Inequalities for the Schmidt Number of Bipartite States

TL;DR

This paper establishes new inequalities relating the Schmidt number of bipartite states to their ranks and constructs PPT states with specified Schmidt numbers, advancing understanding of quantum entanglement and PPT state properties.

Contribution

It introduces novel bounds for the Schmidt number based on state ranks and presents a family of PPT states with controllable Schmidt numbers, addressing open problems in quantum entanglement theory.

Findings

Derived lower bounds for the Schmidt number using rank inequalities.

Constructed PPT states with any desired Schmidt number within a certain range.

Demonstrated the diversity of quantum entanglement through two contrasting state construction methods.

Abstract

In this short note we show two completely opposite methods of constructing entangled states. Given a bipartite state $\gamma\in M_k\otimes M_k$, define $\gamma_S=(Id+F)\gamma (Id+F)$, $\gamma_A=(Id-F)\gamma(Id-F)$, where $F\in M_k\otimes M_k$ is the flip operator. In the first method, entanglement is a consequence of the inequality $\text{rank}(\gamma_S)<\sqrt{\text{rank}(\gamma_A)}$. In the second method, there is no correlation between $\gamma_S$ and $\gamma_A$. These two methods show how diverse is quantum entanglement. We prove that any bipartite state $\gamma\in M_k\otimes M_k$ satisfies $\displaystyle SN(\gamma)\geq\max \left\{ \frac{\text{rank}(\gamma_L)}{\text{rank}(\gamma)}, \frac{\text{rank}(\gamma_R)}{\text{rank}(\gamma)}, \frac{SN(\gamma_S)}{2}, \frac{SN(\gamma_A)}{2} \right\},$ where $SN(\gamma)$ stands for the Schmidt number of $\gamma$ and $\gamma_L,\gamma_R$ are the marginal states of $\gamma$. We also present a family of PPT states in $M_k\otimes M_k$, whose members have Schmidt number equal to $n$, for any given $1\leq n\leq \left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.

Equations

$$SN(\gamma )\ge \max \left\{\frac{\text{rank}({\gamma }_L)}{\text{rank}(\gamma )},\frac{\text{rank}({\gamma }_R)}{\text{rank}(\gamma )},\frac{SN({\gamma }_S)}{2},\frac{SN({\gamma }_A)}{2}\right\},$$

$$SN(\gamma )\ge \max \left\{\frac{SN({\gamma }_S)}{2},\frac{SN({\gamma }_A)}{2}\right\},$$

$$\text{rank}(\gamma )SN(\gamma )\ge \max \{\text{rank}({\gamma }_L),\text{rank}({\gamma }_R)\}.$$

$$\text{rank}({\gamma }_S)\ge \max \left\{\frac{r}{2},\frac{2}{r}\text{rank}({\gamma }_A)\right\},$$

$$SN(\gamma )\ge \frac{\text{rank}(({\gamma }_S{)}_L)}{2\text{rank}({\gamma }_S)}\text{ and }SN(\gamma )\ge \frac{\text{rank}(({\gamma }_A{)}_L)}{2\text{rank}({\gamma }_A)}.$$

$$SR(v_i)=SR(w_i\pm F{w}_i)\le 2SR(w_i)\le 2SN(\gamma ).$$

$$\gamma =Id+F+ϵ(v{\overline{v}}^t)\in M_k\otimes M_k.$$

$$SN(ϵ(v{\overline{v}}^t))=SR(v)=n\text{ and }SN({\gamma }_A)=SR(a)=2n.$$

Full text

We prove that any bipartite state $\gamma\in M_{k}\otimes M_{k}$ satisfies

[TABLE]

where $SN(\gamma)$ stands for the Schmidt number of $\gamma$ and $\gamma_{L},\gamma_{R}$ are the marginal states of $\gamma$.

We also present a family of PPT states in $M_{k}\otimes M_{k}$, whose members have Schmidt number equal to $n$, for any given $1\leq n\leq\left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.

1. Introduction

The separability problem in Quantum Information Theory asks for a deterministic criterion to distinguish the entangled states from the separable states [Guhne]. This problem is known to be a hard problem even for bipartite mixed states [gurvits2003, gurvits2004].

