Download On the minimum quantum dimension for a given quantum correlation

On the minimum quantum dimension
for a given quantum correlation
Zhaohui Wei (NTU and CQT, Singapore)
Joint work with Jamie Sikora and Antonios Varvitsiotis
arXiv:1507.00213

We will show a new application of PSD-rank in quantum
mechanics


We will show a new application of PSD-rank in quantum
mechanics
PSD-rank has been shown to be related to quantum
information.


We will show a new application of PSD-rank in quantum
mechanics
PSD-rank has been shown to be related to quantum
information.
◦ Communication to compute a function in expectation *1
*1. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015.


We will show a new application of PSD-rank in quantum
mechanics
PSD-rank has been shown to be related to quantum
information.
◦ Communication to compute a function in expectation *1
◦ Correlation complexity of distribution *2
*1. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015.
*2. Jain, Shi, Wei, Zhang, IEEE T INFORM THEORY, 2013.






An introduction to quantum information
An introduction to device-independent style
The target problem and known results
Our main result and its proof
Applications
Further work and open problems

Quantum state: A positive semidefinite (psd) matrix
with the trace

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when
is rank 1, i.e.,
we call
pure

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when is rank 1, i.e.,
◦ Composite systems:
we call
pure
. Usually they are huge.

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when is rank 1, i.e.,
◦ Composite systems:

we call
pure
. Usually they are huge.
Quantum measurement: A measurement is formed by
psd matrices
with

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when is rank 1, i.e.,
◦ Composite systems:

we call
pure
. Usually they are huge.
Quantum measurement: A measurement is formed by
psd matrices
with
◦ The outcome is not deterministic:

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when is rank 1, i.e.,
◦ Composite systems:

we call
pure
. Usually they are huge.
Quantum measurement: A measurement is formed by
psd matrices
with
◦ The outcome is not deterministic:
◦ Measurement will disturb the state

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when is rank 1, i.e.,
◦ Composite systems:

Quantum measurement: A measurement is formed by
psd matrices
with
◦ The outcome is not deterministic:
◦ Measurement will disturb the state

we call
pure
. Usually they are huge.
Quantum advantages:

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when is rank 1, i.e.,
◦ Composite systems:

Quantum measurement: A measurement is formed by
psd matrices
with
◦ The outcome is not deterministic:
◦ Measurement will disturb the state

we call
pure
. Usually they are huge.
Quantum advantages:
◦ More secure: QKD

Quantum state: A positive semidefinite (psd) matrix
with the trace
◦ Pure state: when is rank 1, i.e.,
◦ Composite systems:

Quantum measurement: A measurement is formed by
psd matrices
with
◦ The outcome is not deterministic:
◦ Measurement will disturb the state

we call
pure
. Usually they are huge.
Quantum advantages:
◦ More secure: QKD
◦ More efficient: Shor’s algorithm

The Bell setting: A source S distribute two physical
systems to two separated players Alice and Bob:

The Bell setting: A source S distribute two physical
systems to two separated players Alice and Bob:
◦ After departure, no communication any more

The Bell setting: A source S distribute two physical
systems to two separated players Alice and Bob:
◦ After departure, no communication any more
◦ Alice chooses a measurement
, and gets outcome

The Bell setting: A source S distribute two physical
systems to two separated players Alice and Bob:
◦ After departure, no communication any more
◦ Alice chooses a measurement
, and gets outcome
◦ Bob has similar
and
; the outputs are immediate

The Bell setting: A source S distribute two physical
systems to two separated players Alice and Bob:
◦
◦
◦
◦
After departure, no communication any more
Alice chooses a measurement
, and gets outcome
Bob has similar
and
; the outputs are immediate
After repeating many times, they record the statistics
,
and we call this a correlation – the correlation between the
(classical) inputs and the (classical) outputs

Classical correlation:

Classical correlation:

Quantum correlation:

Classical correlation:

Quantum correlation:

A major physical discovery:

Classical correlation:

