Schmidt Norms for Quantum Statesnjohns01home.webfactional.com/wp-content/uploads/... · Kossakowski in early 2009 to develop a test for k-positivity of linear maps. I A conjecture

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



Conclusions

1. Introduction

2. Schmidt Vector Norms

3. Schmidt Operator Norms

4. Applications in Quantum Information Theory

5. Conclusions




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.




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Motivation








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Motivation








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Motivation








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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...




Conclusions


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...




Conclusions


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.




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Notation









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Conclusions


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 .




Conclusions




〈v |X |v〉 ≥ 0 ∀ |v〉 with SR(|v〉) ≤ k .







Conclusions




〈v |X |v〉 ≥ 0 ∀ |v〉 with SR(|v〉) ≤ k .







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〉∥∥




Conclusions




s(k):= sup|w〉

{∣∣〈w |v〉∣∣ : SR(|w〉) ≤ k}.



s(n)=∥∥|v〉∥∥.

I∥∥|v〉∥∥

s(1)≤∥∥|v〉∥∥

s(2)≤ · · · ≤

∥∥|v〉∥∥s(n−1)

≤∥∥|v〉∥∥




Conclusions




s(k):= sup|w〉

{∣∣〈w |v〉∣∣ : SR(|w〉) ≤ k}.



s(n)=∥∥|v〉∥∥.

I∥∥|v〉∥∥

s(1)≤∥∥|v〉∥∥

s(2)≤ · · · ≤

∥∥|v〉∥∥s(n−1)

≤∥∥|v〉∥∥




Conclusions




s(k):= sup|w〉

{∣∣〈w |v〉∣∣ : SR(|w〉) ≤ k}.



s(n)=∥∥|v〉∥∥.

I∥∥|v〉∥∥

s(1)≤∥∥|v〉∥∥

s(2)≤ · · · ≤

∥∥|v〉∥∥s(n−1)

≤∥∥|v〉∥∥




Conclusions


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 .




Conclusions


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 .




Conclusions


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∥∥




Conclusions




∥∥S(k)

:= sup|v〉,|w〉

{∣∣〈w |X |v〉∣∣ : SR(|v〉),SR(|w〉) ≤ k}.



=∥∥X∥∥.

I∥∥X∥∥S(1)≤∥∥X∥∥S(2)≤ · · · ≤

∥∥X∥∥S(n−1)

≤∥∥X∥∥




Conclusions




∥∥S(k)

:= sup|v〉,|w〉

{∣∣〈w |X |v〉∣∣ : SR(|v〉),SR(|w〉) ≤ k}.



=∥∥X∥∥.

I∥∥X∥∥S(1)≤∥∥X∥∥S(2)≤ · · · ≤

∥∥X∥∥S(n−1)

≤∥∥X∥∥




Conclusions




∥∥S(k)

:= sup|v〉,|w〉

{∣∣〈w |X |v〉∣∣ : SR(|v〉),SR(|w〉) ≤ k}.



=∥∥X∥∥.

I∥∥X∥∥S(1)≤∥∥X∥∥S(2)≤ · · · ≤

∥∥X∥∥S(n−1)

≤∥∥X∥∥




Conclusions


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.




Conclusions


Computation

Good luck!






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Computation

Good luck!






Conclusions


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.




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).




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Testing k-Block Positivity

Because the general result is quite technical, we present only someof the new tests that follow from it.




Conclusions



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.




Conclusions








λmin

λmax≥ 1− n

k.




Conclusions








λmin

λmax≥ 1− n

k.




Conclusions



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.




Conclusions



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.




Conclusions



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.




Conclusions



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.




Conclusions



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.




Conclusions


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.




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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)?




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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!




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Conclusions


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.




Conclusions


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.




Conclusions


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.




Conclusions



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.




Conclusions



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 .




Conclusions




k .



2 .


12 ],






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k .



2 .


12 ],






Conclusions



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...




Conclusions



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!




Conclusions



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!




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.




Conclusions


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.




Conclusions


Further Reading

N. Johnston, D. W. Kribs, Schmidt Norms for QuantumStates, preprint (2009). arXiv:0909.3907 [quant-ph]


Documents

Schmidt Norms for Quantum Statesnjohns01home.webfactional.com/wp-content/uploads/... · Kossakowski in early 2009 to develop a test for k-positivity of linear maps. I A conjecture