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IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Schmidt Norms for Quantum States
Nathaniel JohnstonCollaborative work with David Kribs
University of Guelph
December 2009
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
1. Introduction
2. Schmidt Vector Norms
3. Schmidt Operator Norms
4. Applications in Quantum Information Theory
5. Conclusions
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Motivation
What we refer to as Schmidt norms have popped up in theliterature several times over the past few years...
I The Schmidt 2-norms have appeared in papers regardingNPPT bound entanglement – e.g., a 2000 paper ofDiVincenzo et al and a 2007 paper of Pankowski et al.
I The Schmidt vector norms were used by Chruscinski andKossakowski in early 2009 to develop a test for k-positivity oflinear maps.
I A conjecture of Brandao (which would implyQMA(k) = QMA(2) for k > 2) involves the Schmidt operator1-norm.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Motivation
What we refer to as Schmidt norms have popped up in theliterature several times over the past few years...
I The Schmidt 2-norms have appeared in papers regardingNPPT bound entanglement – e.g., a 2000 paper ofDiVincenzo et al and a 2007 paper of Pankowski et al.
I The Schmidt vector norms were used by Chruscinski andKossakowski in early 2009 to develop a test for k-positivity oflinear maps.
I A conjecture of Brandao (which would implyQMA(k) = QMA(2) for k > 2) involves the Schmidt operator1-norm.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Motivation
What we refer to as Schmidt norms have popped up in theliterature several times over the past few years...
I The Schmidt 2-norms have appeared in papers regardingNPPT bound entanglement – e.g., a 2000 paper ofDiVincenzo et al and a 2007 paper of Pankowski et al.
I The Schmidt vector norms were used by Chruscinski andKossakowski in early 2009 to develop a test for k-positivity oflinear maps.
I A conjecture of Brandao (which would implyQMA(k) = QMA(2) for k > 2) involves the Schmidt operator1-norm.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Motivation
What we refer to as Schmidt norms have popped up in theliterature several times over the past few years...
I The Schmidt 2-norms have appeared in papers regardingNPPT bound entanglement – e.g., a 2000 paper ofDiVincenzo et al and a 2007 paper of Pankowski et al.
I The Schmidt vector norms were used by Chruscinski andKossakowski in early 2009 to develop a test for k-positivity oflinear maps.
I A conjecture of Brandao (which would implyQMA(k) = QMA(2) for k > 2) involves the Schmidt operator1-norm.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Motivation
Nonetheless, these norms have not been systematically studied inthe past. We develop the basic mathematical properties of thesenorms, and then study their implications in quantum informationtheory.
But first...
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Motivation
Nonetheless, these norms have not been systematically studied inthe past. We develop the basic mathematical properties of thesenorms, and then study their implications in quantum informationtheory.
But first...
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Notation
I H is a (finite-dimensional) Hilbert space. Hn is ann-dimensional Hilbert space.
I L(H) is the space of linear operators acting on H.
I idn represents the identity map on L(Hn).
I |v〉 ∈ H is a unit (column) vector represented using Diracnotation. 〈v | := |v〉∗ is the dual (row) vector. The associatedprojection |v〉〈v | represents a pure state.
I SR(|v〉) is the Schmidt rank (a.k.a. the tensor rank) of thebipartite pure state |v〉. SN(ρ) is the Schmidt number of thebipartite density operator ρ.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Notation
I H is a (finite-dimensional) Hilbert space. Hn is ann-dimensional Hilbert space.
I L(H) is the space of linear operators acting on H.
I idn represents the identity map on L(Hn).
I |v〉 ∈ H is a unit (column) vector represented using Diracnotation. 〈v | := |v〉∗ is the dual (row) vector. The associatedprojection |v〉〈v | represents a pure state.
I SR(|v〉) is the Schmidt rank (a.k.a. the tensor rank) of thebipartite pure state |v〉. SN(ρ) is the Schmidt number of thebipartite density operator ρ.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Notation
I H is a (finite-dimensional) Hilbert space. Hn is ann-dimensional Hilbert space.
I L(H) is the space of linear operators acting on H.
I idn represents the identity map on L(Hn).
I |v〉 ∈ H is a unit (column) vector represented using Diracnotation. 〈v | := |v〉∗ is the dual (row) vector. The associatedprojection |v〉〈v | represents a pure state.