Quantum correlation:

A major physical discovery:

The main idea to achieve this: Bell inequality:

Classical correlation:

Quantum correlation:

A major physical discovery:

The main idea to achieve this: Bell inequality:
is a convex polytope, thus described by linear inequalities,
and each of them is called a Bell inequality

Classical correlation:

Quantum correlation:

A major physical discovery:

The main idea to achieve this: Bell inequality:
is a convex polytope, thus described by linear inequalities,
and each of them is called a Bell inequality
◦ For a same scenario, points in
can violate some inequality

Classical correlation:

Quantum correlation:

A major physical discovery:

The main idea to achieve this: Bell inequality:
is a convex polytope, thus described by linear inequalities,
and each of them is called a Bell inequality
◦ For a same scenario, points in
can violate some inequality
◦ An example - CHSH inequality:
vs.

Classical correlation:

Quantum correlation:

A major physical discovery:

The main idea to achieve this: Bell inequality:
is a convex polytope, thus described by linear inequalities,
and each of them is called a Bell inequality
◦ For a same scenario, points in
can violate some inequality
◦ An example - CHSH inequality:
vs.
They can be verified by physical experiments!

Suppose
, and
. A correlation
in this scenario can be expressed as the following block matrix:

Suppose
, and
. A correlation
in this scenario can be expressed as the following block matrix:

Suppose
, and
. A correlation
in this scenario can be expressed as the following block matrix:
with each block being

The difficulties in physical realizations:

The difficulties in physical realizations:
◦ Measurement disturbs states, cannot clone unknown
information: error-correcting is hard, though possible

The difficulties in physical realizations:
◦ Measurement disturbs states, cannot clone unknown
information: error-correcting is hard, though possible
◦ Quantum states are fragile; memory is short

The difficulties in physical realizations:
◦ Measurement disturbs states, cannot clone unknown
information: error-correcting is hard, though possible
◦ Quantum states are fragile; memory is short
◦ The accuracy of quantum operations is limited

The difficulties in physical realizations:
◦ Measurement disturbs states, cannot clone unknown
information: error-correcting is hard, though possible
◦ Quantum states are fragile; memory is short
◦ The accuracy of quantum operations is limited

Quantum control is hard. Especially,

The difficulties in physical realizations:
◦ Measurement disturbs states, cannot clone unknown
information: error-correcting is hard, though possible
◦ Quantum states are fragile; memory is short
◦ The accuracy of quantum operations is limited

Quantum control is hard. Especially,
◦ The internal working of a quantum device is hard to monitor

Question: Can we know nontrivial internal properties of
a quantum system using very limited classical in-out?


Question: Can we know nontrivial internal properties of
a quantum system using very limited classical in-out?
Challenge1: The cost to describe a quantum system
classically is huge
◦ exponential


Question: Can we know nontrivial internal properties of
a quantum system using very limited classical in-out?
Challenge1: The cost to describe a quantum system
classically is huge
◦ exponential

Challenge2: One single measurement only reveals very
limited information of the system
◦ Typically quantum state collapses


Question: Can we know nontrivial internal properties of
a quantum system using very limited classical in-out?
Challenge1: The cost to describe a quantum system
classically is huge
◦ exponential

Challenge2: One single measurement only reveals very
limited information of the system
◦ Typically quantum state collapses

The answer is YES: the idea of Device-independent (DI)

How to understand?

How to understand?
◦ It is helpful to think the quantum system as a quantum box.
The correctness of DI comes from quantum mechanics, and has
nothing to do with how to realize the box specifically

How to understand?
◦ It is helpful to think the quantum system as a quantum box.
The correctness of DI comes from quantum mechanics, and has
nothing to do with how to realize the box specifically

The values of DI:

How to understand?
◦ It is helpful to think the quantum system as a quantum box.
The correctness of DI comes from quantum mechanics, and has
nothing to do with how to realize the box specifically

