Preserver problems in quantum information science

Preserver problems in quantum information science

Chi-Kwong Li Department of Mathematics The College of William and Mary

Based on recent work with:

Shmuel Friedland (University of Illinois - Chicago), Yiu-Tung Poon (Iowa State University), Nung-Sing Sze (Hong Kong Polytechnic University),

Preserver problems in quantum information science

Chi-Kwong Li Department of Mathematics The College of William and Mary

Based on recent work with:

Shmuel Friedland (University of Illinois - Chicago), Yiu-Tung Poon (Iowa State University), Nung-Sing Sze (Hong Kong Polytechnic University),

Preserver problems in quantum information science

Chi-Kwong Li Department of Mathematics The College of William and Mary

Based on recent work with:

Shmuel Friedland (University of Illinois - Chicago),

Yiu-Tung Poon (Iowa State University), Nung-Sing Sze (Hong Kong Polytechnic University),

Preserver problems in quantum information science

Chi-Kwong Li Department of Mathematics The College of William and Mary

Based on recent work with:

Shmuel Friedland (University of Illinois - Chicago), Yiu-Tung Poon (Iowa State University),

Nung-Sing Sze (Hong Kong Polytechnic University),

Preserver problems in quantum information science

Chi-Kwong Li Department of Mathematics The College of William and Mary

Based on recent work with:

Shmuel Friedland (University of Illinois - Chicago),

Dedication

It is my great pleasure to join the colleagues from 14 regions:

Australia, Brazil, Canada, Estonia, Germany, Israel, Korea, Malaysia, Poland, Slovenia, South Africa, USA, Taiwan, Thailand,

inAfrica, Asia, Australia, Europe, (North and South) America to honor Professor Pjek-Hwee Lee on the occasion of his retirement.

In fact, I am also affiliated with the following institutions:

Hong Kong University of Science & Technology (2011 Fulbright Fellow) Taiyuan University of Technology (Shanxi 100 Talent Program) Shanghai University, University of Hong Kong (Honorary Professor).

Dedication

It is my great pleasure to join the colleagues from 14 regions:

Australia, Brazil, Canada, Estonia, Germany, Israel, Korea, Malaysia, Poland, Slovenia, South Africa, USA, Taiwan, Thailand,

inAfrica, Asia, Australia, Europe, (North and South) America to honor Professor Pjek-Hwee Lee on the occasion of his retirement.

In fact, I am also affiliated with the following institutions:

Hong Kong University of Science & Technology (2011 Fulbright Fellow) Taiyuan University of Technology (Shanxi 100 Talent Program) Shanghai University, University of Hong Kong (Honorary Professor).

Dedication

It is my great pleasure to join the colleagues from 14 regions:

Australia, Brazil, Canada, Estonia, Germany, Israel, Korea, Malaysia, Poland, Slovenia, South Africa, USA, Taiwan, Thailand,

inAfrica, Asia, Australia, Europe, (North and South) America

to honor Professor Pjek-Hwee Lee on the occasion of his retirement.

In fact, I am also affiliated with the following institutions:

Hong Kong University of Science & Technology (2011 Fulbright Fellow) Taiyuan University of Technology (Shanxi 100 Talent Program) Shanghai University, University of Hong Kong (Honorary Professor).

Dedication

It is my great pleasure to join the colleagues from 14 regions:

Australia, Brazil, Canada, Estonia, Germany, Israel, Korea, Malaysia, Poland, Slovenia, South Africa, USA, Taiwan, Thailand,

inAfrica, Asia, Australia, Europe, (North and South) America to honor Professor Pjek-Hwee Lee on the occasion of his retirement.

In fact, I am also affiliated with the following institutions:

Hong Kong University of Science & Technology (2011 Fulbright Fellow) Taiyuan University of Technology (Shanxi 100 Talent Program) Shanghai University, University of Hong Kong (Honorary Professor).

