Degree of entanglement: a distance to the closest separable state 

K. ˙ Z, M. Ku´s, 2001 

(entanglement of formation) 


Discrete Time Evolution & Quantum Maps 

Quantum operation: linear, completely positive trace preserving map 

Enviromental form 

ρ ′ = Φ(ρ) = TrE [U (ρ ⊗ ωE ) U † ] . 

where ωE is an initial state of the environment while UU † = . 

Kraus form 

ρ ′ = Φ(ρ) = 

i AiρA † 

i , 

where the Kraus operators satisfy 

i A† 

i Ai = . 


A model discrete quantum dynamics 

a) unitary dynamics (rotation), ρ ′ = UρU † 

b) decoherence (contraction), ρ ′′ = k 

i Aiρ ′ A † 

i 

Two qubit model : N = 2 × 2 = 4 

a) free evolution: U = exp(itH) where H = σx ⊗ σy 

(non-local unitary dynamics !) 

variant b1) bistochastic channel: Φ(/N) = /N, 

One–qubit Pauli channel: k = 4, A1 = √ 1 − ɛ ⊗ , 

A2 = ɛ/3 ⊗ σx, A3 = ɛ/3 ⊗ σy , A4 = ɛ/3 ⊗ σz. 

variant b2) non bistochastic channel: 

One qubit amplitude damping channel, (decaying channel), k = 2, 

where A1 = ⊗ B1 and A2 = ⊗ B2 

with B1 = 

1 0 

0 √ p 

 

and B2 = 

0 √ 1 − p 

0 0 


Dynamics of entanglement 

Entanglement of formation E as a function of time tn 

for some initially pure states of a two–qubit system. 

revivals of entanglement 



Entanglement of formation E as a function of time tn 

for some initially pure states of a two–qubit system. 

sudden death of entanglement 

K. ˙Z, P.Horodecki, M.Horodecki, R.Horodecki, PRA 2001 

the name coined by Yau and Eberly, 

who independently reported this effect in 2003. 



A general, geometric picture 

The effects of 

a) unitary evolution (rotation of the body of mixed states) induces 

oscillations of entanglement (revivals!) 

b) non–unitary evolution (decoherence): an increase of the degree of 

mixing (von Neumann entropy) sends the trajectory into the separable 

central region of the set of mixed states. 


Operator theory: Numerical Range (Field of Values) 

Definition 

For any operator A acting on HN one defines its NUMERICAL RANGE 

(Wertevorrat) as a subset of the complex plane defined by: 

Hermitian case 

Λ(A) = {〈x|A|x〉 : |x〉 ∈ H N , 〈x|x〉 = 1}. (1) 

For any hermitian operator A = A † with spectrum λ1 ≤ λ2 ≤ · · · ≤ λN its 

numerical range forms an interval: the set of all possible expectation 

values of the observable A among arbitrary pure states, Λ(A) = [λ1, λN]. 


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 for random matrices of order N = 6 

a) normal, b) generic (non-normal) 

 

 

 

0.4 

0.2 

 

0.5 0.5 1.0 Re 

0.2 

0.4 

0.6 

Im 

 

 

 

 

 

2 1 1 2 

 

K ˙Z (ChInka) Quantum entanglement & Numerical shadow 23.01.2012 12 / 32 

2 

1 

1 

2 

Im 

 

 

Re

Numerical range & quantum states: 

Set of pure states of a finite size N: 

Let ΩN = P N−1 denotes the set of all pure states in HN. 

Example: for N = 2 one obtains Bloch sphere, Ω2 = P 1 . 

Probability distributions on complex plane supported in 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’) 

PA(z) := 

 

ΩN 

is supported on Λ(A). How it looks like? 

dψ δ z − 〈ψ|A|ψ〉 


Normal case: a ’shadow’ of the classical simplex... 

Normal matrices of size N = 2 with spectrum {λ1, λ2} 

Distribution PA(z) covers uniformly the interval [λ1, λ2] on the complex 

plane, 

Examples for diagonal matrices A of size two and three 

1.0 

0.5 

0.0 

−0.5 

Density plot of numerical range of operator [1, 0; 0, i] 

−1.0 

−1.0 −0.5 0.0 0.5 1.0 

1.0 

0.5 

0.0 

−0.5 

Density plot of numerical range of operator diag(1, i, −i) 

−1.0 

−1.0 −0.5 0.0 0.5 1.0 

Normal matrices of order N = 3 with spectrum {λ1, λ2, λ3} 

Distribution PA(z) covers uniformly the triangle ∆(λ1, λ2, λ3) on the 


Normal case: a ’shadow’ of the classical simplex... 