The values of DI:
◦ Theoretical values on its own

How to understand?
◦ It is helpful to think the quantum system as a quantum box.
The correctness of DI comes from quantum mechanics, and has
nothing to do with how to realize the box specifically

The values of DI:
◦ Theoretical values on its own
◦ A huge convenience for quantum tasks

How to understand?
◦ It is helpful to think the quantum system as a quantum box.
The correctness of DI comes from quantum mechanics, and has
nothing to do with how to realize the box specifically

The values of DI:
◦ Theoretical values on its own
◦ A huge convenience for quantum tasks

Known applications of DI:

How to understand?
◦ It is helpful to think the quantum system as a quantum box.
The correctness of DI comes from quantum mechanics, and has
nothing to do with how to realize the box specifically

The values of DI:
◦ Theoretical values on its own
◦ A huge convenience for quantum tasks

Known applications of DI:
◦ quantum key distribution(QKD)
◦ Entropy
◦ Entanglement

Task: Two separated quantum players want to prepare
and share a secret classical key

Task: Two separated quantum players want to prepare
and share a secret classical key
◦ They have quantum and classical channels, but unsafe

Task: Two separated quantum players want to prepare
and share a secret classical key
◦ They have quantum and classical channels, but unsafe

A DI flavor scheme*1:
*1. Artur Ekert, Phys. Rev. Lett, 1991.

Task: Two separated quantum players want to prepare
and share a secret classical key
◦ They have quantum and classical channels, but unsafe

A DI flavor scheme*1:
◦ They share a lot of EPR pairs
*1. Artur Ekert, Phys. Rev. Lett, 1991.

Task: Two separated quantum players want to prepare
and share a secret classical key
◦ They have quantum and classical channels, but unsafe

A DI flavor scheme*1:
◦ They share a lot of EPR pairs
◦ The choose the following binary POVMs randomly to measure the
qubits they have:
*1. Artur Ekert, Phys. Rev. Lett, 1991.

Task: Two separated quantum players want to prepare
and share a secret classical key
◦ They have quantum and classical channels, but unsafe

A DI flavor scheme*1:
◦ They share a lot of EPR pairs
◦ The choose the following binary POVMs randomly to measure the
qubits they have:
Alice
*1. Artur Ekert, Phys. Rev. Lett, 1991.

Task: Two separated quantum players want to prepare
and share a secret classical key
◦ They have quantum and classical channels, but unsafe

A DI flavor scheme*1:
◦ They share a lot of EPR pairs
◦ The choose the following binary POVMs randomly to measure the
qubits they have:
Alice
Bob
*1. Artur Ekert, Phys. Rev. Lett, 1991.

Task: Two separated quantum players want to prepare
and share a secret classical key
◦ They have quantum and classical channels, but unsafe

A DI flavor scheme*1:
◦ They share a lot of EPR pairs
◦ The choose the following binary POVMs randomly to measure the
qubits they have:
Alice
Bob
◦ After all measurements, they announce the choices and outcomes
*1. Artur Ekert, Phys. Rev. Lett, 1991.

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR
◦ Eve cannot be entangled to any qubit of EPR

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR
◦ Eve cannot be entangled to any qubit of EPR
◦ If the value is smaller than
, start it over

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR
◦ Eve cannot be entangled to any qubit of EPR
◦ If the value is smaller than
, start it over

The secrete key: If the shared states are EPR, the first
group of POVMs always give the same random outcomes

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR
◦ Eve cannot be entangled to any qubit of EPR
◦ If the value is smaller than
, start it over


The secrete key: If the shared states are EPR, the first
group of POVMs always give the same random outcomes
Advantage:

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR
◦ Eve cannot be entangled to any qubit of EPR
◦ If the value is smaller than
, start it over


The secrete key: If the shared states are EPR, the first
group of POVMs always give the same random outcomes
Advantage:
◦ It works even if the channel and the EPRs are prepared by enemy

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR
◦ Eve cannot be entangled to any qubit of EPR
◦ If the value is smaller than
, start it over