I SR(|v〉) is the Schmidt rank (a.k.a. the tensor rank) of thebipartite pure state |v〉. SN(ρ) is the Schmidt number of thebipartite density operator ρ.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Notation
I H is a (finite-dimensional) Hilbert space. Hn is ann-dimensional Hilbert space.
I L(H) is the space of linear operators acting on H.
I idn represents the identity map on L(Hn).
I |v〉 ∈ H is a unit (column) vector represented using Diracnotation. 〈v | := |v〉∗ is the dual (row) vector. The associatedprojection |v〉〈v | represents a pure state.
I SR(|v〉) is the Schmidt rank (a.k.a. the tensor rank) of thebipartite pure state |v〉. SN(ρ) is the Schmidt number of thebipartite density operator ρ.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
Notation
I H is a (finite-dimensional) Hilbert space. Hn is ann-dimensional Hilbert space.
I L(H) is the space of linear operators acting on H.
I idn represents the identity map on L(Hn).
I |v〉 ∈ H is a unit (column) vector represented using Diracnotation. 〈v | := |v〉∗ is the dual (row) vector. The associatedprojection |v〉〈v | represents a pure state.
I SR(|v〉) is the Schmidt rank (a.k.a. the tensor rank) of thebipartite pure state |v〉. SN(ρ) is the Schmidt number of thebipartite density operator ρ.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
k-Block Positive Operators
An operator X = X ∗ ∈ L(Hn)⊗ L(Hn) is said to be k-blockpositive if
〈v |X |v〉 ≥ 0 ∀ |v〉 with SR(|v〉) ≤ k .
I The n-block positive operators are exactly the positiveoperators, since SR(|v〉) ≤ n for all vectors |v〉.
I These operators are sometimes referred to ask-entanglement witnesses because SN(ρ) ≤ k if and only if
Tr(Xρ) ≥ 0 ∀ k-block positive X .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
k-Block Positive Operators
An operator X = X ∗ ∈ L(Hn)⊗ L(Hn) is said to be k-blockpositive if
〈v |X |v〉 ≥ 0 ∀ |v〉 with SR(|v〉) ≤ k .
I The n-block positive operators are exactly the positiveoperators, since SR(|v〉) ≤ n for all vectors |v〉.
I These operators are sometimes referred to ask-entanglement witnesses because SN(ρ) ≤ k if and only if
Tr(Xρ) ≥ 0 ∀ k-block positive X .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
MotivationNotationk-Block Positive Operators
k-Block Positive Operators
An operator X = X ∗ ∈ L(Hn)⊗ L(Hn) is said to be k-blockpositive if
〈v |X |v〉 ≥ 0 ∀ |v〉 with SR(|v〉) ≤ k .
I The n-block positive operators are exactly the positiveoperators, since SR(|v〉) ≤ n for all vectors |v〉.
I These operators are sometimes referred to ask-entanglement witnesses because SN(ρ) ≤ k if and only if
Tr(Xρ) ≥ 0 ∀ k-block positive X .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Vector Norms
Let |v〉 ∈ Hn ⊗Hn and let 1 ≤ k ≤ n. Then we define theSchmidt vector k-norm of |v〉 by∥∥|v〉∥∥
s(k):= sup|w〉
{∣∣〈w |v〉∣∣ : SR(|w〉) ≤ k}.
I Yes, these are actually norms.
I If k = n, then this is just the standard Euclidean norm. Thatis,∥∥|v〉∥∥
s(n)=∥∥|v〉∥∥.
I∥∥|v〉∥∥
s(1)≤∥∥|v〉∥∥
s(2)≤ · · · ≤
∥∥|v〉∥∥s(n−1)
≤∥∥|v〉∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Vector Norms
Let |v〉 ∈ Hn ⊗Hn and let 1 ≤ k ≤ n. Then we define theSchmidt vector k-norm of |v〉 by∥∥|v〉∥∥
s(k):= sup|w〉
{∣∣〈w |v〉∣∣ : SR(|w〉) ≤ k}.
I Yes, these are actually norms.
I If k = n, then this is just the standard Euclidean norm. Thatis,∥∥|v〉∥∥
s(n)=∥∥|v〉∥∥.
