The Separability Problem and
its Variants in Quantum
Entanglement Theory
Nathaniel Johnston
Institute for Quantum Computing
University of Waterloo
Overview
•
What is Quantum Entanglement?
•
The Separability Problem
•
The Bound Entanglement Problem
•
The Separability from Spectrum Problem
Overview
•
What is Quantum Entanglement?
•
Weird physical phenomenon
•
Linear algebra works!
•
The Separability Problem
•
The Bound Entanglement Problem
•
The Separability from Spectrum Problem
What is Quantum Entanglement?
Physicist
Mathematician
•
Particles can be “linked”
•
•
Always get correlated
measurement results
Tensor product of finitedimensional vector spaces
•
Tensors of rank > 1 exist
That’s weird!
•
That’s obvious!
•
What is Quantum Entanglement?
Pure quantum state:
i.e.,
with
Dual (row) vector:
Inner product:
with
What is Quantum Entanglement?
Tensor product:
=
=
What is Quantum Entanglement?
Outer product
tensor product:
Obtained via “stacking columns”:
What is Quantum Entanglement?
Definition
A pure state
is separable if it can be written as
Otherwise, it is entangled.
rank 1
rank 2 (thus entangled)
What is Quantum Entanglement?
Mixed quantum state:
•
Trace 1
•
Positive semidefinite
equivalent
Pure state (again):
•
Rank 1
•
Trace 1
•
Positive semidefinite
equivalent
What is Quantum Entanglement?
Definition
A mixed state
with each
is separable if it can be written as
separable. Otherwise, it is entangled.
for some
Equivalent:
convex combination
separable
positive semidefinite
is the “maximally mixed” state.
Overview
•
What is Quantum Entanglement?
•
The Separability Problem
•
How to determine separability?
•
Positive matrix-valued maps
•
Funky matrix norms
•
The Bound Entanglement Problem
•
The Separability from Spectrum Problem
The Separability Problem
Recall:
is separable if we can write
for some
Definition
Given
the separability problem is the problem
of determining whether ρ is separable or entangled.
This is an
a hard
NP-hard
problem!
problem! (Gurvits, 2003)
The Separability Problem
Separable states
ρ
All states
The Separability Problem
Method 1: “Partial” transpose
Define a linear map Γ on
In matrices:
by
The Separability Problem
Apply Γ to a separable state:
is positive semidefinite
We say that ρ has positive partial transpose (PPT).
Not true for some entangled states:
which has eigenvalues 1, 1, 1, and -1.
The Separability Problem
Separable states
PPT states
ρ
All states
The Separability Problem
Theorem (Størmer, 1963; Woronowicz, 1976; Peres, 1996)
Let
be a quantum state. If ρ is separable then
Furthermore, the converse holds if and only if mn ≤ 6.
•
Separability problem is completely solved when mn ≤ 6
•
Higher dimensions?
The Separability Problem
Method 1.1: Positive maps
Given
, define a linear map
by
In matrices:
on
The Separability Problem
Definition
is positive if
whenever
Transpose map:
positive semidefinite
Theorem (Horodecki3, 1996)
A quantum state
for all positive maps
is separable if and only if
The Separability Problem
Separable states
Transpose map
ρ
All states
The Separability Problem
The problem:
•
Coming up with positive maps is hard!
•
Proving that a map is positive is NP-hard
Current status:
•
Dozens of papers
•
Only a handful of known positive maps
The Separability Problem
Method 2: Norms
Definition
The operator norm and trace norm of a matrix are defined by:
where
•
•
are the singular values of X.
The Separability Problem
Separable states ≈ unit ball of
All states
≈ unit ball of
The Separability Problem
Definition
Given
define the S(1)-norm via
Separable version of
dual
dual
Separable version of
The Separability Problem
Theorem (Rudolph, 2000)
Let
be the dual of the S(1)-norm, defined by
A quantum state ρ is separable if and only if
The Separability Problem
Separable states ≈ unit ball of
All states
≈ unit ball of
The Separability Problem
The goal: derive bounds for
•
“Swap” operator:
•
“Realignment” map:
because
= 1 if ρ separable
The Separability Problem
Theorem (Chen–Wu, 2003)
If
then ρ is entangled.
The goal:
•
Come up with more bounds on
•
Lower bounds
entanglement
•
Upper bounds
separability
σ
ρ
Overview
•
What is Quantum Entanglement?
•
The Separability Problem
•
The Bound Entanglement Problem
•
•
Not all entanglement is “useful”
•
Partial transpose is awesome
The Separability from Spectrum Problem
Bound Entanglement
Can we turn mixed entanglement into pure entanglement?
