$k$-Uniform states and quantum information masking

Multipartite entanglement has many applications in quantum information such as quantum teleportation bennett1993teleporting ; bouwmeester1997experimental , quantum key distribution ekert1991quantum ; gisin2002quantum ; bennett1992quantum , superdense coding bennett1992communication and quantum error correcting codes scott2004multipartite . However, it is difficult to quantify the level of entanglement in an arbitrary multipartite system. A very striking class of pure states called absolutely maximally entangled (AME) states has attracted much attention in recent years. These states are maximally entangled on every bipartition. AME states can be used to design holographic quantum codes and perfect tensors pastawski2015holographic . AME states can also be used for threshold quantum secret sharing schemes, for parallel and open-destination teleportation protocols helwig2012absolute ; helwig2013absolutely . However, AME states are very rare for given local dimensions AMEtable . Taking qubits as an example, AME states exist only for $2$-, $3$-, $5$-, and $6$-qubits scott2004multipartite ; rains1999quantum ; huber2017absolutely .

A much more general concept is called $k$-uniform states, which is defined to be a multipartite pure state of $N$-qudits whose all reductions to $k$ parties are maximally mixed scott2004multipartite . AME states are special kinds of $k$-uniform states, more exactly, the $\lfloor\frac{N}{2}\rfloor$-uniform states. Goyeneche et al. first associated the $k$-uniform states with orthogonal arrays goyeneche2014genuinely . Moreover, $k$-uniform states can also be constructed from Latin squares, symmetric matrices, graph states, quantum error correcting codes and classical error correcting codes goyeneche2014genuinely ; feng2017multipartite ; goyeneche2015absolutely ; scott2004multipartite ; goyeneche2018entanglement ; helwig2013absolutelygraph . Very recently, the authors in Ref. raissi2019constructing derived a new method to construct $k$-uniform states. Although there are several methods to construct $k$-uniform states, a complete solution to the existence of $k$-uniform states is far from reach. It has been known that a $1$-uniform state exists for any $d\geq 2$ and $N\geq 2$, by the existence of $\mathop{\rm GHZ}$ states. Pang et al. showed the existence of $2,3$-uniform states for almost every $N$-qudits pang2019two . However, there are a few results on the existence of $k$-uniform states of $N$-qudits when $k\geq 4$. In this paper, we shall give general constructions of $k$-uniform states for $k\geq 4$ by using the linear codes.

Recently, Modi et al proposed the concept of quantum information masking modi2018masking . This is a physical process that encodes quantum information into a bipartite system, while the information is completely unknown to each local system. They obtained a no-masking theorem: an arbitrary quantum state cannot be masked. In li2019k-uniform , the authors showed that quantum information masking in multipartite systems is possible. In their masking protocol, it also required that the original information is inaccessible to each local system. However, collusion between some subsystems would then reveal the encoded quantum information modi2018masking . Thus, a stronger version of quantum information masking is desirable. In this paper, we propose the concept of $k$-uniform quantum information masking in multipartite systems. It requires that the original information is inaccessible to each $k$ subsystems. Specially, we refer to the $\lfloor\frac{N}{2}\rfloor$-uniform quantum information masking as the strong quantum information masking. We show that if there exists a $(k+1)$-uniform state of $(N+1)$-qudits, then all states of one-qudit can be $k$-uniformly masked in an $N$-qudits system. We also show that when $N$ is odd, all states of one-qudit can be strongly masked in an $N$-qudits system provided that an AME state of $(N+1)$-qudits exists. However, when $N$ is even, we show that the strong quantum information masking is impossible, as a generalized no-masking theorem. In the $k$-uniform quantum information masking scheme, if the reduction states of $k$ parties are proportional to identity, we show that a pure $((N,d,k+1))_{d}$ quantum error-correcting code is equivalent to that all states of one-qudit can be $k$-uniformly masked in an $N$-qudits system.

The rest of this paper is organized as follows. In Sec. II, we introduce the connections among $k$-uniform states, orthogonal arrays and linear codes. We also review the existence of $1,2,3$-uniform states. In Sec. III, by using linear codes, we show that when $d\geq 4k-2$ (resp. $d\geq 2k-1$) is a prime power, there exists a $k$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ for any $N\geq 2k$ (resp. $2k\leq N\leq d+1$). Specially, we give the existence of $4,5$-uniform states for almost every $N$-qudits (see Table 1 and Table 2), by using the technique derived in Sec. III and some previously known results. In Sec. IV, we propose the concept of $k$-uniform quantum information masking in multipartite systems, and we show that $k$-uniform states and quantum error-correcting codes can be used for $k$-uniform quantum information masking. Finally, we conclude in Sec. V.