I∥∥|v〉∥∥
s(1)≤∥∥|v〉∥∥
s(2)≤ · · · ≤
∥∥|v〉∥∥s(n−1)
≤∥∥|v〉∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Vector Norms
Let |v〉 ∈ Hn ⊗Hn and let 1 ≤ k ≤ n. Then we define theSchmidt vector k-norm of |v〉 by∥∥|v〉∥∥
s(k):= sup|w〉
{∣∣〈w |v〉∣∣ : SR(|w〉) ≤ k}.
I Yes, these are actually norms.
I If k = n, then this is just the standard Euclidean norm. Thatis,∥∥|v〉∥∥
s(n)=∥∥|v〉∥∥.
I∥∥|v〉∥∥
s(1)≤∥∥|v〉∥∥
s(2)≤ · · · ≤
∥∥|v〉∥∥s(n−1)
≤∥∥|v〉∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Vector Norms
Let |v〉 ∈ Hn ⊗Hn and let 1 ≤ k ≤ n. Then we define theSchmidt vector k-norm of |v〉 by∥∥|v〉∥∥
s(k):= sup|w〉
{∣∣〈w |v〉∣∣ : SR(|w〉) ≤ k}.
I Yes, these are actually norms.
I If k = n, then this is just the standard Euclidean norm. Thatis,∥∥|v〉∥∥
s(n)=∥∥|v〉∥∥.
I∥∥|v〉∥∥
s(1)≤∥∥|v〉∥∥
s(2)≤ · · · ≤
∥∥|v〉∥∥s(n−1)
≤∥∥|v〉∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Computation
From now on, when we refer to the Schmidt coefficients {αi} of avector, we will assume that they are written in non-increasing order(i.e., α1 ≥ α2 ≥ · · · ≥ αn ≥ 0).
The Schmidt vector norms are actually very simple to compute, asthe following theorem demonstrates:
TheoremLet |v〉 ∈ Hn ⊗Hn have Schmidt coefficients
{αi
}. Then
∥∥|v〉∥∥s(k)
=
√√√√ k∑i=1
α2i .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Computation
From now on, when we refer to the Schmidt coefficients {αi} of avector, we will assume that they are written in non-increasing order(i.e., α1 ≥ α2 ≥ · · · ≥ αn ≥ 0).
The Schmidt vector norms are actually very simple to compute, asthe following theorem demonstrates:
TheoremLet |v〉 ∈ Hn ⊗Hn have Schmidt coefficients
{αi
}. Then
∥∥|v〉∥∥s(k)
=
√√√√ k∑i=1
α2i .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Operator Norms
Let X ∈ L(Hn)⊗ L(Hn) and let 1 ≤ k ≤ n. Then we define theSchmidt operator k-norm of X by∥∥X
∥∥S(k)
:= sup|v〉,|w〉
{∣∣〈w |X |v〉∣∣ : SR(|v〉),SR(|w〉) ≤ k}.
I Yes, these too are actually norms.
I If k = n, then this is the standard operator norm. That is,∥∥X∥∥S(n)
=∥∥X∥∥.
I∥∥X∥∥S(1)≤∥∥X∥∥S(2)≤ · · · ≤
∥∥X∥∥S(n−1)
≤∥∥X∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Operator Norms
Let X ∈ L(Hn)⊗ L(Hn) and let 1 ≤ k ≤ n. Then we define theSchmidt operator k-norm of X by∥∥X
∥∥S(k)
:= sup|v〉,|w〉
{∣∣〈w |X |v〉∣∣ : SR(|v〉),SR(|w〉) ≤ k}.
I Yes, these too are actually norms.
I If k = n, then this is the standard operator norm. That is,∥∥X∥∥S(n)
=∥∥X∥∥.
I∥∥X∥∥S(1)≤∥∥X∥∥S(2)≤ · · · ≤
∥∥X∥∥S(n−1)
≤∥∥X∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Operator Norms
Let X ∈ L(Hn)⊗ L(Hn) and let 1 ≤ k ≤ n. Then we define theSchmidt operator k-norm of X by∥∥X
∥∥S(k)
:= sup|v〉,|w〉
{∣∣〈w |X |v〉∣∣ : SR(|v〉),SR(|w〉) ≤ k}.
I Yes, these too are actually norms.
I If k = n, then this is the standard operator norm. That is,∥∥X∥∥S(n)
=∥∥X∥∥.
