Asymptotic properties of quantum states and channels Ion Nechita, - - PowerPoint PPT Presentation

Asymptotic properties of quantum states and channels

Ion Nechita, Łukasz Pawela, Zbigniew Puchała, Karol Życzkowski

Institute of Theoretical and Applied Informatics, Polish Academy of Sciences

18 June 2017

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 1 / 31

Outline

1

Random matrix theory and free probability crash course

2

Random states and channels

3

Eigenvalues of random quatnum states and channels

4

Distances between random quantum states

5

The diamond norm

6

Final remarks

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 2 / 31

Random matrix theory and free probability crash course

Contents

1

Random matrix theory and free probability crash course

2

Random states and channels

3

Eigenvalues of random quatnum states and channels

4

Distances between random quantum states

5

The diamond norm

6

Final remarks

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 3 / 31

Random matrix theory and free probability crash course

Some useful random matrix ensembles

Ginibre matrices Matrices G such that Gij ∼ NC(0, 1/2)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 4 / 31

Random matrix theory and free probability crash course

Some useful random matrix ensembles

Ginibre matrices Matrices G such that Gij ∼ NC(0, 1/2) Wishart matrices W = GG †

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 4 / 31

Random matrix theory and free probability crash course

Free probability introduction

Why free probability Let us consider two sequences AN and BN of selfadjoint N × N matrices

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31

Random matrix theory and free probability crash course

Free probability introduction

Why free probability Let us consider two sequences AN and BN of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f (AN, BN) for some non-trivial selfadjoint function f.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31

Random matrix theory and free probability crash course

Free probability introduction

Why free probability Let us consider two sequences AN and BN of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f (AN, BN) for some non-trivial selfadjoint function f. Freenes

1 N tr [p1(An)q1(BN)p2(AN)q2(BN) . . .] → 0.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31

Random matrix theory and free probability crash course

Free probability introduction

Why free probability Let us consider two sequences AN and BN of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f (AN, BN) for some non-trivial selfadjoint function f. Freenes

1 N tr [p1(An)q1(BN)p2(AN)q2(BN) . . .] → 0.

If AN and BN have almost surely an asymptotic eigenvalue distribution for N → ∞;

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31

Random matrix theory and free probability crash course

Free probability introduction

Why free probability Let us consider two sequences AN and BN of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f (AN, BN) for some non-trivial selfadjoint function f. Freenes

1 N tr [p1(An)q1(BN)p2(AN)q2(BN) . . .] → 0.

If AN and BN have almost surely an asymptotic eigenvalue distribution for N → ∞; AN and BN are independent;

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31

Random matrix theory and free probability crash course

Free probability introduction

Why free probability Let us consider two sequences AN and BN of selfadjoint N × N matrices We are interested in the asymptotic eigenvalue distribution of the sequence f (AN, BN) for some non-trivial selfadjoint function f. Freenes

1 N tr [p1(An)q1(BN)p2(AN)q2(BN) . . .] → 0.

If AN and BN have almost surely an asymptotic eigenvalue distribution for N → ∞; AN and BN are independent; BN is a unitarily invariant ensemble. Then, AN and BN are almost surely asymptotically free.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 5 / 31

Random matrix theory and free probability crash course

Free convolution

If x, y are free then moments of x + y are uniquely determined by the moments of x and y.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31

Random matrix theory and free probability crash course

Free convolution

If x, y are free then moments of x + y are uniquely determined by the moments of x and y. Notation We say that x + y is the free convolution of the distribution of x and distribution

• f y. We write:

µx+y = µx ⊞ µy (1)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31

Random matrix theory and free probability crash course

Free convolution

If x, y are free then moments of x + y are uniquely determined by the moments of x and y. Notation We say that x + y is the free convolution of the distribution of x and distribution

• f y. We write:

µx+y = µx ⊞ µy (1) R transform Consider a random variable x and its Cauchy transform G(z). Then 1 G(z) + R[G(z)] = z (2)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31

