Download @let@token Polarized Ensembles of Random Quantum States

INTRO
RANDOM STATES
Polarized Ensembles of Random Quantum States
Summer School - Randomness in Physics and Mathematics
Fabio Deelan Cunden
Università degli Studi di Bari
Dipartimento di Matematica
August 5-17, 2013
INTRO
RANDOM STATES
Intro
Some preliminaries
A physical system is specified by the observables 𝐴, 𝐵, 𝐶, ⋅ ⋅ ⋅ ∈ 𝒜 that
we can measures. The observables form an algebra 𝒜.
In classical mechanics 𝒜 is commutative. In quantum mechanics 𝒜 is not.
A state of a system is an assignement of a number 𝜑(𝒜) for each
element 𝒜 of the algebra of observables. Such a number 𝜑(𝒜) is the
outcome of a measurement of the observable 𝒜.
𝜑:𝐴→ℂ
Remark
The set of states 𝒮 is convex.
𝜑 linear, 𝜑 ≥ 0, 𝜑(1) = 1.
INTRO
RANDOM STATES
Intro
Some preliminaries
A physical system is specified by the observables 𝐴, 𝐵, 𝐶, ⋅ ⋅ ⋅ ∈ 𝒜 that
we can measures. The observables form an algebra 𝒜.
In classical mechanics 𝒜 is commutative. In quantum mechanics 𝒜 is not.
A state of a system is an assignement of a number 𝜑(𝒜) for each
element 𝒜 of the algebra of observables. Such a number 𝜑(𝒜) is the
outcome of a measurement of the observable 𝒜.
𝜑:𝐴→ℂ
𝜑 linear, 𝜑 ≥ 0, 𝜑(1) = 1.
Remark
The set of states 𝒮 is convex.
What happens if the state 𝜑 of the system is random?
What about its typical properties?
INTRO
RANDOM STATES
Intro
Quantum mechanics
In a quantum mechanical setting:
∙
∙
𝒜 = ℬ(ℋ), ℋ Hilbert space;
𝜑 = tr(𝜌 ⋅), where 𝜌 = 𝜌† , 𝜌 ≥ 0, tr𝜌 = 1.
Examples ℋ = ℂ𝑁
𝜌=
1
𝑁
maximally mixed state
𝜌 = ∣𝜓⟩ ⟨𝜓∣ pure state (a rank-1 projection operator).
Remark
∙
∙
The pure states ∣𝜓⟩ are normalized vectors ∥𝜓∥2 = 1.
The set of quantum states 𝒮(ℋ) is convex, the pure states being its
extremal points.
INTRO
RANDOM STATES
Composite Quantum Systems
𝑆 =𝐴+𝐵
ℋ𝑆 = ℋ𝐴 ⊗ ℋ 𝐵
dim ℋ𝐴 = 𝑁
dim ℋ𝐵 = 𝑀
“...the quantum theory subscribes to the view that the whole is greater
than the sum of its parts.” (H.Weyl)
Given a state 𝜑𝐴𝐵 of the whole system, the state of subsystem 𝐴 is
given by the reduced state 𝜑𝐴 :
Tr
𝜑𝐴𝐵 ∈ 𝒮(ℋ𝐴 ⊗ ℋ𝐵 ) −−−𝐵
−→ 𝜑𝐴 ∈ 𝒮(ℋ𝐴 )
Definition (Entanglement)
A pure state 𝜑𝐴𝐵 ∈ 𝒮(ℋ𝐴 ⊗ ℋ𝐵 ) is separable if 𝜑𝐴 ∈ 𝒮(ℋ𝐴 ) is pure.
A non separable state is said entangled.
INTRO
RANDOM STATES
The problem
Typicality in Hilbert Spaces
Let us consider a bipartite quantum system:
𝑆 =𝐴+𝐵 .
Let us choose randomly a state ∣𝜓⟩𝐴𝐵 of system 𝑆:
What about the typical properties of the subsystem 𝐴?