In this section, we introduce the preliminary knowledge and facts. First, we introduce the concept of $k$-uniform states scott2004multipartite .

We denote $\otimes_{\ell=1}^{N}{\cal H}_{A_{\ell}}$ as $(\mathbb{C}^{d})^{\otimes N}$ for simplicity. Due to the Schmidt decomposition of bipartite pure state, $k$ satisfies $k\leq\lfloor\frac{N}{2}\rfloor$. If $k=\lfloor\frac{N}{2}\rfloor$, then $|\psi\rangle$ is also called an absolutely maximally entangled ($\mathop{\rm AME}$) state. A $k$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ corresponds to a pure $((N,1,k+1))_{d}$ quantum error-correcting code (QECC) scott2004multipartite . See Appendix A for the definition of QECCs. For a pure $((N,1,k+1))_{d_{1}}$ code $\mathcal{C}_{1}$ and a pure $((N,1,k+1))_{d_{2}}$ code $\mathcal{C}_{2}$, the tensor product $\mathcal{C}_{1}\otimes\mathcal{C}_{2}$ is a pure $((N,1,k+1))_{d_{1}d_{2}}$ code rains1999nonbinary . By the relation between $k$-uniform states and QECCs, we have the following lemma.

Lemma 2 provides a simple but very useful construction. If we can construct a $k$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ for any prime $d$, then we can construct a $k$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ for any local dimension $d$. Since $\mathop{\rm GHZ}$ states are $1$-uniform states, it means that a $1$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ exists for any $d\geq 2$ and $N\geq 2$. In pang2019two , there are several unsolved cases for the existence of $2$-uniform states and $3$-uniform states. The unsolved cases for $2$-uniform states in $(\mathbb{C}^{d})^{\otimes N}$ are $d=2,6$ and $N=4$; $d=10$ and $N=5$. By highu , we know that $2$-uniform states in $(\mathbb{C}^{2})^{\otimes 4}$ do not exist. By using Lemma 2, we can construct a $2$-uniform state in $(\mathbb{C}^{10})^{\otimes 5}$ from a $2$-uniform state in $(\mathbb{C}^{2})^{\otimes 5}$ and a $2$-uniform state in $(\mathbb{C}^{5})^{\otimes 5}$. Thus the only unsolved case for $2$-uniform states in $(\mathbb{C}^{d})^{\otimes N}$ is just $d=6$ and $N=4$. The unsolved cases for $3$-uniform states in $(\mathbb{C}^{d})^{\otimes N}$ are that $d\geq 6$ is not a prime power and $N=9,10,11$; $d=2,3$, $d\geq 6$ is not a prime power and $N=6$; $d=2,3,4,5$, $d\geq 6$ is not a prime power and $N=7$. By using Lemma 2, we can construct a $3$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ for any $d\geq 6$ not a prime power and $N=9,10,11$. A $3$-uniform state in $(\mathbb{C}^{d})^{\otimes 6}$ exists for any $d\geq 2$ by rains1999nonbinary . Since $3$-uniform states in $(\mathbb{C}^{2})^{\otimes 7}$ do not exist huber2017absolutely , and there exists a $3$-uniform state in $(\mathbb{C}^{d})^{\otimes 7}$ for any $d=3,5$ by AMEtable , $d=4$ by raissi2019constructing , we know that a $3$-uniform state in $(\mathbb{C}^{d})^{\otimes 7}$ exists for any $d\geq 2$ except for $d=2(2k+1)$ and $k\geq 1$ by Lemma 2. Thus the left unsolved cases for $3$-uniform states in $(\mathbb{C}^{d})^{\otimes N}$ are $d\geq 6,d=2\pmod{4}$ and $N=7$. See Table 3 for a summary of $1,2,3$-uniform states.

