The Independence of Distinguishability and the Dimension of the System

In quantum information theory, the distinguishability of states is of central importance. If general POVMs are allowed, states can be distinguished if and only if they are orthogonal1 (1). However, in realistic tasks, multipartite states are often shared by separated owners, who can not employ general POVMs. Fortunately, technologies of classical communications have been well-developed and can be employed easily. In spirit of this, distinguishing states by local operators and classical communications (LOCC) becomes available and significant. On the other hand, since the distinguishability via LOCC POVMs implies the distinguishability via SEP POVMs, which implies the distinguishability via PPT POVMs, while PPT and SEP POVMs have more simple properties than LOCC ones, they are also considerable.

There are substantial researches on distinguishabilities, especially LOCC distinguishability. It has been shown that two orthogonal pure states are LOCC distinguishable1 (1). An innocent intuition might be that the more entanglement a set has, the harder it can be distinguished by LOCC, which, however, is not true in general. Although entanglement indeed gives bounds to the LOCC distinguishability3 (3, 4), a LOCC indistinguishable set of nine orthogonal product states in $C^{3}\otimes C^{3}$ exists2 (2), which of course has no entanglements.

The local distinguishability of maximally entangled states might be the most-researched one. In $C^{3}\otimes C^{3}$, three orthogonal maximally entangled states are LOCC distinguishable5 (5) while there exist three LOCC₁ indistinguishable orthogonal maximally entangled states in $C^{d}\otimes C^{d}$, for $d\geqslant 4$ be even or $d=3k+2$6 (6). The result was weakened but extended to all $d\geqslant 4$, namely there are 4 LOCC₁ indistinguishable orthogonal maximally entangled states in $C^{d}\otimes C^{d}$ when $d\geqslant 4$7 (7). More results were given for Bell states and generalized Bell states. In $C^{2}\otimes C^{2}$, three Bell states are LOCC indistinguishable8 (8), while in $C^{d}\otimes C^{d}$ with $d\geqslant 3$, three generalized Bell states are LOCC distinguishable9 (9). A result shows that if $d$ is a prime and $l(l-1)\leqslant 2d$, then $l$ generalized Bell states are LOCC distinguishable10 (10). Note that $d+1$ maximally entangled states in $C^{d}\otimes C^{d}$ are LOCC indistinguishable4 (4). Strangely, a set of generalized Bell states is LOCC distinguishable by two copies11 (11).

On the other hand, LOCC distinguishability of orthogonal product states is also interesting. It has been shown that an unextendible product basis is LOCC indistinguishable12 (12) while the LOCC distinguishabllity of a completed product basis is equivalent to its LPCC distinguishability19 (19). Constructing LOCC indistinguishable product states is also studied13 (13, 14, 15, 16, 17). We mention that in $C^{3}\otimes C^{2}$, there are four LOCC₁ indistinguishable orthogonal product states when Alice goes firstly, while in $C^{3}\otimes C^{3}$, there are five LOCC₁ indistinguishable orthogonal product states no matter who goes firstly18 (18). Other methods include 20 (20, 21, 22, 23).

As for LOCC indistinguishable sets, auxiliary resources might be employed distinguishing24 (24, 25, 26, 27, 28). Another scheme is distinguishing with an inconclusive result, known as the unambiguous discrimination23 (23, 29, 30, 31, 32, 33, 34, 35). Finally, there are works for asymptotically LOCC distinguishability36 (36, 37).