The secrete key: If the shared states are EPR, the first
group of POVMs always give the same random outcomes
Advantage:
◦ It works even if the channel and the EPRs are prepared by enemy
◦ Do not have to check the internal working of quantum devices

The main idea: the second group of POVMs are used to
calculate the value of the CHSH inequality:
◦ If the value is
, the shared states must be EPR
◦ Eve cannot be entangled to any qubit of EPR
◦ If the value is smaller than
, start it over


The secrete key: If the shared states are EPR, the first
group of POVMs always give the same random outcomes
Advantage:
◦ It works even if the channel and the EPRs are prepared by enemy
◦ Do not have to check the internal working of quantum devices
◦ The base of realistic DI QKD

Question: What is the optimum size of quantum state
needed to generate a given quantum correlation?

Question: What is the optimum size of quantum state
needed to generate a given quantum correlation?
◦ Size means dimension - a DI style problem

Question: What is the optimum size of quantum state
needed to generate a given quantum correlation?
◦ Size means dimension - a DI style problem

For a given
, if there exists a quantum state
on
, POVMs
and
s.t.
then
admits a d-dimensional representation.
We denote by
the minimum such d.

Question: What is the optimum size of quantum state
needed to generate a given quantum correlation?
◦ Size means dimension - a DI style problem

For a given
, if there exists a quantum state
on
, POVMs
and
s.t.
then
admits a d-dimensional representation.
We denote by
the minimum such d.

For an arbitrarily given
, , how to estimate
?

Question: What is the optimum size of quantum state
needed to generate a given quantum correlation?
◦ Size means dimension - a DI style problem

For a given
, if there exists a quantum state
on
, POVMs
and
s.t.
then
admits a d-dimensional representation.
We denote by
the minimum such d.

For an arbitrarily given
, , how to estimate
◦ Dimension is a most fundamental quantum property
?

Question: What is the optimum size of quantum state
needed to generate a given quantum correlation?
◦ Size means dimension - a DI style problem

For a given
, if there exists a quantum state
on
, POVMs
and
s.t.
then
admits a d-dimensional representation.
We denote by
the minimum such d.

For an arbitrarily given
, , how to estimate
◦ Dimension is a most fundamental quantum property
◦ Dimension is a kind of computational resource
?

Question: What is the optimum size of quantum state
needed to generate a given quantum correlation?
◦ Size means dimension - a DI style problem

For a given
, if there exists a quantum state
on
, POVMs
and
s.t.
then
admits a d-dimensional representation.
We denote by
the minimum such d.

For an arbitrarily given
, , how to estimate
◦ Dimension is a most fundamental quantum property
◦ Dimension is a kind of computational resource
?

For a fixed Bell scenario, the set of all quantum
correlations is convex. However, if restricting
dimension, it is usually not.


For a fixed Bell scenario, the set of all quantum
correlations is convex. However, if restricting
dimension, it is usually not.
By allowing free classical correlations, the set of
quantum correlations for a given dimension becomes
convex:


For a fixed Bell scenario, the set of all quantum
correlations is convex. However, if restricting
dimension, it is usually not.
By allowing free classical correlations, the set of
quantum correlations for a given dimension becomes
convex:
◦ The setting is changed a little bit


For a fixed Bell scenario, the set of all quantum
correlations is convex. However, if restricting
dimension, it is usually not.
By allowing free classical correlations, the set of
quantum correlations for a given dimension becomes
convex:
◦ The setting is changed a little bit
◦ The standard convex analysis method applies


For a fixed Bell scenario, the set of all quantum
correlations is convex. However, if restricting
dimension, it is usually not.
By allowing free classical correlations, the set of
quantum correlations for a given dimension becomes
convex:
◦ The setting is changed a little bit
◦ The standard convex analysis method applies

The main known method: dimension witness
◦ Others: entropy method

A d-dimensional witness is a linear function of the
correlation
described by a vector
s.t.
is valid for all correlations admitting d-dimensional
representation, and can be violated by some others*1
*1. Brunner et al., Phys. Rev. Lett, 2008.