I∥∥X∥∥S(1)≤∥∥X∥∥S(2)≤ · · · ≤
∥∥X∥∥S(n−1)
≤∥∥X∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Schmidt Operator Norms
Let X ∈ L(Hn)⊗ L(Hn) and let 1 ≤ k ≤ n. Then we define theSchmidt operator k-norm of X by∥∥X
∥∥S(k)
:= sup|v〉,|w〉
{∣∣〈w |X |v〉∣∣ : SR(|v〉),SR(|w〉) ≤ k}.
I Yes, these too are actually norms.
I If k = n, then this is the standard operator norm. That is,∥∥X∥∥S(n)
=∥∥X∥∥.
I∥∥X∥∥S(1)≤∥∥X∥∥S(2)≤ · · · ≤
∥∥X∥∥S(n−1)
≤∥∥X∥∥
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Computation
Good luck!
I Computing Schmidt operator norms (even just for positiveoperators) is equivalent to the problem of determining k-blockpositivity of an operator.
I Determining k-block positivity has been extensively studied bymathematicians for some 30 years, but no complete method isknown. This suggests that computing Schmidt operatornorms is very difficult.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Computation
Good luck!
I Computing Schmidt operator norms (even just for positiveoperators) is equivalent to the problem of determining k-blockpositivity of an operator.
I Determining k-block positivity has been extensively studied bymathematicians for some 30 years, but no complete method isknown. This suggests that computing Schmidt operatornorms is very difficult.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Computation
Good luck!
I Computing Schmidt operator norms (even just for positiveoperators) is equivalent to the problem of determining k-blockpositivity of an operator.
I Determining k-block positivity has been extensively studied bymathematicians for some 30 years, but no complete method isknown. This suggests that computing Schmidt operatornorms is very difficult.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
DefinitionComputation
Computation
Nonetheless, we can derive many inequalities to bound theSchmidt operator norms (or even compute them exactly) in certainsituations. These results lead to a multitude of tests for k-blockpositivity, which help us tackle the NPPT bound entanglementproblem.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block PositivityMany tests have been derived over the years for determiningwhether or not an operator is k-block positive in certain situations.
I These tests mostly seem to be completely unrelated and eachprovide “bits-and-pieces” that seem to only work in veryspecific situations.
I We derive a general result for testing k-block positivity basedon Schmidt operator norms. We then use our inequalities toderive “computable” tests for k-block positivity.
I Known tests that follow from our general result includeTakesaki and Tomiyama (1983), Benatti, Floreanini, and Piani(2004), Kuah and Sudarshan (2005), and Chruscinski andKossakowski (2009).
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block PositivityMany tests have been derived over the years for determiningwhether or not an operator is k-block positive in certain situations.
I These tests mostly seem to be completely unrelated and eachprovide “bits-and-pieces” that seem to only work in veryspecific situations.
I We derive a general result for testing k-block positivity basedon Schmidt operator norms. We then use our inequalities toderive “computable” tests for k-block positivity.
I Known tests that follow from our general result includeTakesaki and Tomiyama (1983), Benatti, Floreanini, and Piani(2004), Kuah and Sudarshan (2005), and Chruscinski andKossakowski (2009).
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block PositivityMany tests have been derived over the years for determiningwhether or not an operator is k-block positive in certain situations.
I These tests mostly seem to be completely unrelated and eachprovide “bits-and-pieces” that seem to only work in veryspecific situations.
I We derive a general result for testing k-block positivity basedon Schmidt operator norms. We then use our inequalities toderive “computable” tests for k-block positivity.
I Known tests that follow from our general result includeTakesaki and Tomiyama (1983), Benatti, Floreanini, and Piani(2004), Kuah and Sudarshan (2005), and Chruscinski andKossakowski (2009).
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block PositivityMany tests have been derived over the years for determiningwhether or not an operator is k-block positive in certain situations.
I These tests mostly seem to be completely unrelated and eachprovide “bits-and-pieces” that seem to only work in veryspecific situations.
I We derive a general result for testing k-block positivity basedon Schmidt operator norms. We then use our inequalities toderive “computable” tests for k-block positivity.
I Known tests that follow from our general result includeTakesaki and Tomiyama (1983), Benatti, Floreanini, and Piani(2004), Kuah and Sudarshan (2005), and Chruscinski andKossakowski (2009).