The Schmidt number of a state ($SN(\gamma)$ - Definition 2.1) is a measure of how entangled a state is [Terhal, Sperling2011]. If its Schmidt number is 1 then the state is separable. If its Schmidt number is greater than 1 then the state is entangled. A method to compute the Schmidt Number is unknown.

Denote by $M_{k}$ the set of complex matrices of order $k$. The separability problem has been completely solved in $M_{2}\otimes M_{2}$. A state in $M_{2}\otimes M_{2}$ is separable if and only if it is positive under partial transposition or simply PPT (Definition 2.1) [peres, horodeckifamily]. Therefore, the Schmidt number of a PPT state in $M_{2}\otimes M_{2}$ is equal to 1. Recently, the Schmidt number of every PPT state of $M_{3}\otimes M_{3}$ has been proved to be less or equal to 2 [Yang, Chen].

The authors of [Marcus] left an open problem to determine the best possible Schmidt number for PPT states. They also presented a construction of PPT states in $M_{k}\otimes M_{k}$ whose Schmidt numbers are greater or equal to $\left\lceil\frac{k-1}{4}\right\rceil$. This was the first explicit example of a family of PPT states achieving a Schmidt number that scales linearly in the local dimension.

We investigate this matter. We present an explicit construction of PPT states in $M_{k}\otimes M_{k}$, whose Schmidt numbers are equal to $n$, for any given $1\leq n\leq\left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to their open problem.

We manage to compute the Schmidt number of these PPT states using the following inequality

[TABLE]

where $\gamma_{S}=(Id+F)\gamma(Id+F)$, $\gamma_{A}=(Id-F)\gamma(Id-F)$ and $F\in M_{k}\otimes M_{k}$ is the flip operator $($i.e., $F(a\otimes b)=b\otimes a,$ for every $a,b\in\mathbb{C}^{k})$.

We believe this is one of the simplest constructions of an entangled PPT state made so far.

Another inequality that we present here extends a result that was previously known for separable states $($[smolin, Theorem 1]$)$ to every state of $M_{k}\otimes M_{m}$. Denote by $\gamma_{L}$ and $\gamma_{R}$ the marginal states of a state $\gamma\in M_{k}\otimes M_{m}$ (Definition 2.1).

We show that every state $\gamma$ of $M_{k}\otimes M_{m}$ satisfies

[TABLE]

We can use this inequality to obtain a lower bound for the Schmidt number of low rank states.

Next, through a series of very technical results, the author of [Cariello_LAA] obtained the following lower bounds for the $\text{rank}(\gamma_{S})$ of any separable state $\gamma\in M_{k}\otimes M_{k}$

[TABLE]

where $r$ is the marginal rank of $\gamma+F\gamma F$.

These inequalities can be combined into one inequality:

$\displaystyle\text{rank}(\gamma_{S})\geq\frac{2}{r}\text{rank}(\gamma_{A})\geq\frac{\text{rank}(\gamma_{A})}{\text{rank}(\gamma_{S})}.$

Hence, $\text{rank}(\gamma_{S})\geq\sqrt{\text{rank}(\gamma_{A})}$ for every separable state $\gamma\in M_{k}\otimes M_{k}$. Therefore, if $\text{rank}(\gamma_{S})<\sqrt{\text{rank}(\gamma_{A})}$ then $\gamma$ is entangled.

Next, we can combine equations 1.1 and 1.2 in order to obtain

[TABLE]

We can easily create entangled states by satisfying $\displaystyle\frac{\text{rank}((\gamma_{S})_{L})}{\text{rank}(\gamma_{S})}>2$ or $\displaystyle\frac{\text{rank}((\gamma_{A})_{L})}{\text{rank}(\gamma_{A})}>2$ and no correlation between $\gamma_{S}$ and $\gamma_{A}$ is required.

These two methods of creating entangled states are completely opposite. One depends on a correlation between $\gamma_{S},\gamma_{A}$ and the other does not. They show how diverse is quantum entanglement.

This paper is organized as follows.