A d-dimensional witness is a linear function of the
correlation
described by a vector
s.t.
is valid for all correlations admitting d-dimensional
representation, and can be violated by some others*1
◦ If violation happens, then the dimension must be larger than d
*1. Brunner et al., Phys. Rev. Lett, 2008.

A d-dimensional witness is a linear function of the
correlation
described by a vector
s.t.
is valid for all correlations admitting d-dimensional
representation, and can be violated by some others*1
◦ If violation happens, then the dimension must be larger than d
◦ For different Bell scenarios, the values of
‘s are different
*1. Brunner et al., Phys. Rev. Lett, 2008.

Cannot handle the case without public randomness


Cannot handle the case without public randomness
Even for the case with classical correlation, it is not a
direct function or bound of quantum correlations.



Cannot handle the case without public randomness
Even for the case with classical correlation, it is not a
direct function or bound of quantum correlations.
The quantum region for a fixed quantum dimension is
very hard to characterize:



Cannot handle the case without public randomness
Even for the case with classical correlation, it is not a
direct function or bound of quantum correlations.
The quantum region for a fixed quantum dimension is
very hard to characterize:
◦ The values
‘s are hard to compute



Cannot handle the case without public randomness
Even for the case with classical correlation, it is not a
direct function or bound of quantum correlations.
The quantum region for a fixed quantum dimension is
very hard to characterize:
◦ The values
‘s are hard to compute
◦ Dimension witnesses were found only on some very small
quantum systems

We go back to the initial setting, i.e., no free shared
randomness is allowed, thus not convex any more.

We go back to the initial setting, i.e., no free shared
randomness is allowed, thus not convex any more.
◦ The convex analysis approach fails

We go back to the initial setting, i.e., no free shared
randomness is allowed, thus not convex any more.
◦ The convex analysis approach fails
◦ The idea of PSD-factorization plays the key role

We go back to the initial setting, i.e., no free shared
randomness is allowed, thus not convex any more.
◦ The convex analysis approach fails
◦ The idea of PSD-factorization plays the key role

We provide an easy-to-compute lower bound for
,
which is composed by simple functions of the entries
of
:
Recall that a correlation is a block matrix:

Let
be a nonnegative
matrix, then a PSD
factorization of size is given by two sets of PSD
matrices
and
satisfying that
*1*2
*1. Gouveia, Parrilo, Thomas, Math. Oper. Res., 2013
*2. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015.

Let
be a nonnegative
matrix, then a PSD
factorization of size is given by two sets of PSD
matrices
and
satisfying that
*1*2

The PSD-rank of
smallest integer
of size .
, denoted by
, is the
such that
has a PSD factorization
*1. Gouveia, Parrilo, Thomas, Math. Oper. Res., 2013
*2. Fiorini, Massar, Pokutta, Tiwary, de Wolf, J. ACM, 2015.

Proof:

Proof:
◦ Suppose it is generated by
,
and

Proof:
◦ Suppose it is generated by
◦ Purify the state onto
,
and
with

Proof:
◦ Suppose it is generated by
◦ Purify the state onto
◦ Set
and
,
and
with

Proof:
◦
◦
◦
◦
Suppose it is generated by
Purify the state onto
Set
and
Choose
,
and
with

Proof:
◦
◦
◦
◦
Suppose it is generated by
Purify the state onto
Set
and
Choose
,
and
with

Proof:
◦
◦
◦
◦
Suppose it is generated by
Purify the state onto
Set
and
Choose
◦ Then
,
and
with

A new characterization for quantum correlations*1
*1. Sikora, Varvitsiotis, 2015.

A new characterization for quantum correlations*1
*1. Sikora, Varvitsiotis, 2015.


A new characterization for quantum correlations*1
Thus,
is the minimum d to make the above
conditions satisfied
*1. Sikora, Varvitsiotis, 2015.