Rephrasing:
∙ Fix an ensemble of random pure quantum states ∣𝜓⟩𝐴𝐵 of a bipartite space
ℋ𝑆 = ℋ𝐴 ⊗ ℋ𝐵 .
∙ Consider the consequent ensemble of reduced density matrices 𝜌𝐴 ∈ 𝒮(ℋ𝐴 ).
∙ What are the typical properties of 𝜌𝐴 ?
INTRO
RANDOM STATES
Entanglement characterization
Thm (A consequence of SVD)
The pure state ∣𝜓⟩𝐴𝐵 is separable iff rank𝜌𝐴 = 1.
H0,1,0L
Let (𝜆1 , . . . , 𝜆𝑑𝐴 ) = (1, 0, . . . , 0) be the eigenvalues of 𝜌𝐴 .
If (𝜆1 , . . . , 𝜆𝑑𝐴 ) = (1, 0, . . . , 0) then ∣𝜓⟩ is separable.
If at least two eigenvalues are ∕= 0 the state ∣𝜓⟩ is entangled.
N=3
H
H1,0,0L
How to quantify the non-separability of ∣𝜓⟩?
Local Purity:
𝜋𝐴𝐵 (𝜓) =
∑
𝜆2𝑖
𝑖
The lower the local purity the more entangled is ∣𝜓⟩.
1/𝑑𝐴 ≤ 𝜋𝐴𝐵 ≤ 1
1 1 1
, , L
3 3 3
H0,0,1L
INTRO
RANDOM STATES
Random Pure States
Unbiased ensemble
There exists a “uniform” measure 𝜇Haar on pure states (from topological reasons)
maximal simmetry in the Hilbert space ℋ
⇕
minimal prior knowledge of system 𝑆.
This ensemble has been intensively investigated (lot of RMT technology).
Induced measure on density matrices
The unitarily invariant measure on pure states 𝜇Haar induces a measure 𝜈𝐴 on the
reduced density matrices (the push-forward measure of 𝜇Haar w.r.t. the map Tr𝐵 ):
𝜈𝐴 = (Tr𝐵 )∗ 𝜇Haar = 𝑓 × 𝜇
INTRO
RANDOM STATES
Joint pdf of the eigenvalues
The eigenvalues are distribuited according to the following density:
𝑓𝑁,𝑀 (𝜆1 , . . . , 𝜆𝑁 ) = 𝐶𝑁,𝑀
∏
𝑖<𝑗
(𝜆𝑖 − 𝜆𝑗 )2
∏
−𝑁
𝜆𝑀
𝑙
𝑙
dim ℋ𝐴 = 𝑁
dim ℋ𝐵 = 𝑀
∏
𝐶𝑁,𝑀
:
normalization constant
2
:
levels repulsion
−𝑁
𝜆𝑀
𝑙
:
subsystem 𝐴 is strongly correlated with 𝐵
(𝜆𝑖 − 𝜆𝑗 )
𝑖<𝑗
∏
𝑙
The pdf is supported in the domain defined by
∑
𝜆𝑖 ≥ 0 ,
𝜆𝑘 = 1 .
INTRO
RANDOM STATES
Want to study different physically motivated measures on the space of pure states.
The setup
A bipartite quantum system ℋ = ℋ𝐴 ⊗ ℋ𝐵 , with 𝑑𝐴 = dim ℋ𝐴 ≤ dim ℋ𝐵 = 𝑑𝐵 .
Suppose that the state of the quantum system has the form (up to normalization)
∣𝜓⟩ = 𝛼 ∣𝜓1 ⟩ + 𝛽 ∣𝜓2 ⟩ ,
where ∣𝜓1 ⟩ , ∣𝜓2 ⟩ ∈ ℋ are random states.
Once the probability distributions of ∣𝜓1 ⟩ and ∣𝜓2 ⟩ are specified,
the random state ∣𝜓⟩ is characterized by a well defined distribution.
(1)
INTRO
RANDOM STATES
∣𝜓⟩ = superposition of a fixed pure state and an unbiased random one.