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
Because the general result is quite technical, we present only someof the new tests that follow from it.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
We can derive some simple tests for k-block positivity based on thenegative eigenvalues of X :
Theorem (number of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive. Then ithas at most (n − k)2 negative eigenvalues.
Theorem (magnitude of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive. Denotethe maximal and minimal eigenvalues of X by λmax and λmin,respectively. Then
λmin
λmax≥ 1− n
k.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
We can derive some simple tests for k-block positivity based on thenegative eigenvalues of X :
Theorem (number of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive. Then ithas at most (n − k)2 negative eigenvalues.
Theorem (magnitude of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive. Denotethe maximal and minimal eigenvalues of X by λmax and λmin,respectively. Then
λmin
λmax≥ 1− n
k.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
We can derive some simple tests for k-block positivity based on thenegative eigenvalues of X :
Theorem (number of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive. Then ithas at most (n − k)2 negative eigenvalues.
Theorem (magnitude of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive. Denotethe maximal and minimal eigenvalues of X by λmax and λmin,respectively. Then
λmin
λmax≥ 1− n
k.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
The following result “interpolates” between the two results on theprevious slide.
Theorem (number and magnitude of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive with rnegative eigenvalues. Denote the maximal and minimal eigenvaluesof X by λmax and λmin, respectively. Then
λmin
λmax≥ 1−
⌈(n −√
r − 1)⌉
kand
λmin
λmax≥ 1− n2(n − 1)
(k − 1)n2 + (n − k)r.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
The following result “interpolates” between the two results on theprevious slide.
Theorem (number and magnitude of negative eigenvalues)
Suppose X = X ∗ ∈ L(Hn)⊗ L(Hn) is k-block positive with rnegative eigenvalues. Denote the maximal and minimal eigenvaluesof X by λmax and λmin, respectively. Then
λmin
λmax≥ 1−
⌈(n −√
r − 1)⌉
kand
λmin
λmax≥ 1− n2(n − 1)
(k − 1)n2 + (n − k)r.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
The Schmidt operator norms provide a necessary and sufficientcondition for an operator with exactly two distinct eigenvalues tobe k-block positive.
TheoremLet X = X ∗ ∈ L(Hn)⊗ L(Hn) have exactly two distincteigenvalues λ1 > λ2. Let P− be the projection onto the negativepart of X . Then X is k-block positive if and only if
‖P−‖S(k) ≤λ1
λ1 − λ2.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
The Schmidt operator norms provide a necessary and sufficientcondition for an operator with exactly two distinct eigenvalues tobe k-block positive.
TheoremLet X = X ∗ ∈ L(Hn)⊗ L(Hn) have exactly two distincteigenvalues λ1 > λ2. Let P− be the projection onto the negativepart of X . Then X is k-block positive if and only if
‖P−‖S(k) ≤λ1
λ1 − λ2.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Testing k-Block Positivity
The result on the previous slide is useful for helping tackle theproblem of whether or not there exist NPPT bound entangledstates. The rest of this talk will concern this problem, so we willnow introduce Werner states and bound entanglement.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
A state ρ ∈ L(Hn)⊗ L(Hn) is said to be bound entangled if(idn ⊗ T )(ρ)⊗` ∈ L(H⊗`n )⊗ L(H⊗`n ) is 2-block positive for all` ≥ 1, where T denotes the transpose operation.
I Equivalently, a state is bound entangled if it has zerodistillable entanglement.
I This means that bound entangled states are states that cannot be transformed via local operations and classicalcommunication into a pure maximally entangled state.
I Separable states are clearly bound entangled.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
A state ρ ∈ L(Hn)⊗ L(Hn) is said to be bound entangled if(idn ⊗ T )(ρ)⊗` ∈ L(H⊗`n )⊗ L(H⊗`n ) is 2-block positive for all` ≥ 1, where T denotes the transpose operation.
I Equivalently, a state is bound entangled if it has zerodistillable entanglement.
I This means that bound entangled states are states that cannot be transformed via local operations and classicalcommunication into a pure maximally entangled state.