Normal matrices of size N = 4 with spectrum {λ1, λ2, λ3, λ4} 

Distribution PA(z) covers the quadrangle (λ1, λ2, λ3, λ4) with the density 

determined by projection of the regular tetrahedron (3–simplex) on the 

complex plane 

1.0 

0.5 

0.0 

−0.5 

Density plot of numerical range of operator diag(1, exp(iπ/3), exp(i2π/3), −1) 

−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. K ˙Z (ChInka) Quantum entanglement & Numerical shadow 23.01.2012 15 / 32

Numerical range for matrices of order N = 2. 

with spectrum {λ1, λ2} 

a) normal matrix A ⇒ Λ(A) = closed interval [λ1, λ2] 

b) not normal matrix A ⇒ Λ(A) = elliptical disk with λ1, λ2 as focal 

points and minor axis, d = TrAA † − |λ1| 2 − |λ2| 2 (Murnaghan, 1932; 

Li, 1996). 

example Jordan matrix J = 

 

0 

0 

1 

0 

a circular disk, Λ(J) = D(0, r = 1/2). 

The set of N = 2 pure quantum states 

 

. Then its numerical range forms 

A projection of the Bloch sphere S 2 = P 1 onto a plane forms an 

ellipse, (which could be degenerated to an interval). 


Numerical shadow for N = 2 

Non–normal matrices of size N = 2 

Distribution PA(z) covers entire (eliptical) disk on the complex plane with 

non-uniform (!) density determined by projection of (empty!) Bloch 

sphere, S 2 = P 1 on the complex plane. 

1.0 

0.5 

0.0 

−0.5 

Density plot of numerical range of operator [0, 1; 0, 1/2] 

−1.0 

−1.0 −0.5 0.0 0.5 1.0 

2.0 

1.5 

1.0 

0.5 

0.0 

−0.5 

−1.0 

−1.5 

Density plot of numerical range of operator [0, 1; 0, 1] 

−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 P 1 = S 2 = set of quantum pure states for N = 2. 


Shadows of three dimensional objects... 


Quantum States and Numerical Range/Shadow 

Classical States & normal matrices 

Proposition 1.Let CN denote the set of classical states of size N, which 

forms the regular simplex ∆N−1 in N−1 . Then the set of similar images 

of orthogonal projections of CN on a 2–plane is equivalent to the set of all 

possible numerical ranges Λ(A) of all normal matrices A of order N 

(such that AA ∗ = A ∗ A). 

Quantum States & non–normal matrices 

Proposition 2.Let MN denotes the set of quantum states size N 

embedded in N2 −1 with respect to Euclidean geometry induced by 

Hilbert-Schmidt distance. Then the set of similar images of orthogonal 

projections MN on a 2-plane is equivalent to the set of all possible 

numerical ranges Λ(A) of all matrices A of order N. 


Normal case: a ’shadow’ of the 4-D simplex... 

Normal matrices of size N = 5 with spectrum (λ1, λ2, λ3, λ4, λ5) 

Distribution PA(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 


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 

im 

1.0 

0.5 

0.0 

−0.5 

Density plot of numerical range of operator O5 subject to complex states 

−1.0 −0.5 0.0 0.5 

re 

im 

1.0 

0.5 

0.0 

−0.5 

Density plot of numerical range of operator O5 subject to real states 

−1.0 −0.5 0.0 0.5 

re 

’shadow’ of the projective space Ω3 = P 2 (left) and P 2 (right) for a 

N = 3 non-normal matrix A = [0, 1, 1; 0, i, 1; 0, 0, −1] 


Numerical range for matrices of order N = 3. 

Classification by Keeler, Rodman, Spitkovsky 1997 

Numerical range Λ of a 3 × 3 matrix A forms: 

a) Λ(A) is a compact set of an ’ovular’ shape 

(which contains three eigenvalues!) – the generic case, 

b) a compact set with one flat part (e.g. convex hull of a cardioid), 

c) a compact set with two flat parts 

(e.g. convex hull of an ellipse and a point outside it), 

d) triangle of eigenvalues, Λ(A) = ∆(λ1, λ2, λ3) 

for any normal matrix A. 

These four cases describe the shape of possible projections of the complex 

projective space P 2 (embedded into 8 ) onto a 2–plane. 


Shadows of typical matrices of size N = 3 

correspond to the four classes of N = 3 numerical ranges by Keeler et al. 