Random matrix theory and free probability crash course

Free convolution

If x, y are free then moments of x + y are uniquely determined by the moments of x and y. Notation We say that x + y is the free convolution of the distribution of x and distribution

• f y. We write:

µx+y = µx ⊞ µy (1) R transform Consider a random variable x and its Cauchy transform G(z). Then 1 G(z) + R[G(z)] = z (2) Rx+y(z) = Rx(z) + Ry(z) (3)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 6 / 31

Random states and channels

Contents

1

Random matrix theory and free probability crash course

2

Random states and channels

3

Eigenvalues of random quatnum states and channels

4

Distances between random quantum states

5

The diamond norm

6

Final remarks

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 7 / 31

Random states and channels

Random states in ΩN

From pure states

1

Consider a random pure state |φ ∈ X ⊗ Y.

2

Trace out one of the systems ρ = trY|φφ|.

3

If dim(X) = dim(Y), we get the Hilbert-Schmidt distribution of ρ.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 8 / 31

Random states and channels

Random states in ΩN

From pure states

1

Consider a random pure state |φ ∈ X ⊗ Y.

2

Trace out one of the systems ρ = trY|φφ|.

3

If dim(X) = dim(Y), we get the Hilbert-Schmidt distribution of ρ. From Ginibre matrices Let G be a N × K Ginibre matrix (independent normal complex entries). Then, the matrix ρ = GG † trGG † , (4) is a random mixed state. If N = K we recover the flat Hilbert-Schmidt distribution.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 8 / 31

Random states and channels

Geometry of the set ΩN

Ω2 ⊂ R3 - Bloch ball, pure states reside on the sphere

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 9 / 31

Random states and channels

Geometry of the set ΩN

Ω2 ⊂ R3 - Bloch ball, pure states reside on the sphere For N > 2, ΩN is not a Ball!

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 9 / 31

Random states and channels

Geometry of the set ΩN

Ω2 ⊂ R3 - Bloch ball, pure states reside on the sphere For N > 2, ΩN is not a Ball!

Ω3

∂Ω3

Figure: Sketch of a qutrit

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 9 / 31

Eigenvalues of random quatnum states and channels

Contents

1

Random matrix theory and free probability crash course

2

Random states and channels

3

Eigenvalues of random quatnum states and channels

4

Distances between random quantum states

5

The diamond norm

6

Final remarks

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 10 / 31

Eigenvalues of random quatnum states and channels

Marchenko-Pastur distribution

Consider ρ =

GG † trGG † , where G is N × K. Then, the asymptotic eigenvalue

distribution is MPc(x) =

1 2πx

• x − (1 − √c)2

(1 + √c)2 − x, where x = Nλ and c = N/K. When c = 1: MP1(x) = √

4/x−1 2π

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 11 / 31

Eigenvalues of random quatnum states and channels

Symmetrized Marchenko-Pastur distribution

Trace distance Dtr(ρ, σ) = 1 2ρ − σ1 = 1 2tr|ρ − σ| (5) We need the eigenvalue distribution of ρ − σ.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 12 / 31

Eigenvalues of random quatnum states and channels

Symmetrized Marchenko-Pastur distribution

Trace distance Dtr(ρ, σ) = 1 2ρ − σ1 = 1 2tr|ρ − σ| (5) We need the eigenvalue distribution of ρ − σ. Free additive convolution Symmetrized Marchenko-Pastur distribution SMPc(x) = MPc(x) ⊞ MPc(−x). In the case of HS measure (c = 1), we get: SMP1(x) = −1 − 3x2 +

• 1 + 3x

√ 3 + 33x2 − 3x4 + 6x 2/3 2 √ 3πx

• 1 + 3x

√ 3 + 33x2 − 3x4 + 6x 1/3 . (6)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 12 / 31