Dedication

It is my great pleasure to join the colleagues from 14 regions:

Australia, Brazil, Canada, Estonia, Germany, Israel, Korea, Malaysia, Poland, Slovenia, South Africa, USA, Taiwan, Thailand,

inAfrica, Asia, Australia, Europe, (North and South) America to honor Professor Pjek-Hwee Lee on the occasion of his retirement.

In fact, I am also affiliated with the following institutions:

Hong Kong University of Science & Technology (2011 Fulbright Fellow)

Taiyuan University of Technology (Shanxi 100 Talent Program) Shanghai University, University of Hong Kong (Honorary Professor).

Dedication

It is my great pleasure to join the colleagues from 14 regions:

Australia, Brazil, Canada, Estonia, Germany, Israel, Korea, Malaysia, Poland, Slovenia, South Africa, USA, Taiwan, Thailand,

inAfrica, Asia, Australia, Europe, (North and South) America to honor Professor Pjek-Hwee Lee on the occasion of his retirement.

In fact, I am also affiliated with the following institutions:

Hong Kong University of Science & Technology (2011 Fulbright Fellow) Taiyuan University of Technology (Shanxi 100 Talent Program)

Shanghai University, University of Hong Kong (Honorary Professor).

Dedication

It is my great pleasure to join the colleagues from 14 regions:

Australia, Brazil, Canada, Estonia, Germany, Israel, Korea, Malaysia, Poland, Slovenia, South Africa, USA, Taiwan, Thailand,

inAfrica, Asia, Australia, Europe, (North and South) America to honor Professor Pjek-Hwee Lee on the occasion of his retirement.

In fact, I am also affiliated with the following institutions:

Hong Kong University of Science & Technology (2011 Fulbright Fellow) Taiyuan University of Technology (Shanxi 100 Talent Program) Shanghai University, University of Hong Kong (Honorary Professor).

Preserver problems

Preserver problemsconcern the characterization of maps on matrices, operators, or other algebraic objects with special properties.

(Frobenius,1897) AlinearmapT :Mn→Mnsatisfies det(L(A)) = det(A) for allA∈Mn

if and only if there are matricesM, N∈Mnwith det(M N) = 1 such thatLhas the form

A7→M AN or A7→M A^tN.

(Dieudonné,1949) LetS be the set of singular matrices. Aninvertible linearmapL:Mn→MnsatisfiesT(S)⊆ S. if and only if there are M, N ∈GLnsuch thatLhas the form

A7→M AN or A7→M A^tN.

Preserver problems

Preserver problemsconcern the characterization of maps on matrices, operators, or other algebraic objects with special properties.

(Frobenius,1897) AlinearmapT :Mn→Mnsatisfies det(L(A)) = det(A) for allA∈Mn

if and only if there are matricesM, N∈Mnwith det(M N) = 1 such thatLhas the form

A7→M AN or A7→M A^tN.

(Dieudonné,1949) LetS be the set of singular matrices. Aninvertible linearmapL:Mn→MnsatisfiesT(S)⊆ S. if and only if there are M, N ∈GLnsuch thatLhas the form

A7→M AN or A7→M A^tN.

Preserver problems

Preserver problemsconcern the characterization of maps on matrices, operators, or other algebraic objects with special properties.

(Frobenius,1897) AlinearmapT :Mn→Mnsatisfies det(L(A)) = det(A) for allA∈Mn

if and only if

there are matricesM, N∈Mnwith det(M N) = 1 such thatLhas the form

A7→M AN or A7→M A^tN.

(Dieudonné,1949) LetS be the set of singular matrices. Aninvertible linearmapL:Mn→MnsatisfiesT(S)⊆ S. if and only if there are M, N ∈GLnsuch thatLhas the form

A7→M AN or A7→M A^tN.

Preserver problems

Preserver problemsconcern the characterization of maps on matrices, operators, or other algebraic objects with special properties.