Instead of considering states in a fixed system as most previous researches, we consider the nature problem that whether a set of indistinguishable states can become distinguishable by viewing them in a larger system without employing other resources. It is considerable for at least three reasons. Firstly, the local distinguishability of states is bounded by the dimension of the system. In a bipartite system, a necessary condition of a local distinguishable set is the total Schmidt rank not larger than the system dimension4 (4), which however, can be removed by viewing states in a larger system. Hence, whether the states remain indistinguishable is still suspectable. Secondly, by employing extra resources such as entanglement, a local indistinguishable set may become distinguishable24 (24, 25, 26, 27, 28), while, an universal resource might only exist in a larger system27 (27). This gives a feeling that the local indistinguishability of states might depend on the system. Finally, the distinguishability of points in a Hilbert space could be described by distances. For example, setting the distance of two different points be 1, and otherwise be 0. However, the distance of points may depend on the chosen space. For instance, the usual distance of two diagonal points of a unit square is 2 in the 1-dimensional space consisting of the edges of the square, while it is $\sqrt{2}$ in a 2-dimensional space consisting of the plane of the square. Hence, it is worth suspecting that the distinguishability of states may depend on the chosen space.

In this paper, we demonstrate that LOCC₁, unambiguous LOCC, PPT and SEP distinguishabilities are properties of states themselves, namely independent of the system dimension, by proving that an indistinguishable set in $\otimes_{k=1}^{K}C^{d_{k}}$ remains indistinguishable via POVMs of the same kind even being viewed in $\otimes_{k=1}^{K}C^{d_{k}+h_{k}}$. Our result solves the problem of searching the maximal number of local distinguishable states once for all and provides a LOCC indistinguishable product basis in general systems. Note that the result is suitable for general states in general systems and both perfect and unambiguous discriminations.

In mathematics, a quantum system shared by $K$ owners, namely partite $A^{(s)},s=1,2,...,K$, can be described as a Hilbert space $\otimes_{k=1}^{K}C^{d_{k}}$, while a general state can be described as a density operator (positive semi-definited with unit trace). A state set $S$ is LOCC(LOCC₁, PPT, SEP) distinguishable means that there is a LOCC(LOCC₁, PPT, SEP) POVM $\left\{\ M_{j}\right\}_{j=1,2,...,J}$ such that for any $j$, $Tr(M_{j}\rho)\neq 0$ for at most one state $\rho$ in $S$.

A LOCC_r POVM is described as follow. Partite $A^{(s)}$, $s=1,2,...,K$, provide local measurements, depending on the previously published results, on their partita and publish the results, in order. A round is that every member measures and publishes the result once. A POVM $\left\{\ M_{j}\right\}_{j=1,2,...,J}$ generated by such a procedure after $r$ rounds is defined to be a LOCC_r POVM. For example, a $LOCC_{1}$ POVM is given as follow. $A^{(1)}$ provides a POVM $\left\{\ A^{(1)}_{j}\right\}_{j=1,2,...,J}$ on his partita and gets an outcome $j_{1}$. For $s=2,3,...,K$, each $A^{(s)}$ provides a POVM $\left\{\ A^{(s)}_{j_{1},j_{2},...,j_{(s-1)},j}\right\}_{j=1,2,...,J_{j_{1},j_{2},...,j_{(s-1)}}}$ on his partita depending on the classical communications of others measured before him, and gets an outcome $j_{s}$. Hence, the LOCC₁ POVM is $\left\{\ M_{j_{1},j_{2},...,j_{(K-1)},j_{K}}\right\}_{1\leqslant j_{s}\leqslant J_{j_{1},j_{2},...,j_{(s-1)}}}$, where $M_{j_{1},j_{2},...,j_{(K-1)},j_{K}}=\otimes_{s=1}^{K}A^{(s)}_{j_{1},j_{2},...,j_{(s-1)},j_{s}}$ and $J_{j_{1},j_{2},...,j_{(s-1)}}=J$ for $s=1$. On the other hand, $\left\{\ M_{j}\right\}_{j=1,2,...,J}$ is said to be a PPT POVM if every $M_{j}$ is positive semi-definited after a partial transposition, while it is said to be a SEP POVM if every $M_{j}$ can be written as a tensor product of local operators.