Eigenvalues of random quatnum states and channels

Examples

−2 2

y

0.00 0.15 0.30 P(y) a) −2 2

y

0.0 0.5 1.0 P(y) b) −3 3

y

0.0 0.2 0.4 P(y) c) −6 6

y

0.00 0.07 0.14 P(y) d)

Figure: Symmetrized Marchenko–Pastur distribution for c = 0.2 (a), 0.5 (b), 1.0 (c), 4.0 (d) denoted by the solid (red) line.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 13 / 31

Eigenvalues of random quatnum states and channels

Tetilla law

For c = 1 - free commutator of two semicircular distribution, studied by Nica and Speicher (1998) and called the tetilla law (Deya and Nourdin, 2012).

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 14 / 31

Eigenvalues of random quatnum states and channels

Tetilla law

For c = 1 - free commutator of two semicircular distribution, studied by Nica and Speicher (1998) and called the tetilla law (Deya and Nourdin, 2012).

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 14 / 31

Eigenvalues of random quatnum states and channels

Tetilla law

For c = 1 - free commutator of two semicircular distribution, studied by Nica and Speicher (1998) and called the tetilla law (Deya and Nourdin, 2012). For c → ∞ we obtain a rescaled semircicle.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 14 / 31

Distances between random quantum states

Contents

1

Random matrix theory and free probability crash course

2

Random states and channels

3

Eigenvalues of random quatnum states and channels

4

Distances between random quantum states

5

The diamond norm

6

Final remarks

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 15 / 31

Distances between random quantum states

Average distances between 2 random states

Take ρ and σ sampled from the flat (HS) measure, c = 1. As N → ∞, the trace distance tends to an integral over the symmetrized Marchenko-Pastur distribution: Dtr → 1 2

• SMP1(x)|x|dx = ˜

D = 1 4 + 1 π ≈ 0.5683 (7)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 16 / 31

Distances between random quantum states

Average distances between 2 random states

Take ρ and σ sampled from the flat (HS) measure, c = 1. As N → ∞, the trace distance tends to an integral over the symmetrized Marchenko-Pastur distribution: Dtr → 1 2

• SMP1(x)|x|dx = ˜

D = 1 4 + 1 π ≈ 0.5683 (7) Average distances of random state ρ to the maximally mixed state ρ∗ Dtr(ρ, ρ∗) → 1 2

• |t − 1|MP(t)dt = a = 3

√ 3 4π ≈ 0.4135 (8) the closest pure state, Dtr(ρ, |φφ|) → 1 the closest boundary state ˜ ρ, Dtr(ρ, ˜ ρ) → 0

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 16 / 31

Distances between random quantum states

Te set ΩN for large N

The HS measure is concentrated in an ε neighborhood of the unitary orbit, UρU†, where U a Haar unitary and ρ is a random state with spectrum distributed according to MP. Here, d is the diameter given by the distance between two diagonal matrices with opposite order of the eigenvalues d = DTr(p↑, p↓) = 4

0 x sign(x − M)MP1(x)dx ≃ 0.7875, where M denotes the

median, M

0 MP1(x)dx = 1/2.

ΩN

∂ΩN

|φ1 |φ2

ρ↑ ˜ ρ ρ↓ ρ∗ σ

˜ D a d 1 1 1 ε

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 17 / 31

Distances between random quantum states

Asymptotic distances

Given two random states ρ, σ of dimension N. For large N (N ≫ 1), we have: relative entropy S(ρ||σ) = trρ log ρ − ρ log σ S(ρ||σ) →

• dt
• ds(t log t − t log s)MP(t)MP(s) = 3

2

quantum Sanov theorem: Performing n measurements on ρ, we obtain result compatible with σ with probability p ∼ exp( −3n

2 ).

Chernoff information Q(ρ, σ) = mins∈[0,1] trρsσ1−s. We get the Chernoff bound for generic quantum states: Q(ρ, σ) = trρ

1 2 σ 1 2 →

√tMP(t)dt = 8

3π

2 = 0.72 = Q∗ Performing n measurements on ρ and σ we get the probability of error p ∼ exp(−Q∗n).