• •

In Section II, we prove that $SN(\gamma)\geq\max\left\{\frac{SN(\gamma_{S})}{2},\frac{SN(\gamma_{A})}{2}\right\}$ (Proposition 2.2) and we construct a PPT state whose Schmidt number is equal to $n$, for any given $n\in\{1,\ldots,\left\lceil\frac{k-1}{2}\right\rceil\}$ (Proposition 2.3).

• •

In Section III, we prove our main inequality $\text{rank}(\gamma)SN(\gamma)\geq\max\{\text{rank}(\gamma_{L}),\text{rank}(\gamma_{R})\}$ (Theorem 3.1) and two corollaries $\displaystyle SN(\gamma)\geq\frac{\text{rank}((\gamma_{S})_{L})}{2\ \text{rank}(\gamma_{S})}$ and $\displaystyle SN(\gamma)\geq\frac{\text{rank}((\gamma_{A})_{L})}{2\ \text{rank}(\gamma_{A})}$ (Corollaries 3.2 and 3.3).

Notation: Given $x\in\mathbb{R}$, define $\lceil x\rceil=\min\{n\in\mathbb{Z},n\geq x\}$. Identify $M_{k}\otimes M_{m}\simeq M_{km}$ and $\mathbb{C}^{k}\otimes\mathbb{C}^{m}\simeq\mathbb{C}^{km}$ via Kronecker product. Let us call a positive semidefinite Hermitian matrix of $M_{km}$ a (non-normalized bipartite) state of $M_{k}\otimes M_{m}$. Let $\Im(\delta)$ denote the image of $\delta\in M_{k}\otimes M_{m}$ within $\mathbb{C}^{k}\otimes\mathbb{C}^{m}$. Given $w\in\mathbb{C}^{k}\otimes\mathbb{C}^{m}$ denote by $SR(w)$ its Schmidt rank $($or tensor rank$)$. Let the trace of a matrix $A\in M_{k}$ be denoted by $tr(A)$.

2. Preliminary Inequalities

In this section we present two preliminary inequalities (Proposition 2.2). They have independent interest as we can see in Proposition 2.3. There we construct a family of PPT states in $M_{k}\otimes M_{k}$ whose members have Schmidt number equal to $n$, for any given $1\leq n\leq\left\lceil\frac{k-1}{2}\right\rceil$.

Definition 2.1**.**

Given a state $\delta=\sum_{i=1}^{n}A_{i}\otimes B_{i}\in M_{k}\otimes M_{m}$, define

• •

*the Schmidt number of $\delta$ as *

$\displaystyle SN(\delta)=\min\left\{\max_{j}\left\{SR(w_{j})\right\},\ \delta=\sum_{j=1}^{m}w_{j}\overline{w_{j}}^{t}\right\}$* *$($This minimum is taken over all decompositions of $\delta$ as $\sum_{j=1}^{m}w_{j}\overline{w_{j}}^{t})$.

• •

the partial transposition of $\delta$ as $\delta^{\Gamma}=\sum_{i=1}^{n}A_{i}\otimes B_{i}^{t}$ . Moreover, let us say that $\delta$ is positive under partial transposition or simply a PPT state if and only if $\delta$ and $\delta^{\Gamma}$ are states.

• •

*the marginal states of $\delta$ as $\delta_{L}=\sum_{i=1}^{n}A_{i}tr(B_{i})$ and $\delta_{R}=\sum_{i=1}^{n}B_{i}tr(A_{i})$. *

Proposition 2.2**.**

Every state $\gamma\in M_{k}\otimes M_{k}$ satisfies $\displaystyle SN(\gamma)\geq\max\left\{\frac{SN(\gamma_{S})}{2},\frac{SN(\gamma_{A})}{2}\right\}$.

Proof.

By definition 2.1, there is a subset $\{w_{1},\ldots,w_{n}\}\subset\mathbb{C}^{k}\otimes\mathbb{C}^{k}$ such that $\gamma=\sum_{i=1}^{n}w_{i}\overline{w_{i}}^{t}$ and $SR(w_{i})\leq SN(\gamma)$, for every $i$.