(Frobenius,1897) AlinearmapT :Mn→Mnsatisfies det(L(A)) = det(A) for allA∈Mn

if and only if there are matricesM, N∈Mnwith det(M N) = 1 such thatLhas the form

A7→M AN or A7→M A^tN.

(Dieudonné,1949) LetS be the set of singular matrices. Aninvertible linearmapL:Mn→MnsatisfiesT(S)⊆ S. if and only if there are M, N ∈GLnsuch thatLhas the form

A7→M AN or A7→M A^tN.

Preserver problems

Preserver problemsconcern the characterization of maps on matrices, operators, or other algebraic objects with special properties.

(Frobenius,1897) AlinearmapT :Mn→Mnsatisfies det(L(A)) = det(A) for allA∈Mn

if and only if there are matricesM, N∈Mnwith det(M N) = 1 such thatLhas the form

A7→M AN or A7→M A^tN.

(Dieudonné,1949) LetS be the set of singular matrices. Aninvertible linearmapL:Mn→MnsatisfiesT(S)⊆ S. if and only if there are M, N ∈GLnsuch thatLhas the form

There were study and results on matrix pairs preserving

adjacency, i.e., rank(A−B) = 1,partial orders, i.e.,A≤B, disjointness / having zero product, say,AB= 0,AB^∗= 0, [A, B] =AB−BA= 0,{A, B}=AB+BA= 0, etc.

The were also study connected to (Lie) derivations, isometry, effect algebra, Jordan homorophism,C^∗-homomorphisms, etc.

ProfessorPjek-Hwee Leeand many other colleagues in the audience have nice results on this subject.

There were study and results on matrix pairs preserving adjacency, i.e., rank(A−B) = 1,

partial orders, i.e.,A≤B, disjointness / having zero product, say,AB= 0,AB^∗= 0, [A, B] =AB−BA= 0,{A, B}=AB+BA= 0, etc.

The were also study connected to (Lie) derivations, isometry, effect algebra, Jordan homorophism,C^∗-homomorphisms, etc.

ProfessorPjek-Hwee Leeand many other colleagues in the audience have nice results on this subject.

There were study and results on matrix pairs preserving adjacency, i.e., rank(A−B) = 1,partial orders, i.e.,A≤B,

disjointness / having zero product, say,AB= 0,AB^∗= 0, [A, B] =AB−BA= 0,{A, B}=AB+BA= 0, etc.

The were also study connected to (Lie) derivations, isometry, effect algebra, Jordan homorophism,C^∗-homomorphisms, etc.

ProfessorPjek-Hwee Leeand many other colleagues in the audience have nice results on this subject.

There were study and results on matrix pairs preserving adjacency, i.e., rank(A−B) = 1,partial orders, i.e.,A≤B, disjointness / having zero product,

say,AB= 0,AB^∗= 0, [A, B] =AB−BA= 0,{A, B}=AB+BA= 0, etc.

The were also study connected to (Lie) derivations, isometry, effect algebra, Jordan homorophism,C^∗-homomorphisms, etc.

ProfessorPjek-Hwee Leeand many other colleagues in the audience have nice results on this subject.

There were study and results on matrix pairs preserving adjacency, i.e., rank(A−B) = 1,partial orders, i.e.,A≤B, disjointness / having zero product, say,AB= 0,AB^∗= 0, [A, B] =AB−BA= 0,{A, B}=AB+BA= 0, etc.

The were also study connected to (Lie) derivations, isometry, effect algebra, Jordan homorophism,C^∗-homomorphisms, etc.

ProfessorPjek-Hwee Leeand many other colleagues in the audience have nice results on this subject.

There were study and results on matrix pairs preserving adjacency, i.e., rank(A−B) = 1,partial orders, i.e.,A≤B, disjointness / having zero product, say,AB= 0,AB^∗= 0, [A, B] =AB−BA= 0,{A, B}=AB+BA= 0, etc.

The were also study connected to (Lie) derivations, isometry, effect algebra, Jordan homorophism,C^∗-homomorphisms, etc.