We get the following one-parameter ensemble
[
]
√
∣𝜓⟩ = 𝜖1𝐴𝐵 + 1 − 𝜖2 𝑈𝐴𝐵 ∣𝜙0 ⟩
(2)
where:
∙ the normalized state ∣𝜙0 ⟩ ∈ ℋ is fixed;
∙ 𝜖 ∈ [0, 1] is a tunable parameter;
∙
1𝐴𝐵
is the identity operator;
∙ 𝑈𝐴𝐵 ∈ 𝒰 (ℋ) is a random unitary (Haar distributed).
∣𝜓⟩ = 𝜖 ∣𝜙0 ⟩ +
√
1 − 𝜖2 ∣𝜙⟩
(3)
INTRO
RANDOM STATES
Typical local purity
We study the local purity of one of its party
𝜋𝐴𝐵 (𝜓) = Tr𝜌2𝐴 ∈ [1/𝑑𝐴 , 1] .
The lower the local purity the more entangled is ∣𝜓⟩.
(4)
INTRO
RANDOM STATES
Typical local purity
We study the local purity of one of its party
𝜋𝐴𝐵 (𝜓) = Tr𝜌2𝐴 ∈ [1/𝑑𝐴 , 1] .
(4)
The lower the local purity the more entangled is ∣𝜓⟩.
The typical purity of the random state (2) is the average value of (4):
(
)
𝔼[𝜋𝐴𝐵 (𝜓)] = 𝜖4 𝜋0 + 1 − 𝜖4 𝜋unb + 𝑂
where 𝜋unb =
𝑑𝐴 +𝑑𝐵
𝑑𝐴 𝑑𝐵
(
1
𝑑𝐴 𝑑𝐵
)
, 𝜋0 = 𝜋𝐴𝐵 (∣𝜙0 ⟩) is the local purity of the bias ∣𝜙0 ⟩.
(5)
INTRO
RANDOM STATES
Typical local purity
We study the local purity of one of its party
𝜋𝐴𝐵 (𝜓) = Tr𝜌2𝐴 ∈ [1/𝑑𝐴 , 1] .
(4)
The lower the local purity the more entangled is ∣𝜓⟩.
The typical purity of the random state (2) is the average value of (4):
(
)
𝔼[𝜋𝐴𝐵 (𝜓)] = 𝜖4 𝜋0 + 1 − 𝜖4 𝜋unb + 𝑂
where 𝜋unb =
𝑑𝐴 +𝑑𝐵
𝑑𝐴 𝑑𝐵
(
1
𝑑𝐴 𝑑𝐵
)
(5)
, 𝜋0 = 𝜋𝐴𝐵 (∣𝜙0 ⟩) is the local purity of the bias ∣𝜙0 ⟩.
The average value (5) is typical (concentration of measure phenomenon):
(
)
{
}
𝑑𝐴 𝑑𝐵 𝛼2
Pr 𝜋𝐴𝐵 (𝜓) − 𝔼[𝜋𝐴𝐵 (𝜓)] > 𝛼 ≤ 2 exp −
32(1 − 𝜖2 )
(6)
INTRO
RANDOM STATES
Isopurity manifolds
[
∣𝜓⟩ = 𝜖𝑈𝐴 ⊗ 𝑈𝐵 +
√
]
1 − 𝜖2 𝑈𝐴𝐵 ∣𝜙0 ⟩
(7)
4
(
4
∣𝜙0 ⟩ = ∣𝜙sep ⟩
⇒
𝜋0 = 1
(8)
⇒
𝜋0 = 1/𝑑𝐴
(9)
𝜋𝐴𝐵 ≃ 𝜖 𝜋0 + 1 − 𝜖
)
𝜋unb
Separable bias:
Maximally entangled bias:
∣𝜙0 ⟩ = ∣𝜙ent ⟩𝐴𝐵
Π AB
N=30 M=30
1.0
0.8
0.6
0.4 Maximally Entangled
Φ ent >
0.2
0.0
1
0.5
Separable
Φ sep >
0
Ε
4
0.5
1
INTRO
RANDOM STATES
Generation of random pure states with fixed purity
An inexpensive strategy for generating random pure states with fixed value of 𝜋𝐴𝐵 :
1. Choose 𝜖 ∈ [0, 1] such that
(
)
𝜋𝐴𝐵 = 𝜖4 𝜋0 + 1 − 𝜖4 𝜋unb ,
(10)
where 𝜋0 = 1 or 𝜋0 = 1/𝑁 if the desired value of 𝜋𝐴𝐵 is, respectively, larger or
smaller than the unbiased typical value 𝜋Haar .