Unitary dynamics in the shadow (as a backgraound) 

Examplary unitary dynamics of N = 3 pure state in P 2 

|ψ(t)〉 = U|ψ(0)〉 = e iHt |ψ(0)〉 

(green line) 

z(t) := 〈ψ(t)|A|ψ(t)〉 = 〈ψ(0)|U ∗ AU|ψ(t)〉 

superimposed on the numerical shadow of three exemplary matrices 

A1, A2 and A3 of size N = 3. 


Mixed states shadow 

Let MN denotes the set of mixed states of size N endowed with the 

(flat) Hilbert–Schmidt measure µ. 

For any operator A we define its mixed states shadow 

 

dµ(ρ)δ z − TrρA . (2) 

P µ 

A (z) := 

MN 

For N = 2 the mixed states shadow is located in an ellipse with a density 

obtained by a projection of a full 3–ball onto a plane. 

In general we show that for any A of size N its mixed states shadow is 

equal to the (standard) shadow of the extended operator A ⊗ N, 

P µ 

A (z) = PA⊗N (z) = PN⊗A(z). (3) 


Shadows of N 2 − 1 dim dimensional objects (sketch) 


Numerical range & quantum entanglement: 

Shadow with respect to a restricted set ΩR of quantum states 

Let ΩSep (ΩEnt) denotes the set of all separable (entangled) pure states 

in HN ⊗ HN. 

For an arbitrary operator A of size N 2 one defines its numerical shadow 

with respect to separable (entangled) states 

P S A(z) := 

 

ΩSep 

dψ δ z − 〈ψ|A|ψ〉 

The distribution P S A (z) is supported on Λ⊗(A). How does it look ? 


Standard numerical shadow 

Shadows of (various) operators of size N = 4 

ℑ 

0.8 

0.4 

0.0 

−0.4 

−0.8 

−0.8 −0.4 0.0 

ℜ 

0.4 0.8 

ℑ 

0.8 

0.4 

0.0 

−0.4 

−0.8 

−0.8 −0.4 0.0 

ℜ 

0.4 0.8 

ℑ 

0.8 

0.4 

0.0 

−0.4 

−0.8 

−0.8 −0.4 0.0 

ℜ 

0.4 0.8 

characterize projections of 6-dim complex projective space 

P 3 onto a 2–plane 

The projection plane and direction is detemined by the particular matrix A 

of order N = 4. 

(three exemplary matrices selected here) 


Numerical shadow restricted to 

(complex) separable states 

Shadows of (various) operators of size N = 4 

ℑ 

0.8 

0.4 

0.0 

−0.4 

−0.8 

−0.8 −0.4 0.0 

ℜ 

0.4 0.8 

ℑ 

0.8 

0.4 

0.0 

−0.4 

−0.8 

−0.20.0 0.2 

ℜ 

ℑ 

0.8 

0.4 

0.0 

−0.4 

−0.8 

−0.8 −0.4 0.0 

ℜ 

0.4 0.8 

characterize projections of 4-dim manifold 

P 1 × P 1 = S 2 × S 2 onto a 2–plane 

The projection plane and direction is detemined by the particular matrix A 

of order N = 4. 

(three exemplary matrices selected here) 


Dynamics of Entanglement and separable shadow 

Trajectories of quantum dynamics on the complex plane 

ℑ 

0.2 

0.0 

−0.2 

z(t) = 〈ψ(t)|A|ψ(t)〉 

−0.4 0.0 0.4 

ℜ 

ℑ 

0.2 

0.0 

−0.2 

−0.8 −0.4 0.0 

ℜ 

0.4 0.8 

Figures obtained for former example of 2 × 2 system 

with initial separable pure state |ψ(0)〉 

and suitably chosen (non–Hermitian !) operator A 

visualize possible behaviour of quantum entanglement... 


Dynamics of entanglement II 

A general, geometric picture 

The effects of 

a) revivals of entanglement = oscillations induced by unitary evolution 

b) entanglement sudden death 

can be explained by 

a geometric approach based on numerical shadow 

hskip 2.0cm and separable numerical shadow. 


Concluding Remarks 

Different scenarios of Dynamics of quantum entanglement 

(revivals, sudded death) can be understood by investigating the 

geometry of the set of separable states: 

A key role is played by a ball of separable states located at the center 

of the body of mixed states. 

We introduce a concept of numerical shadow of an operator. 

The set of all possible projections of the N 2 − 1 dimensional set of 

quantum states of order N onto a 2–plane is equivalent (up to shift 

and rescaling) to the set of numerical ranges of matrices of order N. 

Numerical shadow restricted to separable/entangled states is 

useful for investigations of the geometry of quantum entanglement ! 

One can trace quantum trajectories with the Numerical shadow 

restricted to separable states in the background...

Dynamics of Quantum Entanglement & Separable Numerical ...

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