I Separable states are clearly bound entangled.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
A state ρ ∈ L(Hn)⊗ L(Hn) is said to be bound entangled if(idn ⊗ T )(ρ)⊗` ∈ L(H⊗`n )⊗ L(H⊗`n ) is 2-block positive for all` ≥ 1, where T denotes the transpose operation.
I Equivalently, a state is bound entangled if it has zerodistillable entanglement.
I This means that bound entangled states are states that cannot be transformed via local operations and classicalcommunication into a pure maximally entangled state.
I Separable states are clearly bound entangled.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
A state ρ ∈ L(Hn)⊗ L(Hn) is said to be bound entangled if(idn ⊗ T )(ρ)⊗` ∈ L(H⊗`n )⊗ L(H⊗`n ) is 2-block positive for all` ≥ 1, where T denotes the transpose operation.
I Equivalently, a state is bound entangled if it has zerodistillable entanglement.
I This means that bound entangled states are states that cannot be transformed via local operations and classicalcommunication into a pure maximally entangled state.
I Separable states are clearly bound entangled.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
I Positive partial transpose (PPT) states are also boundentangled. That is, if ρ ≥ 0 and (idn ⊗ T )(ρ) ≥ 0 then ρ isbound entangled.
I It has been an open question for over a decade whether or notother bound entangled states exist.
I That is, does there exist a bound entangled state ρ withnon-positive partial transpose (NPPT)?
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
I Positive partial transpose (PPT) states are also boundentangled. That is, if ρ ≥ 0 and (idn ⊗ T )(ρ) ≥ 0 then ρ isbound entangled.
I It has been an open question for over a decade whether or notother bound entangled states exist.
I That is, does there exist a bound entangled state ρ withnon-positive partial transpose (NPPT)?
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
I Positive partial transpose (PPT) states are also boundentangled. That is, if ρ ≥ 0 and (idn ⊗ T )(ρ) ≥ 0 then ρ isbound entangled.
I It has been an open question for over a decade whether or notother bound entangled states exist.
I That is, does there exist a bound entangled state ρ withnon-positive partial transpose (NPPT)?
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
I Mathematically, does there exist a density operator ρ suchthat (idn ⊗ T )(ρ) � 0 yet (idn ⊗ T )(ρ)⊗` is 2-block positivefor all ` ≥ 1?
I It has been shown that it is enough to consider Werner states– that is, there exist NPPT bound entangled states if and onlyif there exist NPPT bound entangled Werner states.
I So let’s look at Werner states!
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
I Mathematically, does there exist a density operator ρ suchthat (idn ⊗ T )(ρ) � 0 yet (idn ⊗ T )(ρ)⊗` is 2-block positivefor all ` ≥ 1?
I It has been shown that it is enough to consider Werner states– that is, there exist NPPT bound entangled states if and onlyif there exist NPPT bound entangled Werner states.
I So let’s look at Werner states!
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled States
I Mathematically, does there exist a density operator ρ suchthat (idn ⊗ T )(ρ) � 0 yet (idn ⊗ T )(ρ)⊗` is 2-block positivefor all ` ≥ 1?
I It has been shown that it is enough to consider Werner states– that is, there exist NPPT bound entangled states if and onlyif there exist NPPT bound entangled Werner states.
I So let’s look at Werner states!
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Werner States
Let S ∈ L(Hn)⊗ L(Hn) be the swap operator that maps |a〉 ⊗ |b〉to |b〉 ⊗ |a〉. Then Werner states are the density operators of thefollowing form, which are parametrized by a single real variableα ∈ [−1, 1]:
ρα :=1
n2 − αn(I − αS) ∈ L(Hn)⊗ L(Hn).
I From now on we will ignore the scaling factor of 1n2−αn in
front of the Werner state, since positive scalars don’t affectblock positivity.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Werner States
Let S ∈ L(Hn)⊗ L(Hn) be the swap operator that maps |a〉 ⊗ |b〉to |b〉 ⊗ |a〉. Then Werner states are the density operators of thefollowing form, which are parametrized by a single real variableα ∈ [−1, 1]:
ρα :=1
n2 − αn(I − αS) ∈ L(Hn)⊗ L(Hn).
I From now on we will ignore the scaling factor of 1n2−αn in
front of the Werner state, since positive scalars don’t affectblock positivity.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
The partial transpose of a Werner state is of the form
(idn ⊗ T )(ρα) = I − αnE ,
where E := 1n
∑ni ,j=1 |i〉〈j | ⊗ |i〉〈j | is the projection onto the
“standard” pure maximally entangled state.