Orthogonal arrays are essential in statistics and have wide applications in computer science and cryptography. An $r\times N$ array $A$ with entries taken from a set $S$ with $d$ elements is said to be an orthogonal array hedayat1999orthogonal with $r$ runs, $N$ factors, $d$ levels, strength $k$, and index $\lambda$, denoted by $\mathop{\rm OA}(r,N,d,k)$, if every $r\times k$ subarray of $A$ contains each $k$-tuple of symbols from $S$ exactly $\lambda$ times as a row. It is easy to see that an $\mathop{\rm OA}(r,N,d,k)$ must be an $\mathop{\rm OA}(r,N,d,k-1)$, and it is an $\mathop{\rm OA}(r,N-n,d,k)$ when we delete any $n$ columns.

An orthogonal array $\mathop{\rm OA}(r,N,d,k)$ is called irredundant goyeneche2014genuinely , denoted by $\mathop{\rm IrOA}(r,N,d,k)$, if after removing from the array any $k$ columns all remaining $r$ rows, containing $N-k$ symbols each, are different. It is easy to see that the $\mathop{\rm OA}(9,4,3,2)$ in Example 3 is an $\mathop{\rm IrOA}(9,4,3,2)$.

By Example 3 and Proposition 4, we can construct a $2$-uniform state in $(\mathbb{C}^{3})^{\otimes 4}$. In pang2019two , the authors provided a way to check whether an orthogonal array is irredundant. For any two vectors ${\bf u}$ and ${\bf v}$ of length $N$, the Hamming distance $d_{H}({\bf u},{\bf v})$ is the number of positions in which they differ. Given an $r\times N$ array $A$, the minimum distance of $A$, denoted by $d_{H}(A)$ is the minimum Hamming distance between any two distinct rows.

The set of rows of an $r\times N$ array $A$ over $\mathbb{Z}_{d}$ is often referred to a code $\mathcal{C}$ in $\mathbb{Z}_{d}^{N}$, where $\mathbb{Z}_{d}^{N}$ means $\overbrace{\mathbb{Z}_{d}\times\cdots\times\mathbb{Z}_{d}}^{N\text{ times}}$. If $d_{H}(A)=w$, then we say that the code $\mathcal{C}$ has minimum distance $w$, and denote it by an $(N,r,w)_{d}$ code. Further, if $d$ is a prime power, and $\mathcal{C}$ forms a subspace of $\mathbb{F}_{d}^{N}$ of dimension $t$ ($\mathbb{F}_{d}$ is a Galois Field with order $d$), then it is called a linear code, and denoted by an $[N,t,w]_{d}$ code. In this case, the dual code of $\mathcal{C}$ is defined by $\mathcal{C}^{\bot}=\{{\bf u}\in\mathbb{F}_{d}^{N}|{\bf u}^{\mathrm{T}}\cdot{\bf v}=0,\forall\ {\bf v}\in\mathcal{C}\}$, and the minimum distance of $\mathcal{C}^{\bot}$ is called the dual distance of $\mathcal{C}$, denoted by $w^{\bot}$. If $\mathcal{C}=\mathcal{C}^{\bot}$, then $\mathcal{C}$ is called self-dual. The next lemma gives the relation between linear codes and orthogonal arrays.

Example 3 is a $[4,2,3]_{3}$ code with dual distance $3$, then we obtain an $\mathop{\rm OA}(9,4,3,2)$ by Lemma 6. If an orthogonal array is constructed from a linear code, we call it a linear orthogonal array. The strategy for constructing $k$-uniform states in this paper is as follows (see Fig. 1). Using some known linear codes and the correspondence in Lemma 6, we can obtain an orthogonal array. Then by computing the minimum distance of the orthogonal array, we might obtain a series of irredundant orthogonal arrays by Lemma 5. Finally, by Proposition 4, we obtain a series of $k$-uniform states. From Lemma 6, we can give a recursion for $\mathop{\rm IrOA}$s by the recursion for linear codes.

In this section, we give a recursion for $\mathop{\rm IrOA}$s in Lemma 7. Then we construct some $k$-uniform states by linear codes in Theorem 8. Further, we give the existence of $4,5$-uniform states in Tables 1 and 2. First, we give the key recursion for $\mathop{\rm IrOA}$s. For any two positive integers $a$ and $b$ ($a\leq b$), let $[a,b]$ denote the set $\{a,a+1,\ldots,b\}$.

By repeatedly using the above lemma, we can deduce that if there exists a linear $\mathop{\rm IrOA}(r_{1},N,d,k)$ for any $N\in[N_{1},2N_{1}-1]$, then there exists a linear $\mathop{\rm IrOA}(r_{1},N,d,k)$ for any integer $N\geq N_{1}$. By some known linear codes, we construct some $k$-uniform states in the following.