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 18 / 31

Distances between random quantum states

Rate of convergence

4 8 12 16 20 N 0.5 0.6 0.7 0.8 ˜ D

√ 2 2

DB DTr

Figure: Dependence of average distance between two generic states on the dimension N. Dashed (red) line shows the Bures distance and solid (black) line shows the trace

• distance. The horizontal lines mark the asymptotic values.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 19 / 31

Distances between random quantum states

Asymptotic entanglement

Consider |φ ∈ X ⊗ X and ρ = trX |φφ|. For a partially transposed matrix, ρTA, its eigenvalues have the shifted semicircle as the limiting distribution (Aubrun 2012), λ(ρTA) ∼ 1 2π

• 4 − (x − 1)2.

(9) We get:

1

the fraction of negative eigenvalues tends to

−1

1 2π

• 4 − (x − 1)2dx = 1

3 − √ 3 4π , (10)

2

the average negativity tends to N →

−1

|x| 2π

• 4 − (x − 1)2dx ≈ 0.080.

(11) The G-concurrence of a state G(|φ) = N(det ρ)

1 N , converges:

G(|φ) → exp 4 log tMP(t)dt

• = 1

e ≈ 0.368 (12)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 20 / 31

The diamond norm

Contents

1

Random matrix theory and free probability crash course

2

Random states and channels

3

Eigenvalues of random quatnum states and channels

4

Distances between random quantum states

5

The diamond norm

6

Final remarks

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 21 / 31

The diamond norm

Random quantum channels

From Wishart matrices Let W be a N2 × N2 Wishart matrix. Then, the matrix DΦ =

• 1

lN ⊗ 1 √tr1W

• W
• 1

lN ⊗ 1 √tr1W

• (13)

is a random Choi matrix for some channel Φ : L(X) → L(X) and dim(X) = N.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 22 / 31

The diamond norm

Random quantum channels

From Wishart matrices Let W be a N2 × N2 Wishart matrix. Then, the matrix DΦ =

• 1

lN ⊗ 1 √tr1W

• W
• 1

lN ⊗ 1 √tr1W

• (13)

is a random Choi matrix for some channel Φ : L(X) → L(X) and dim(X) = N. Random state vs random channel Consider the Jamiołkowski state corresponding to the channel Φ, JΦ = DΦ/trDΦ and a quantum state ρ.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 22 / 31

The diamond norm

Random quantum channels

From Wishart matrices Let W be a N2 × N2 Wishart matrix. Then, the matrix DΦ =

• 1

lN ⊗ 1 √tr1W

• W
• 1

lN ⊗ 1 √tr1W

• (13)

is a random Choi matrix for some channel Φ : L(X) → L(X) and dim(X) = N. Random state vs random channel Consider the Jamiołkowski state corresponding to the channel Φ, JΦ = DΦ/trDΦ and a quantum state ρ. ρ = ρ†, trρ = 1. JΦ = J†

Φ,

trJΦ = 1, tr2JΦ = 1 l/N.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 22 / 31

The diamond norm

What is it?

Induced trace norm Given a superoperator Φ : L(X) → L(Y) the induced trace norm is defined as: Φ1 = max{Φ(A)1 : A ∈ L(X), A1 ≤ 1} (14)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 23 / 31

The diamond norm

What is it?

Induced trace norm Given a superoperator Φ : L(X) → L(Y) the induced trace norm is defined as: Φ1 = max{Φ(A)1 : A ∈ L(X), A1 ≤ 1} (14) Diamond norm Given a superoperator Φ : L(X) → L(Y) the diamond norm is defined as: Φ⋄ = Φ ⊗ 1 lL(X)1 (15)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 23 / 31

The diamond norm

What is it?