There is a nature embedding from $C^{d}$ to $C^{d+h}$, viewing a state in $\otimes_{k=1}^{K}C^{d_{k}}$ as a state in $\otimes_{k=1}^{K}C^{d_{k}+h_{k}}$. Precisely, a computational basis in $\otimes_{k=1}^{K}C^{d_{k}}$ can be extended to a computational basis in $\otimes_{k=1}^{K}C^{d_{k}+h_{k}}$, by which all operators are written in matrix form. For a density matrix(state) $\rho$ in $\otimes_{k=1}^{K}C^{d_{k}}$, it can be viewed as the density matrix(state) $\widetilde{\rho}=\begin{pmatrix}\rho&0\\ 0&0\end{pmatrix}$ in $\otimes_{k=1}^{K}C^{d_{k}+h_{k}}$.

These views will be employed in the rest of the paper.

Note that unambiguous LOCC distinguishability is equivalent to unambiguous SEP distinguishability23 (23).

The following corollaries, providing maximal number of distinguishable states, generalize results in special systems to general systems and thus somehow show the abilities of the theorem.

For general pure states, since there exist three LOCC₁ indistinguishable orthogonal states in $C^{2}\otimes C^{2}$, for example, three orthogonal Bell states, Theorem 1 together with the result in 1 (1) imply that:

For orthogonal product states, four LOCC₁ indistinguishable states were constructed in $C^{3}\otimes C^{2}$, providing fixed measurement order, while five LOCC₁ indistinguishable states were constructed in $C^{3}\otimes C^{3}$, providing choice measurement order. Results in bipartite systems also show that three orthogonal product states are LOCC₁ distinguishable in any order while four orthogonal product states are LOCC₁ distinguishable in suitable order18 (18).

Therefore, as a consequence of Theorem 1, we have:

For orthogonal product basis, a LOCC indistinguishable basis in $C^{3}\otimes C^{3}$ was constructed2 (2). Assisted with the result in 19 (19), by using Theorem 1, a LOCC indistinguishable orthogonal product basis in $C^{m}\otimes C^{n}$ can be constructed, where $m,n\geqslant 3$.

The above basis is LOCC indistinguishable, not only LOCC₁ indistinguishable. Details are provided as follow.

The nine Domino states (unnormalized) in 2 (2) form an orthogonal product basis in $C^{3}\otimes C^{3}$. They are $|0\rangle|0\pm 1\rangle,|0\pm 1\rangle|2\rangle,|2\rangle|1\pm 2\rangle,|1\pm 2\rangle|0\rangle,|1\rangle|1\rangle$. It is easy to see that in $C^{m}\otimes C^{n}$ with $m,n\geqslant 3$, the states together with $|i\rangle|j\rangle,i=3,4,...,m-1,j=3,4,...,n-1$, form a completed orthogonal product basis. We will demonstrate that it is LOCC indistinguishable.

We need a lemma which is an easy corollary of the result in 19 (19).

By Theorem 1, the above orthogonal basis is LOCC₁ indistinguishable, since the Domino states are LOCC (and thus LOCC₁) indistinguishable in $C^{3}\otimes C^{3}$. As we are considering an orthogonal product basis, the above lemma shows that it is LOCC indistinguishable.

The construction can be generalized to multipartite systems, by the tensor product of above basis after normalizing and a normalized orthogonal basis of other partite.

We only prove the theorem for perfect discriminations. The proofs for unambiguous discriminations are similar.

We will show that the up-left block of a POVM in $\otimes_{k=1}^{K}C^{d_{k}+h_{k}}$ is a POVM in $\otimes_{k=1}^{K}C^{d_{k}}$ of the same kind for LOCC₁, PPT, SEP or global POVM, while, the condition of distinguishability in a lower dimensional system is the same as in the larger dimensional system, since the low-right block has trace 0.