ProfessorPjek-Hwee Leeand many other colleagues in the audience have nice results on this subject.

There were study and results on matrix pairs preserving adjacency, i.e., rank(A−B) = 1,partial orders, i.e.,A≤B, disjointness / having zero product, say,AB= 0,AB^∗= 0, [A, B] =AB−BA= 0,{A, B}=AB+BA= 0, etc.

The were also study connected to (Lie) derivations, isometry, effect algebra, Jordan homorophism,C^∗-homomorphisms, etc.

ProfessorPjek-Hwee Leeand many other colleagues in the audience have nice results on this subject.

Quantum information science

Quantum information science is about the usequantum mechanical propertiesto help store, transmit, and process information (encoded in quantum bits (qubits) instead of binary bits).

Mathematically,quantum states(withmphysical states) are represented bydensity matricesA∈Mm, i.e., positive semidefinite matrices

(operators) with trace one.

Quantum evolutionof aclosed systemis governed by unitary similarity transforms:

A(t) =UtA(0)U_t^∗.

SupposeA∈Mm andB∈Mn are quantum states. Theirjoint quantum stateisC=A⊗B= (aijB)in the (composite)bipartite system, where the general states are represented bymn×mndensity matrices resulting frommixingordecoherence.

So, in the study of quantum information science, one often deals with tensor products of matrices.

Quantum information science

Quantum information science is about the usequantum mechanical propertiesto help store, transmit, and process information (encoded in quantum bits (qubits) instead of binary bits).

Mathematically,quantum states(withmphysical states) are represented bydensity matricesA∈Mm, i.e., positive semidefinite matrices

(operators) with trace one.

Quantum evolutionof aclosed systemis governed by unitary similarity transforms:

A(t) =UtA(0)U_t^∗.

SupposeA∈Mm andB∈Mn are quantum states. Theirjoint quantum stateisC=A⊗B= (aijB)in the (composite)bipartite system, where the general states are represented bymn×mndensity matrices resulting frommixingordecoherence.

So, in the study of quantum information science, one often deals with tensor products of matrices.

Quantum information science

Quantum information science is about the usequantum mechanical propertiesto help store, transmit, and process information (encoded in quantum bits (qubits) instead of binary bits).

Mathematically,quantum states(withmphysical states) are represented bydensity matricesA∈Mm, i.e., positive semidefinite matrices

(operators) with trace one.

Quantum evolutionof aclosed systemis governed by unitary similarity transforms:

A(t) =UtA(0)Ut^∗.

SupposeA∈Mm andB∈Mn are quantum states. Theirjoint quantum stateisC=A⊗B= (aijB)in the (composite)bipartite system, where the general states are represented bymn×mndensity matrices resulting frommixingordecoherence.

So, in the study of quantum information science, one often deals with tensor products of matrices.

Quantum information science

Quantum information science is about the usequantum mechanical propertiesto help store, transmit, and process information (encoded in quantum bits (qubits) instead of binary bits).

Mathematically,quantum states(withmphysical states) are represented bydensity matricesA∈Mm, i.e., positive semidefinite matrices

(operators) with trace one.

Quantum evolutionof aclosed systemis governed by unitary similarity transforms:

A(t) =UtA(0)Ut^∗. SupposeA∈MmandB∈Mn are quantum states.

Theirjoint quantum stateisC=A⊗B= (aijB)in the (composite)bipartite system, where the general states are represented bymn×mndensity matrices resulting frommixingordecoherence.

So, in the study of quantum information science, one often deals with tensor products of matrices.

Quantum information science

Quantum information science is about the usequantum mechanical propertiesto help store, transmit, and process information (encoded in quantum bits (qubits) instead of binary bits).

Mathematically,quantum states(withmphysical states) are represented bydensity matricesA∈Mm, i.e., positive semidefinite matrices

(operators) with trace one.