Therefore, $(Id\pm F)\gamma(I\pm F)=\sum_{i=1}^{n}v_{i}\overline{v_{i}}^{t}$, where $v_{i}=(Id\pm F)w_{i}$. Notice that, for every $i$,

[TABLE]

Hence, $SN((Id\pm F)\gamma(I\pm F))\leq 2SN(\gamma)$. ∎

Proposition 2.3**.**

Let $v=\sum_{i=1}^{n}a_{i}\otimes b_{i}$, where $\{a_{1},\ldots,a_{n},b_{1},\ldots b_{n}\}$ is a linearly independent subset of $\mathbb{C}^{k}$. Define

[TABLE]

1. (1)

For every $\epsilon>0$, $SN(\gamma)=n$. Notice that $1\leq n\leq\left\lceil\frac{k-1}{2}\right\rceil$. 2. (2)

There is $\epsilon>0$ such that $\gamma$ is positive under partial transposition.

Proof.

(1) Notice that $\gamma_{A}=(Id-F)\gamma(Id-F)=\epsilon(a\overline{a}^{t})$, where $a=\sum_{i=1}^{n}a_{i}\otimes b_{i}-b_{i}\otimes a_{i}$.

Since $\{a_{1},\ldots,a_{n},b_{1},\ldots b_{n}\}$ is linearly independent then $SR(v)=n$ and $SR(a)=2n$. Hence,

[TABLE]

Thus, $\displaystyle SN(\gamma)\geq\frac{SN(\gamma_{A})}{2}=n$, by Proposition 2.2.

Next, the separability of $Id+F\in M_{k}\otimes M_{k}$ is a well known fact, therefore $SN(Id+F)=1$.

Finally, $SN(\gamma)\leq\max\{SN(Id+F),SN(\epsilon(v\overline{v}^{t}))\}=\max\{1,n\}=n$. Therefore, $SN(\gamma)=n$.

(2) Notice that $(Id+F)^{\Gamma}=Id+uu^{t}$, where $u=\sum_{i=1}^{k}e_{i}\otimes e_{i}$ and $\{e_{1},\ldots,e_{k}\}$ is the canonical basis of $\mathbb{C}^{k}$. So $(Id+F)^{\Gamma}$ is positive definite and, for a small $\epsilon$, $(Id+F)^{\Gamma}+\epsilon(v\overline{v}^{t})^{\Gamma}$ is positive definite too. ∎

3. Main Inequality

In this section, we present our main result (Theorem 3.1) and two corollaries (Corollaries 3.2 and 3.3).

Theorem 3.1**.**

If $\gamma\in M_{k}\otimes M_{m}$ is a state then $\text{rank}(\gamma)SN(\gamma)\geq\max\{\text{rank}(\gamma_{L}),\text{rank}(\gamma_{R})\}$.

Proof.

The proof is an induction on $\text{rank}(\gamma)$. The case $\text{rank}(\gamma)=0$ is trivial. If $\text{rank}(\gamma)=1$ then $SN(\gamma)=\text{rank}(\gamma_{L})=\text{rank}(\gamma_{R})$.

Let $\text{rank}(\gamma)>1$ and assume that this result is valid for states $\delta\in M_{k}\otimes M_{m}$ satisfying $\text{rank}(\delta)<\text{rank}(\gamma)$.

Since $\Im(\gamma)\subset\Im(\gamma_{L}\otimes\gamma_{R})$ then $\gamma$ can be embedded in $M_{\text{rank}(\gamma_{L})}\otimes M_{\text{rank}(\gamma_{R})}$. The embedding does not change its rank or its Schmidt number. Thus, we can assume without loss of generality that $\text{rank}(\gamma_{L})=k$ and $\text{rank}(\gamma_{R})=m$.

Let $v\in\Im(\gamma)\setminus\{0\}$ be such that $SR(v)=SN(\gamma).$

• •

If $k\geq m$ then choose $U\in M_{k}$ satisfying $\text{rank}(U)=k-SN(\gamma)$ and $(U\otimes Id)v=0$.

Define $\delta=(U\otimes Id)\gamma(U^{*}\otimes Id)$. Note that $\text{rank}(\delta)\leq\text{rank}(\gamma)-1$, since $\Im(\delta)\subset(U\otimes Id)(\Im(\gamma))$ and $(U\otimes Id)v=0$.