In feng2017multipartite , the authors showed that there exists a constant $M_{k}(d)\geq 2k$, such that for all $N\geq M_{k}(d)$, there exists a $k$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ by probabilistic method. From Theorem 8, we know that $M_{k}(d)$ can be chosen to be $2k$ when $d\geq 4k-2$ is a prime power. Further, $M_{k}(d_{1}d_{2}\dots d_{n})$ can be chosen to be $2k$ when every $d_{i}\geq 4k-2$ is a prime power by Lemma 2.

Next, we focus on the construction of $4,5$-uniform states. First, we consider $4$-uniform states. Some of our constructions are from self-dual codes. For an $[N,\frac{N}{2},w]_{s}$ self-dual code, the dual distance is also $w$. For example, there exists a $[12,6,6]_{4}$ self-dual code Selftable , then there exists a linear $\mathop{\rm IrOA}(r,N,2,4)$ for any $N\in[11,12]$ by Lemmas 5 and 6. See Selftable for tables of self-dual codes. By Theorem 8, when $d\geq 14$ is a prime power, there exists a $4$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ for any $N\geq 8$. By (feng2017multipartite, , Theorem 12), there exists a $4$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ for any prime $d\geq 2$ and $N\geq 12$. Then there exists a $4$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ for any $d\geq 2$ and $N\geq 12$ by Lemma 2. Thus, we only need to consider $d<14$ that is a prime power, and $8\leq N\leq 11$.

1. (i)

   When $d=2$, $4$-uniform states in $(\mathbb{C}^{2})^{\otimes N}$ do not exist for each $N=8,9,10$ by Rains’ bound rains1999quantum .

2. (ii)

   When $d=3$, $4$-uniform states in $(\mathbb{C}^{3})^{\otimes 8}$ do not exist by Shadow bound huber2018bounds , and a $4$-uniform state in $(\mathbb{C}^{3})^{\otimes N}$ exists for any $N=9$ by AMEtable , and $N=10,11$ by feng2017multipartite .

3. (iii)

   When $d=4$, a $4$-uniform state in $(\mathbb{C}^{4})^{\otimes N}$ exists for any $N=9,10$ by AMEtable and $N=11$ by a $[12,6,6]_{4}$ self-dual code in Selftable .

4. (iv)

   When $d=5$, a $4$-uniform state in $(\mathbb{C}^{5})^{\otimes N}$ exists for any $N\in[8,10]$ by AMEtable , $N=11$ by a $[12,6,6]_{5}$ self-dual code in Selftable .

5. (v)

   When $d=7,8$, a $4$-uniform state in $(\mathbb{C}^{7})^{\otimes N}$ exists for any $N\in[8,11]$ by AMEtable .

6. (vi)

   When $d=9$, a $4$-uniform state in $(\mathbb{C}^{9})^{\otimes N}$ exists for any $N\in[9,11]$ by Lemma 2, and $N=8$ by Theorem 5.

7. (vii)

   When $d=11,13$, a $4$-uniform state in $(\mathbb{C}^{d})^{\otimes N}$ exists for any $N\in[8,11]$ by Theorem 8.

By using Lemma 2, we are able to list the existence of $4$-uniform states in $(\mathbb{C}^{d})^{\otimes N}$ in Table 1. For the same discussion as above, we have Table 2 for the existence of $5$-uniform states. The detail of arguments are provided in Appendix C. In the next section, we shall apply results in this section to the quantum information masking.