Quantum evolutionof aclosed systemis governed by unitary similarity transforms:

A(t) =UtA(0)Ut^∗.

SupposeA∈MmandB∈Mn are quantum states. Theirjoint quantum stateisC=A⊗B= (aijB)in the (composite)bipartite system,

where the general states are represented bymn×mndensity matrices resulting frommixingordecoherence.

So, in the study of quantum information science, one often deals with tensor products of matrices.

Quantum information science

Quantum information science is about the usequantum mechanical propertiesto help store, transmit, and process information (encoded in quantum bits (qubits) instead of binary bits).

Mathematically,quantum states(withmphysical states) are represented bydensity matricesA∈Mm, i.e., positive semidefinite matrices

(operators) with trace one.

Quantum evolutionof aclosed systemis governed by unitary similarity transforms:

A(t) =UtA(0)Ut^∗.

SupposeA∈MmandB∈Mn are quantum states. Theirjoint quantum stateisC=A⊗B= (aijB)in the (composite)bipartite system, where the general states are represented bymn×mndensity matrices resulting frommixingordecoherence.

So, in the study of quantum information science, one often deals with tensor products of matrices.

Quantum information science

Quantum information science is about the usequantum mechanical propertiesto help store, transmit, and process information (encoded in quantum bits (qubits) instead of binary bits).

Mathematically,quantum states(withmphysical states) are represented bydensity matricesA∈Mm, i.e., positive semidefinite matrices

(operators) with trace one.

Quantum evolutionof aclosed systemis governed by unitary similarity transforms:

A(t) =UtA(0)Ut^∗.

SupposeA∈MmandB∈Mn are quantum states. Theirjoint quantum stateisC=A⊗B= (aijB)in the (composite)bipartite system, where the general states are represented bymn×mndensity matrices resulting frommixingordecoherence.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY. Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states. Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

A quantum stateρ∈Dmnisentangledifρ∈Dmn\ Sm,n.

To determine whether ρ∈Dmnis separable is an NP hard problem.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY.

Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states. Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

A quantum stateρ∈Dmnisentangledifρ∈Dmn\ Sm,n.

To determine whether ρ∈Dmnis separable is an NP hard problem.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY. Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states. Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

A quantum stateρ∈Dmnisentangledifρ∈Dmn\ Sm,n.

To determine whether ρ∈Dmnis separable is an NP hard problem.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY. Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states. Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

A quantum stateρ∈Dmnisentangledifρ∈Dmn\ Sm,n.

To determine whether ρ∈Dmnis separable is an NP hard problem.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY. Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states.

Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

A quantum stateρ∈Dmnisentangledifρ∈Dmn\ Sm,n.

To determine whether ρ∈Dmnis separable is an NP hard problem.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY. Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states.

Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

A quantum stateρ∈Dmnisentangledifρ∈Dmn\ Sm,n.

To determine whether ρ∈Dmnis separable is an NP hard problem.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY. Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states.

Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

A quantum stateρ∈Dmnisentangledifρ∈Dmn\ Sm,n.

To determine whether ρ∈Dmnis separable is an NP hard problem.

Separable States

Notation

Mn: the algebra ofn×ncomplex matrices.

X⊗Y = (xijY): thetensor productof the matricesX= (xij) andY. Hn: the real linear space ofn×nHermitian matrices.

Dn: the set ofn×ndensity matrices, i.e., positive semidefinite matrices with trace one.

Pm: set of rank one matrices inDm, the set ofpure states.

Sm,n= conv{Pm⊗Pn}is a subset ofDmnknown as the set of separable states.

ρ∈D ρ∈D \ S

Problems of interest

Our question

What are the maps which will preserve the set of separable states? Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary. Ifm=n, one can swap the two component states: A⊗B7→B⊗A. Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2. In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is true if(m, n)∈ {(2,2),(2,3),(3,2)}. Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary. Ifm=n, one can swap the two component states: A⊗B7→B⊗A. Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2. In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is true if(m, n)∈ {(2,2),(2,3),(3,2)}. Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary. Ifm=n, one can swap the two component states: A⊗B7→B⊗A. Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2. In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is true if(m, n)∈ {(2,2),(2,3),(3,2)}. Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary.