I This operator has only 2 distinct eigenvalues! This means thatour k-block positivity test from earlier applies in this situation.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
The partial transpose of a Werner state is of the form
(idn ⊗ T )(ρα) = I − αnE ,
where E := 1n
∑ni ,j=1 |i〉〈j | ⊗ |i〉〈j | is the projection onto the
“standard” pure maximally entangled state.
I This operator has only 2 distinct eigenvalues! This means thatour k-block positivity test from earlier applies in this situation.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
TheoremLet ρα be a Werner state. Then (idn ⊗ T )(ρα) is k-block positiveif and only if α ≤ 1
k .
I Two special cases of this result are well-known. First, ρα haspositive partial transpose if and only if α ≤ 1
n . Second, ρα is2-block positive if and only if α ≤ 1
2 .
I This shows that the “interesting region” of α’s is α ∈ ( 1n ,
12 ],
since ρα is PPT for α ≤ 1n and (idn ⊗ T )(ρα) is not 2-block
positive (and hence ρα is not bound entangled) for α > 12 .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
TheoremLet ρα be a Werner state. Then (idn ⊗ T )(ρα) is k-block positiveif and only if α ≤ 1
k .
I Two special cases of this result are well-known. First, ρα haspositive partial transpose if and only if α ≤ 1
n . Second, ρα is2-block positive if and only if α ≤ 1
2 .
I This shows that the “interesting region” of α’s is α ∈ ( 1n ,
12 ],
since ρα is PPT for α ≤ 1n and (idn ⊗ T )(ρα) is not 2-block
positive (and hence ρα is not bound entangled) for α > 12 .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
TheoremLet ρα be a Werner state. Then (idn ⊗ T )(ρα) is k-block positiveif and only if α ≤ 1
k .
I Two special cases of this result are well-known. First, ρα haspositive partial transpose if and only if α ≤ 1
n . Second, ρα is2-block positive if and only if α ≤ 1
2 .
I This shows that the “interesting region” of α’s is α ∈ ( 1n ,
12 ],
since ρα is PPT for α ≤ 1n and (idn ⊗ T )(ρα) is not 2-block
positive (and hence ρα is not bound entangled) for α > 12 .
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
Numerical evidence suggests that ρα is bound entangled for allα ∈ ( 1
n ,12 ]. Actually proving this seems to be out of reach,
although we are able to rephrase the problem in terms of Schmidtoperator norms...
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
TheoremFor n ≥ 4, the state ρ2/n is bound entangled if and only if
lim`→∞∥∥P`∥∥S(2)
= 12 , where P` is the orthogonal projection
defined recursively via
P1 := E ,
P`+1 := E ⊗ (I − P`) + (I − E )⊗ P`, for ` ≥ 1.
So get calculating!
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Testing k-Block PositivityBound Entanglement and Werner States
Bound Entangled Werner States
TheoremFor n ≥ 4, the state ρ2/n is bound entangled if and only if
lim`→∞∥∥P`∥∥S(2)
= 12 , where P` is the orthogonal projection
defined recursively via
P1 := E ,
P`+1 := E ⊗ (I − P`) + (I − E )⊗ P`, for ` ≥ 1.
So get calculating!
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Concluding RemarksFurther Reading
Concluding Remarks
I Schmidt norms are powerful tools for determining k-blockpositivity of Hermitian operators.
I A method of computing these norms on a certain “highlyentangled” family of projections would help solve the NPPTbound entanglement problem.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Concluding RemarksFurther Reading
Concluding Remarks
I Schmidt norms are powerful tools for determining k-blockpositivity of Hermitian operators.
I A method of computing these norms on a certain “highlyentangled” family of projections would help solve the NPPTbound entanglement problem.
Nathaniel Johnston Schmidt Norms for Quantum States
IntroductionSchmidt Vector Norms
Schmidt Operator NormsApplications in Quantum Information Theory
Conclusions
Concluding RemarksFurther Reading
Further Reading
N. Johnston, D. W. Kribs, Schmidt Norms for QuantumStates, preprint (2009). arXiv:0909.3907 [quant-ph]
Nathaniel Johnston Schmidt Norms for Quantum States
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