Induced trace norm Given a superoperator Φ : L(X) → L(Y) the induced trace norm is defined as: Φ1 = max{Φ(A)1 : A ∈ L(X), A1 ≤ 1} (14) Diamond norm Given a superoperator Φ : L(X) → L(Y) the diamond norm is defined as: Φ⋄ = Φ ⊗ 1 lL(X)1 (15) Theorem Given a Hermiticity-preserving superoperator Φ : L(X) → L(Y), it holds that Φ⋄ = max{(Φ ⊗ 1 lL(X))(|φφ|)1, |φ ∈ X ⊗ X} (16)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 23 / 31

The diamond norm

Bounds on the diamond norm For general Φ : L(X) → L(Y) JΦ1 ≤ Φ⋄ ≤

• tr2
• DΦD†

Φ

• ∞

+

• tr2
• D†

ΦDΦ

• ∞

2 , (17)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 24 / 31

The diamond norm

Bounds on the diamond norm For general Φ : L(X) → L(Y) JΦ1 ≤ Φ⋄ ≤

• tr2
• DΦD†

Φ

• ∞

+

• tr2
• D†

ΦDΦ

• ∞

2 , (17) Hermiticity preserving case If Φ is Hermiticity preserving JΦ1 ≤ Φ⋄ ≤ tr2|DΦ|∞ (18)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 24 / 31

The diamond norm

Finally Given two quantum channels sampled from the Hilbert-Schmidt distribution, we can prove that Φ − Ψ⋄ → ∆ = 1 2 + 2 π (19)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 25 / 31

The diamond norm

Finally Given two quantum channels sampled from the Hilbert-Schmidt distribution, we can prove that Φ − Ψ⋄ → ∆ = 1 2 + 2 π (19) Furthermore Ψ − Φdep⋄ → r = 3 √ 3 2π (20) Ψ − ΦU⋄ → a = 2 (21)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 25 / 31

The diamond norm

Finally Given two quantum channels sampled from the Hilbert-Schmidt distribution, we can prove that Φ − Ψ⋄ → ∆ = 1 2 + 2 π (19) Furthermore Ψ − Φdep⋄ → r = 3 √ 3 2π (20) Ψ − ΦU⋄ → a = 2 (21)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 25 / 31

The diamond norm

Key step in proofs

• W −
• 1

lN ⊗ 1 √tr1W

• W
• 1

lN ⊗ 1 √tr1W

• ∞

= O(N−2) (22)

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 26 / 31

The diamond norm

Set of channels

Θ(d, d)

ΨU Ψ Φ Φdep ∆ a r r

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 27 / 31

The diamond norm

Set of bipartite states

r

Ωd2 (a)

|φAB |φA, φB

1 l/d2

σAB r

Ωd

|φA

1 l/d

ρA

(b)

r

Ωd

|φB

1 l/d

ρB

(c)

trB trA

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 28 / 31

Final remarks

Contents

1

Random matrix theory and free probability crash course

2

Random states and channels

3

Eigenvalues of random quatnum states and channels

4

Distances between random quantum states

5

The diamond norm

6

Final remarks

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 29 / 31

Final remarks

Concluding remarks

We derived the symmetric Marchenko-Pastur distribution for the eigenvalue distribution of ρ − σ and found their average trace distance ˜ D = 1

4 + 1 π.

We found the average fidelity and Bures distances, Chernoff bound, relative entropy and Holevo quantity. We found the distribution of eigenvalues of DΦ. We showed that the distance between two quantum channels is given by Φ − Ψ⋄ = 1

2 + 2 π.

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 30 / 31

Final remarks

Thank you!

• Z. Puchała, Ł. Pawela, K. Życzkowski, Distinguishability of generic quantum

states, arXiv:1507.05123

• I. Nechita, Z. Puchała, Ł. Pawela, K. Życzkowski, Almost all quantum

channels are equidistant, arXiv:1612.00401

Ł. Pawela (IITiS PAN) Asymptotic* 18 June 2017 31 / 31