Ifm=n, one can swap the two component states: A⊗B7→B⊗A. Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2. In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is true if(m, n)∈ {(2,2),(2,3),(3,2)}. Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary.

Ifm=n, one can swap the two component states: A⊗B7→B⊗A.

Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2. In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is true if(m, n)∈ {(2,2),(2,3),(3,2)}. Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary.

Ifm=n, one can swap the two component states: A⊗B7→B⊗A.

Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2. In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is true if(m, n)∈ {(2,2),(2,3),(3,2)}. Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary.

Ifm=n, one can swap the two component states: A⊗B7→B⊗A.

Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2.

In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is true if(m, n)∈ {(2,2),(2,3),(3,2)}. Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary.

Ifm=n, one can swap the two component states: A⊗B7→B⊗A.

Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2.

Else, there are examples of entangled states with positive partial transpose.

Problems of interest

Our question

What are the maps which will preserve the set of separable states?

Can we transform the states so that it is easier to check separability?

We can apply changes of bases for each of the systems:

A⊗B7→U^∗AU⊗B, A⊗B7→A⊗V^∗BV, U, V unitary.

Ifm=n, one can swap the two component states: A⊗B7→B⊗A.

Thepartial transpose mapsonHmndefined by

P T1(A⊗B) =A^t⊗B and P T2(A⊗B) =A⊗B^t preserve the set of separable states, i.e.,P Tj(Sm,n) =Sm,n forj= 1,2.

In fact, ifC∈Sm,nthenP Tj(C)∈Dmnforj= 1,2. The converse is

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent. (a) Ψ(Pm⊗Pn) =Pm⊗Pn.

(b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that (c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn, whereψ1has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent.

(a) Ψ(Pm⊗Pn) =Pm⊗Pn. (b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that (c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn, whereψ1has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent.

(a) Ψ(Pm⊗Pn) =Pm⊗Pn.

(b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that (c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn, whereψ1has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent.

(a) Ψ(Pm⊗Pn) =Pm⊗Pn. (b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that (c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn, whereψ1has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent.

(a) Ψ(Pm⊗Pn) =Pm⊗Pn. (b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that

(c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn, whereψ1has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent.

(a) Ψ(Pm⊗Pn) =Pm⊗Pn. (b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that (c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn, whereψ1has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent.

(a) Ψ(Pm⊗Pn) =Pm⊗Pn. (b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that (c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn,

whereψ1has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Result for bipartite systems

Theorem [FLPS, 2010], [Flfsen & Shultz, 2010]

LetΨ :Hmn→Hmnbe a linear map. The following are equivalent.

(a) Ψ(Pm⊗Pn) =Pm⊗Pn. (b) Ψ(Sm,n) =Sm,n.

(c) There are unitaryU∈MmandV ∈Mnsuch that (c.1) Ψ(A⊗B) =ψ1(A)⊗ψ2(B)forA⊗B∈Hmn, or

(c.2)m=nandΨ(A⊗B) =ψ2(B)⊗ψ1(A)forA⊗B∈Hmn, whereψ1 has the formA7→U AU^∗ or A7→U A^tU^∗, andψ2 has the form B7→V BV^∗ or B7→V B^tV^∗.

Remarks

Any scheme for detecting separable states must be invariant under the maps described in the theorem.

If there is a necessary condition for separability, then one can apply the condition toφ(X)for allφdescribed in the theorem.

LetΦ :Dmn→Dmnbe an affine map such thatΦ(Sm,n) =Sm,n. Then Φcan be extended uniquely to an invertible linear mapΨ :Hmn→Hmn. The proof of [FS] (similar to an earlier proof of ours) depends heavily on the convex feature ofSm,nand is not easy to be extended to the multi-partitite case.