ρ
ρ
ρ
Bound Entanglement
Not always!
Theorem (Horodecki3, 1998)
If the quantum state
has positive partial
transpose then it is bound entangled (i.e., many copies of ρ can
not be locally converted into an entangled pure state).
Question: Are there more? Or is this “iff”?
Bound Entanglement
Separable states
PPT states
=
Bound
entangled
states
All states
Bound Entanglement
Let’s phrase the problem mathematically!
•
Recall: for
we have
•
Similarly,
•
“Rank 1” and “full rank” versions of same norm
Bound Entanglement
We now want the “rank 2” version of this norm:
Also need the “maximally entangled state”:
standard basis of
Bound Entanglement
Theorem
Define a family of projections P1, P2, … recursively as follows:
Then there exists non-positive partial transpose bound
entanglement (more or less) if and only if
up to minor technical details (e.g., n ≥ 4 only)
Bound Entanglement
What do we know so far?
•
n = 4, k = 2:
•
•
•
•
•
•
Big gap!
equality when k = 1
•
Overview
•
What is Quantum Entanglement?
•
The Separability Problem
•
The Bound Entanglement Problem
•
The Separability from Spectrum Problem
•
We only know eigenvalues
•
Want to determine separable/entangled
Separability from Spectrum
•
Only given eigenvalues of ρ
•
Can we prove ρ is entangled/separable?
Prove entangled?
No: diagonal
separable
arbitrary eigenvalues, but always
separable
Separability from Spectrum
•
Only given eigenvalues of ρ
•
Can we prove ρ is entangled/separable?
Prove separable?
Sometimes:
If all eigenvalues are equal then
a separable decomposition
Separability from Spectrum
Can also prove separability if ρ is close to
Theorem (Gurvits–Barnum, 2002)
Let
be a mixed state. If
then ρ is separable, where
is the Frobenius norm.
Frobenius norm:
eigenvalues of ρ
Separability from Spectrum
Gurvits–Barnum ball
Separable states
All states
Separability from Spectrum
Definition
A quantum state
is called separable from
spectrum if all quantum states with the same eigenvalues as ρ
are separable.
States in the Gurvits–Barnum ball are separable from spectrum:
But there are more!
only depends on eigenvalues of ρ
Separability from Spectrum
Gurvits–Barnum ball
Separable states
Separable from
spectrum
All states
Separability from Spectrum
The case of two qubits (i.e., m = n = 2) was solved long ago:
Theorem (Verstraete–Audenaert–Moor, 2001)
A state
is separable from spectrum if and only if
What about higher-dimensional systems?
eigenvalues, sorted so that λ1 ≥ λ2 ≥ λ3 ≥ λ4 ≥ 0
Separability from Spectrum
Replace “separable” by “positive partial transpose”.
Definition
A quantum state
is called positive partial
transpose (PPT) from spectrum if all quantum states with the
same eigenvalues as ρ are PPT.
Separability from Spectrum
Gurvits–Barnum ball
Separable states
Separable from
spectrum
All states
PPT from spectrum
Separability from Spectrum
•
PPT from spectrum is completely solved (but complicated)
Theorem (Hildebrand, 2007)
A state
is PPT from spectrum if and only if
•
Recall: separability = PPT when m = 2, n ≤ 3
•
Thus
is separable from spectrum if and only if
Separability from Spectrum
Can PPT from spectrum tell us more about separability from
spectrum?
Yes!
weird when n ≥ 4
obvious when n ≤ 3
Theorem (J., 2013)
A state
it is PPT from spectrum.
is separable from spectrum if and only if
Separability from Spectrum
Gurvits–Barnum ball
Separable states
= Separable from
spectrum
All states
PPT from spectrum
Separability from Spectrum
Sketch of proof.
Write
as a block matrix:
ρ becomes “more positive” as B becomes small compared to A and C
Lemma
If
then ρ is separable.
Separability from Spectrum
Want: every PPT from spectrum
to satisfy
hypotheses of Lemma.
Not true!
Lemma
If
then ρ is separable.
Separability from Spectrum
Instead: for every PPT from spectrum
exists a 2×2 unitary matrix U such that
satisfies hypotheses of Lemma.
works for
Lemma
If
then ρ is separable.
there
Separability from Spectrum
Define
Then
some intermediate
value of t works
Lemma
If
then ρ is separable.
Separability from Spectrum
What about separability from spectrum for
when m, n ≥ 3?
Don’t know!
Gurvits–Barnum ball
Separable states
Separable from
spectrum
All states
PPT from spectrum
Gurvits–Barnum ball
Separable states PPT from spectrum
= Separable from
spectrum
All states