Normalizing the columns of a nonnegative matrix
does not change its PSD-rank


Normalizing the columns of a nonnegative matrix
does not change its PSD-rank
Let
be a nonnegative
column summing to 1, if
exists a PSD factorization
each
has trace 1 and all
*1. Lee, Wei, de Wolf, 2014.
matrix with each
, then there
such that
sum to identity*1


Normalizing the columns of a nonnegative matrix
does not change its PSD-rank
Let
be a nonnegative
column summing to 1, if
exists a PSD factorization
each
has trace 1 and all
matrix with each
, then there
such that
sum to identity*1
◦ Each column corresponds the outcome probability distribution
of one quantum state under the same POVM
*1. Lee, Wei, de Wolf, 2014.

Measurement increase the fidelity:

Measurement increase the fidelity:

Measurement increase the fidelity:

Measurement increase the fidelity:

This is valuable to DI
Suppose
and
, and matrices
with size
satisfy
Suppose
and
, and matrices
with size
satisfy
Wlog we let the summation above be full rank, then
there exists an invertible matrix
such that
Suppose
and
, and matrices
with size
satisfy
Wlog we let the summation above be full rank, then
there exists an invertible matrix
such that
Then for any choice of
valid POVM.
,
is a
Let
factor such that
, where
is a proper
is a valid quantum state.
Let
factor such that
, where
is a proper
is a valid quantum state.
Then we have that
for any fixed
,
distribution of the outcome
the POVM of
.
, which means
is the probability
when
is measured by
Let
factor such that
, where
is a proper
is a valid quantum state.
Then we have that
for any fixed
,
distribution of the outcome
the POVM of
.
, which means
is the probability
when
is measured by
Note that the measurement does not decrease the
fidelity, thus the following is valid for any
Recall that
that
for all
quantum state
, then we have the fact
. Thus for all , we have a valid
.
Recall that
that
for all
quantum state
, then we have the fact
. Thus for all , we have a valid
.
Note that
. Since
we have that for any
,
, then it holds that
is independent in ,
.
Recall that
that
for all
quantum state
, then we have the fact
. Thus for all , we have a valid
.
Note that
. Since
we have that for any
,
Lastly, note that
that
, then it holds that
is independent in ,
.
is a quantum state on
, we have
This means that
This means that
This means that
Thus
This means that
Thus
Notice the fact that
, then
This means that
Thus
Notice the fact that
implying that
, then
This means that
Thus
Notice the fact that
implying that
Recall that for any
, then
Combining the last two together, we have that
Combining the last two together, we have that
Since this is valid for any
, it holds that

If a correlation matrix satisfies the normalization
condition, can it always be generated physically?


If a correlation matrix satisfies the normalization
condition, can it always be generated physically?
Answer: no

If a correlation matrix satisfies the normalization
condition, can it always be generated physically?

Answer: no

The correlation cannot violate relativity:

If a correlation matrix satisfies the normalization
condition, can it always be generated physically?

Answer: no

The correlation cannot violate relativity:
and
are well-defined, i.e., non-signaling

Non-signaling polytope: the set of all correlations that
respect the non-signaling rule. A PR-box is a nonlocal
vertex of this polytope

Non-signaling polytope: the set of all correlations that
respect the non-signaling rule. A PR-box is a nonlocal
vertex of this polytope
◦ Question: can it be quantum? No

Non-signaling polytope: the set of all correlations that
respect the non-signaling rule. A PR-box is a nonlocal
vertex of this polytope
◦ Question: can it be quantum? No

Set
and
PR box can be given by
Then a

Non-signaling polytope: the set of all correlations that
respect the non-signaling rule. A PR-box is a nonlocal
vertex of this polytope
◦ Question: can it be quantum? No


Set
and
PR box can be given by
Then a
The lower bound is infinite, meaning that any finitedimensional quantum system cannot generate it

A magic square is a
odd column sums
Boolean matrix with even row sums and

A magic square is a
odd column sums
◦ No such a square exists
Boolean matrix with even row sums and