Remarks

Any scheme for detecting separable states must be invariant under the maps described in the theorem.

If there is a necessary condition for separability, then one can apply the condition toφ(X)for allφdescribed in the theorem.

LetΦ :Dmn→Dmnbe an affine map such thatΦ(Sm,n) =Sm,n. Then Φcan be extended uniquely to an invertible linear mapΨ :Hmn→Hmn. The proof of [FS] (similar to an earlier proof of ours) depends heavily on the convex feature ofSm,nand is not easy to be extended to the multi-partitite case.

Remarks

Any scheme for detecting separable states must be invariant under the maps described in the theorem.

If there is a necessary condition for separability, then one can apply the condition toφ(X)for allφdescribed in the theorem.

LetΦ :Dmn→Dmnbe an affine map such thatΦ(Sm,n) =Sm,n. Then Φcan be extended uniquely to an invertible linear mapΨ :Hmn→Hmn. The proof of [FS] (similar to an earlier proof of ours) depends heavily on the convex feature ofSm,nand is not easy to be extended to the multi-partitite case.

Remarks

Any scheme for detecting separable states must be invariant under the maps described in the theorem.

If there is a necessary condition for separability, then one can apply the condition toφ(X)for allφdescribed in the theorem.

LetΦ :Dmn→Dmnbe an affine map such thatΦ(Sm,n) =Sm,n. Then Φcan be extended uniquely to an invertible linear mapΨ :Hmn→Hmn.

The proof of [FS] (similar to an earlier proof of ours) depends heavily on the convex feature ofSm,nand is not easy to be extended to the multi-partitite case.

Remarks

Any scheme for detecting separable states must be invariant under the maps described in the theorem.

If there is a necessary condition for separability, then one can apply the condition toφ(X)for allφdescribed in the theorem.

LetΦ :Dmn→Dmnbe an affine map such thatΦ(Sm,n) =Sm,n. Then Φcan be extended uniquely to an invertible linear mapΨ :Hmn→Hmn. The proof of [FS] (similar to an earlier proof of ours) depends heavily on the convex feature ofSm,nand is not easy to be extended to the multi-partitite case.

An auxiliary result

Proposition [FLPS, 2010]

Supposeψ:Hm→Hnis linear and satisfiesψ(Pm)⊆Pn. Then one of the following holds:

there isR∈Pnsuch thatψ has the formA7→(tr A)R.

m≤nand there is aU∈Mm×nwithU U^∗=Imsuch thatψhas the form

A7→U^∗AU or A7→U^∗A^tU.

An auxiliary result

Proposition [FLPS, 2010]

Supposeψ:Hm→Hnis linear and satisfiesψ(Pm)⊆Pn. Then one of the following holds:

there isR∈Pn such thatψ has the formA7→(tr A)R.

m≤nand there is aU∈Mm×nwithU U^∗=Imsuch thatψhas the form

A7→U^∗AU or A7→U^∗A^tU.

An auxiliary result

Proposition [FLPS, 2010]

Supposeψ:Hm→Hnis linear and satisfiesψ(Pm)⊆Pn. Then one of the following holds:

there isR∈Pn such thatψ has the formA7→(tr A)R.

m≤nand there is aU∈Mm×nwithU U^∗=Im such thatψhas the form

A7→U^∗AU or A7→U^∗A^tU.

Extension to multi-partite systems

Theorem [FLPS,2010]

Supposen1≥ · · · ≥nk≥2are positive integers withk >1andN=Qk i=1ni.

The following are equivalent for a linear mapΨ :HN→HN. (a) Ψ ⊗^k_i=1Pn_i

=⊗^k_i=1Pn_i. (b) Ψ conv (⊗^ki=1Pn_i)

= conv ⊗^ki=1Pn_i

.