2. Generate a pure state ∣𝜓⟩ by superposition
√
∣𝜓⟩ = 𝜖 ∣𝜙0 ⟩ + 1 − 𝜖2 ∣𝜙⟩ ,
(11)
where ∣𝜙⟩ ∼ 𝜇Haar and ∣𝜙0 ⟩ is a random separable or maximally entangled pure
state sampled, according to the value of 𝜋0 chosen in (10).
INTRO
RANDOM STATES
Proposals
∙
New ensembles?
INTRO
RANDOM STATES
Proposals
∙
New ensembles? We can think to ensembles with fixed energy
⟨𝜓∣ 𝐻 ∣𝜓⟩ = 𝐸, or fixed entanglement 𝜋𝐴𝐵 (𝜓). . .
INTRO
RANDOM STATES
Proposals
∙
∙
New ensembles? We can think to ensembles with fixed energy
⟨𝜓∣ 𝐻 ∣𝜓⟩ = 𝐸, or fixed entanglement 𝜋𝐴𝐵 (𝜓). . .
Typicality in multipartite systems??
INTRO
RANDOM STATES
Proposals
∙
∙
∙
New ensembles? We can think to ensembles with fixed energy
⟨𝜓∣ 𝐻 ∣𝜓⟩ = 𝐸, or fixed entanglement 𝜋𝐴𝐵 (𝜓). . .
Typicality in multipartite systems??
What is the role of the entanglement of random states in Statistical
Mechanics?
INTRO
RANDOM STATES
Bengtsson I and Życzkowski K 2006 Geometry of quantum states (Cambridge
University Press)
Hayden P, Leung D W and Winter A 2006 Comm. Math. Phys. 265 95
Cunden F D, Facchi P, Florio G and Pascazio S, Eur. Phys. J. Plus 128 48,
(2013) (arXiv:1303.4209 [math-ph]).
Cunden F D, Facchi P and Florio G, J. Phys. A: Math. Theor. 46 315306,
(2013) (arXiv:1304.6219v2 [math-ph]).
INTRO
RANDOM STATES
Robustness of separability
Application: stability of separability of quantum states with respect to random
additive perturbations. If the state of a bipartite system ∣𝜉0 ⟩𝐴 ⊗ ∣𝜒0 ⟩𝐵 is separable,
how much noise 𝜂 is necessary to make the its reduced state 𝜌𝐴 distinguishable from a
pure state?
√
∣𝜓⟩ = 1 − 𝜂 2 ∣𝜉0 ⟩𝐴 ⊗ ∣𝜒0 ⟩𝐵 + 𝜂 ∣𝜙⟩ , 0 ≤ 𝜂 ≤ 1,
(12)
where
∙ ∣𝜙⟩ ∼ 𝜇𝑁 𝑀 is an unbiased random perturbation;
∙ 𝜂 measures the strength of the noise.
In the limit of large system sizes, 𝑑𝐴 , 𝑑𝐵 → ∞, the threshold critical value becomes
(
)
(
)
1
1
+𝑂
,
(13)
𝜂★2 = 1 − √
𝑑𝐴
2
As long as the state ∣𝜓⟩ of the large quantum system has the form (12) with
√
1
𝜂 < 1 − √ ≃ 0.54,
2
the effective dimension is 𝑑eff (𝜌𝐴 ) < 2, and separability will be (approximately)
preserved.
(14)