A magic square is a
odd column sums
Boolean matrix with even row sums and
◦ No such a square exists

Consider the following game: Alice and Bob each gets an input
corresponding to the index of a row and a column, they are
required to give the row and the column themselves, satisfying
the above condition (no communication is allowed)
◦ A Bell setting:

A magic square is a
odd column sums
Boolean matrix with even row sums and
◦ No such a square exists

Consider the following game: Alice and Bob each gets an input
corresponding to the index of a row and a column, they are
required to give the row and the column themselves, satisfying
the above condition (no communication is allowed)
◦ A Bell setting:

No classical scheme can win for sure

A magic square is a
odd column sums
Boolean matrix with even row sums and
◦ No such a square exists

Consider the following game: Alice and Bob each gets an input
corresponding to the index of a row and a column, they are
required to give the row and the column themselves, satisfying
the above condition (no communication is allowed)
◦ A Bell setting:

No classical scheme can win for sure
◦ The best winning probability is 8/9

A magic square is a
odd column sums
Boolean matrix with even row sums and
◦ No such a square exists

Consider the following game: Alice and Bob each gets an input
corresponding to the index of a row and a column, they are
required to give the row and the column themselves, satisfying
the above condition (no communication is allowed)
◦ A Bell setting:

No classical scheme can win for sure
◦ The best winning probability is 8/9

Surprisingly, this game can be wined for sure quantumly
◦ Because of stronger correlations quantum provides

A choice of the state is

A choice of the state is
and by choosing proper POVMs the correlation can be given as

A choice of the state is
and by choosing proper POVMs the correlation can be given as

The lower bound is
which is tight

If
, i.e., each player only has one POVM,
the new lower bound goes back to a lower bound for
the PSD-rank

If
, i.e., each player only has one POVM,
the new lower bound goes back to a lower bound for
the PSD-rank
◦ Has been given: Lee, Wei, de Wolf, 2014.

This algebraic approach still works:

This algebraic approach still works:
◦ Though classical correlation is free, the combination needs more
than one quantum state

This algebraic approach still works:
◦ Though classical correlation is free, the combination needs more
than one quantum state

Suppose the target quantum correlation is generated by
mixing settings with quantum dimensions
,
then the sum of them is lower bounded by the result

We have given a lower bound for

We have given a lower bound for
◦ The bound is easy to compute

We have given a lower bound for
◦ The bound is easy to compute
◦ The bound is tight on a lot of famous examples

We have given a lower bound for
◦ The bound is easy to compute
◦ The bound is tight on a lot of famous examples
◦ Composed by simple functions, thus robust to perturbations

We have given a lower bound for
◦ The bound is easy to compute
◦ The bound is tight on a lot of famous examples
◦ Composed by simple functions, thus robust to perturbations

How to improve this?

We have given a lower bound for
◦ The bound is easy to compute
◦ The bound is tight on a lot of famous examples
◦ Composed by simple functions, thus robust to perturbations

How to improve this?
◦ It can be loose for some cases

We have given a lower bound for
◦ The bound is easy to compute
◦ The bound is tight on a lot of famous examples
◦ Composed by simple functions, thus robust to perturbations

How to improve this?
◦ It can be loose for some cases

Similar lower bound for classical correlations?

We have given a lower bound for
◦ The bound is easy to compute
◦ The bound is tight on a lot of famous examples
◦ Composed by simple functions, thus robust to perturbations

How to improve this?
◦ It can be loose for some cases

Similar lower bound for classical correlations?
◦ The gap will be interesting: one POVM case is known

We have given a lower bound for
◦ The bound is easy to compute
◦ The bound is tight on a lot of famous examples
◦ Composed by simple functions, thus robust to perturbations

How to improve this?
◦ It can be loose for some cases

Similar lower bound for classical correlations?
◦ The gap will be interesting: one POVM case is known

Other physical applications of the PSD-factorization
idea?
◦ Works for the prepare-and-measure setting
Thank you