(c) There is a permutationπon{1, . . . , k}and linear mapsψionHn_i for i= 1, . . . ksuch that

Ψ ⊗^ki=1Ai

=⊗^ki=1ψi Aπ(i)

for ⊗^ki=1Ak∈ ⊗^ki=1Pn_i, whereψihas the form

X7→UiXU_i^∗ or X7→UiX^tU_i^∗, for some unitaryUi∈Mn_i andnπ(i)=nifor i= 1, . . . , k.

Extension to multi-partite systems

Theorem [FLPS,2010]

Supposen1≥ · · · ≥nk≥2are positive integers withk >1andN=Qk i=1ni. The following are equivalent for a linear mapΨ :HN→HN.

(a) Ψ ⊗^k_i=1Pni

=⊗^k_i=1Pni.

(b) Ψ conv (⊗^ki=1Pn_i)

= conv ⊗^ki=1Pn_i

.

(c) There is a permutationπon{1, . . . , k}and linear mapsψionHn_i for i= 1, . . . ksuch that

Ψ ⊗^ki=1Ai

=⊗^ki=1ψi Aπ(i)

for ⊗^ki=1Ak∈ ⊗^ki=1Pn_i, whereψihas the form

X7→UiXU_i^∗ or X7→UiX^tU_i^∗, for some unitaryUi∈Mn_i andnπ(i)=nifor i= 1, . . . , k.

Extension to multi-partite systems

Theorem [FLPS,2010]

Supposen1≥ · · · ≥nk≥2are positive integers withk >1andN=Qk i=1ni. The following are equivalent for a linear mapΨ :HN→HN.

(a) Ψ ⊗^k_i=1Pni

=⊗^k_i=1Pni. (b) Ψ conv (⊗^ki=1Pn_i)

= conv ⊗^ki=1Pn_i

.

(c) There is a permutationπon{1, . . . , k}and linear mapsψionHn_i for i= 1, . . . ksuch that

Ψ ⊗^ki=1Ai

=⊗^ki=1ψi Aπ(i)

for ⊗^ki=1Ak∈ ⊗^ki=1Pn_i, whereψihas the form

X7→UiXU_i^∗ or X7→UiX^tU_i^∗, for some unitaryUi∈Mn_i andnπ(i)=nifor i= 1, . . . , k.

Extension to multi-partite systems

Theorem [FLPS,2010]

Supposen1≥ · · · ≥nk≥2are positive integers withk >1andN=Qk i=1ni. The following are equivalent for a linear mapΨ :HN→HN.

(a) Ψ ⊗^k_i=1Pni

=⊗^k_i=1Pni. (b) Ψ conv (⊗^ki=1Pn_i)

= conv ⊗^ki=1Pn_i

.

(c) There is a permutationπon{1, . . . , k}and linear mapsψionHn_i for i= 1, . . . ksuch that

Ψ ⊗^ki=1Ai

=⊗^ki=1ψi Aπ(i)

for ⊗^ki=1Ak∈ ⊗^ki=1Pn_i,

whereψihas the form

X7→UiXU_i^∗ or X7→UiX^tU_i^∗, for some unitaryUi∈Mn_i andnπ(i)=nifor i= 1, . . . , k.

Extension to multi-partite systems

Theorem [FLPS,2010]

Supposen1≥ · · · ≥nk≥2are positive integers withk >1andN=Qk i=1ni. The following are equivalent for a linear mapΨ :HN→HN.

(a) Ψ ⊗^k_i=1Pni

=⊗^k_i=1Pni. (b) Ψ conv (⊗^ki=1Pn_i)

= conv ⊗^ki=1Pn_i

.

(c) There is a permutationπon{1, . . . , k}and linear mapsψionHn_i for i= 1, . . . ksuch that

Ψ ⊗^ki=1Ai

=⊗^ki=1ψi Aπ(i)

for ⊗^ki=1Ak∈ ⊗^ki=1Pn_i, whereψi has the form

X7→UiXU_i^∗ or X7→UiX^tU_i^∗,