• •

If $k<m$ then choose $U\in M_{m}$ satisfying $\text{rank}(U)=m-SN(\gamma)$ and $(Id\otimes U)v=0$.

Define $\delta=(Id\otimes U)\gamma(Id\otimes U^{*})$. Note that $\text{rank}(\delta)\leq\text{rank}(\gamma)-1$, since $\Im(\delta)\subset(Id\otimes U)(\Im(\gamma))$ and $(Id\otimes U)v=0$.

In any case, by induction hypothesis, $\text{rank}(\delta)SN(\delta)\geq\max\{\text{rank}(\delta_{L}),\text{rank}(\delta_{R})\}$.

• •

If $k\geq m$ then $\delta_{L}=U\gamma_{L}U^{*}$. Since $\gamma_{L}$ is positive definite then $\text{rank}(\delta_{L})=\text{rank}(U)=k-SN(\gamma).$

• •

If $k<m$ then $\delta_{R}=U\gamma_{R}U^{*}$. Since $\gamma_{R}$ is positive definite then $\text{rank}(\delta_{R})=\text{rank}(U)=m-SN(\gamma).$

Since $\text{rank}(\delta)\leq\text{rank}(\gamma)-1$ and $SN(\delta)\leq SN(\gamma)$ then

• •

$(\text{rank}(\gamma)-1)SN(\gamma)\geq k-SN(\gamma)$, if $k\geq m$. Therefore, $\text{rank}(\gamma)SN(\gamma)\geq k$.

• •

$(\text{rank}(\gamma)-1)SN(\gamma)\geq m-SN(\gamma)$, if $k<m$. Therefore, $\text{rank}(\gamma)SN(\gamma)\geq m$.

The induction is complete. ∎

Corollary 3.2**.**

If $\gamma\in M_{k}\otimes M_{k}$ is a state then $\displaystyle SN(\gamma)\geq\frac{\text{rank}((\gamma_{A})_{L})}{2\ \text{rank}(\gamma_{A})}$.

Proof.

First, notice that $(\gamma_{A})_{L}=(\gamma_{A})_{R}$. Therefore, $\text{rank}((\gamma_{A})_{L})=\text{rank}((\gamma_{A})_{R})$.

Next, since $SN(\gamma_{A})\leq 2SN(\gamma)$, by Proposition 2.2, then $\displaystyle\text{rank}(\gamma_{A})SN(\gamma)\geq\frac{1}{2}\text{rank}(\gamma_{A})SN(\gamma_{A})\geq\frac{\text{rank}((\gamma_{A})_{L})}{2}$, by Theorem 3.1. ∎

Corollary 3.3**.**

If $\gamma\in M_{k}\otimes M_{k}$ is a state then $\displaystyle SN(\gamma)\geq\frac{\text{rank}((\gamma_{S})_{L})}{2\ \text{rank}(\gamma_{S})}$.

Proof.

First, notice that $(\gamma_{S})_{L}=(\gamma_{S})_{R}$. Therefore, $\text{rank}((\gamma_{S})_{L})=\text{rank}((\gamma_{S})_{R})$.

Since $SN(\gamma_{S})\leq 2SN(\gamma)$, by Proposition 2.2, then $\displaystyle\text{rank}(\gamma_{S})SN(\gamma)\geq\frac{1}{2}\text{rank}(\gamma_{S})SN(\gamma_{S})\geq\frac{\text{rank}((\gamma_{S})_{L})}{2}$, by Theorem 3.1. ∎

Summary and Conclusion

We presented an inequality that relates the marginal ranks of any bipartite state of $M_{k}\otimes M_{m}$ to its rank and its Schmidt number . Using this inequality, we described a method of constructing entangled states which is not based on any correlation between $\text{rank}(\gamma_{A})$ and $\text{rank}(\gamma_{S})$. This form of entanglement differs completely from the entanglement derived from the inequality $\text{rank}(\gamma_{S})<\sqrt{\text{rank}(\gamma_{A})}$. We also constructed a family of PPT states whose members have Schmidt number equal to $n$, for any given $1\leq n\leq\left\lceil\frac{k-1}{2}\right\rceil$. This is a new contribution to the open problem of finding the best possible Schmidt number for PPT states.