In modi2018masking , the authors proposed the concept of quantum information masking. An operation ${\cal S}$ is said to mask quantum information contained in states $\{|a_{j}\rangle_{A_{1}}\in{\cal H}_{A_{1}}\}$ by mapping them to states $\{|\psi_{j}\rangle\in{\cal H}_{A_{1}}\otimes{\cal H}_{A_{2}}\}$ such that all the reductions to one party of $|\psi_{j}\rangle$ are identical. In li2018masking , quantum information masking in multipartite systems is proposed. It is also required that all the reductions to one party of $\{|\psi_{j}\rangle\in\otimes_{\ell=1}^{N}{\cal H}_{A_{\ell}}\}$ are identical. However, collusion between some parties would then reveal the encoded quantum information. For example, a masker ${\cal S}$ masks quantum information contained in $\{|0\rangle,|1\rangle,|2\rangle\}\in\mathbb{C}^{3}$ into $\{|\psi_{0}\rangle,|\psi_{1}\rangle,|\psi_{2}\rangle\}\in\mathbb{C}^{3}\otimes\mathbb{C}^{3}\otimes\mathbb{C}^{3}$, where $|\psi_{0}\rangle_{ABC}=\frac{1}{\sqrt{3}}(|000\rangle+|111\rangle+|222\rangle)$, $|\psi_{1}\rangle_{ABC}=\frac{1}{\sqrt{3}}(|021\rangle+|102\rangle+|210\rangle)$, and $|\psi_{2}\rangle_{ABC}=\frac{1}{\sqrt{3}}(|012\rangle+|120\rangle+|201\rangle)$. If Alice and Bob are collusive, then $\rho^{(0)}_{AB}={\rm Tr}_{C}|\psi_{0}\rangle\!\langle\psi_{0}|=\frac{1}{3}(|00\rangle\!\langle 00|+|11\rangle\!\langle 11|+|22\rangle\!\langle 22|)$, $\rho^{(1)}_{AB}={\rm Tr}_{C}|\psi_{1}\rangle\!\langle\psi_{1}|=\frac{1}{3}(|02\rangle\!\langle 02|+|10\rangle\!\langle 10|+|21\rangle\!\langle 21|)$, and $\rho^{(2)}_{AB}={\rm Tr}_{C}|\psi_{2}\rangle\!\langle\psi_{2}|=\frac{1}{3}(|01\rangle\!\langle 01|+|12\rangle\!\langle 12|+|20\rangle\!\langle 20|)$. Alice and Bob can easily distinguish $\rho^{(0)}_{AB}$, $\rho^{(1)}_{AB}$, and $\rho^{(2)}_{AB}$, and they would reveal the encoded quantum information. To avoid this collusion, we propose the $k$-uniform quantum information masking as follows.

Our masking protocol also forms the basis for quantum secret sharing zukowski1998quest ; cleve1999how . The $k$-uniform quantum information masking allows secret sharing of quantum information from a “boss” to his “subordinates”, such that every collaboration between $k$ subordinates cannot retrieve the information. When $k=1$, the above definition is the same as modi2018masking ; li2018masking . In fact, ${\cal S}$ can be modeled by a unitary operator $U_{{\cal S}}$ on the system $A_{1}$ plus some ancillary systems $\{A_{2},A_{3},\ldots,A_{N}\}$ and given by

The unitary operator $U_{{\cal S}}$ preserves orthogonality. See Fig. 2 for the sketch map of $k$-uniform quantum information masking. We assume that ${\cal H}_{A_{1}}$ associates with the Hilbert space $\mathbb{C}^{d}$ and $U_{{\cal S}}|j\rangle_{A_{1}}\otimes|b\rangle_{A_{1}^{c}}=|\psi_{j}\rangle$ for $0\leq j\leq d-1$. Then the masking process can be expressed by:

Next, we give the relation between $k$-uniform states and $k$-uniform quantum information masking.

All the $k$-uniform states in Sec. II and Sec. III can be used for $(k-1)$-uniform quantum information masking by Theorem 10.

From Table 3, we know that a $2$-uniform state in $(\mathbb{C}^{d})^{\otimes 4}$ exists for any $d\geq 3$ and $d\neq 6$, then all states in $\mathbb{C}^{d}$ can be (1-uniformly) masked in $(\mathbb{C}^{d})^{\otimes 3}$ for any $d\geq 3$ and $d\neq 6$. Thus (li2018masking, , Corollary 2) is a special case of Theorem 10. All states in $\mathbb{C}^{d}$ can be strongly masked in $(\mathbb{C}^{d})^{\otimes 5}$ for any $d\geq 2$, since an $\mathop{\rm AME}$ state in $(\mathbb{C}^{d})^{\otimes 6}$ exists for any $d\geq 2$ by Table 3. Moreover, since an $\mathop{\rm AME}$ state in $(\mathbb{C}^{d})^{\otimes 8}$ exists for any prime power $d\geq 5$ by Table 1, all states in $\mathbb{C}^{d}$ can be strongly masked in $(\mathbb{C}^{d})^{\otimes 7}$ for any prime power $d\geq 5$. Thus, we have the following corollary.