For a POVM $\left\{\ M_{j}\right\}_{j=1,2,...,J}$ in $\otimes_{k=1}^{K}C^{d_{k}+h_{k}}$, $M_{j}$ can be written as

$M_{j}=\sum_{i}(\otimes_{s=1}^{K}A^{(s)}_{ji})$, where the sum is finite. Write $A^{(s)}_{ji}=\begin{pmatrix}A^{(s)}_{ji1}&A^{(s)}_{ji2}\\ A^{(s)}_{ji3}&A^{(s)}_{ji4}\end{pmatrix}$, $M_{j}=\begin{pmatrix}M_{j1}&M_{j2}\\ M_{j3}&M_{j4}\end{pmatrix}$ in block form, where $A^{(s)}_{ji1}$ is a $d_{s}\times d_{s}$ matrix, $M_{j1}$ is a $\prod_{s=1}^{K}d_{s}\times\prod_{s=1}^{K}d_{s}$ matrix. Then $M_{j1}=\sum_{i}\otimes_{s=1}^{K}(A^{(s)}_{ji1})$. Since $\left\{\ M_{j}\right\}_{j=1,2,...,J}$ is a POVM, $\sum_{j}M_{j}=I_{\prod_{s=1}^{K}(d_{s}+h_{s})}$ while $M_{j}$ is positive semi-definited. Therefore, $\sum_{j}M_{j1}=I_{\prod_{s=1}^{K}d_{s}}$ while $M_{j1}$ is positive semi-definited. Hence, $\left\{\ M_{j1}\right\}_{j=1,2,...,J}$ is a POVM.

$\left\{\ M_{j}\right\}_{j=1,2,...,J}$ is a $PPT$ POVM means that for every j, $M_{j}$ is positive semi-definited after a partial transposition. Without loss generality, for j, assume that the partial transposition is on $A^{(1)}$. Hence, $\sum_{i}[(A^{(1)}_{ji})^{T}\otimes\otimes_{s=2}^{K}(A^{(s)}_{ji})]$ is positive semi-definited and thus, $\sum_{i}[(A^{(1)}_{ji1})^{T}\otimes\otimes_{s=2}^{K}(A^{(s)}_{ji1})]$ is positive semi-definited, which implies that $M_{j1}$ is a $PPT$ operator.

$\left\{\ M_{j}\right\}_{j=1,2,...,J}$ is a $SEP$ POVM means that $M_{j}$ can be written as

On the other hand, as in the setting section, let $\left\{\ M_{j_{1},j_{2},...,j_{(K-1)},j_{K}}\right\}_{1\leqslant j_{s}\leqslant J_{j_{1},j_{2},...,j_{(s-1)}}}$ be a $LOCC_{1}$ POVM, where $M_{j_{1},j_{2},...,j_{(K-1)},j_{K}}=\otimes_{s=1}^{K}A^{(s)}_{j_{1},j_{2},...,j_{(s-1)},j_{s}}$. Write

Let $\left\{\ \rho_{i}|i=1,2,...,N\right\}$ be a set of orthogonal states in $\otimes_{s=1}^{K}C^{d_{s}}$. Assume that it is distinguishable via $LOCC_{1}$(PPT, SEP, global) POVM $\left\{\ M_{j}\right\}_{j=1,2,...,J}$ in $\otimes_{s=1}^{K}C^{d_{s}+h_{s}}$, namely, for every $j$, $Tr(M_{j}\widetilde{\rho_{i}})\neq 0$ for at most one i. Let us prove that it is distinguishable via $LOCC_{1}$(PPT, SEP, global) POVM in $\otimes_{s=1}^{K}C^{d_{s}}$.

Write $M_{j}=\begin{pmatrix}M_{j1}&M_{j2}\\ M_{j3}&M_{j4}\end{pmatrix}$ in block form, where $M_{j1}$ is a $\prod_{s=1}^{K}d_{s}\times\prod_{s=1}^{K}d_{s}$ matrix. Since $\left\{\ M_{j}\right\}_{j=1,2,...,J}$ is a $LOCC_{1}$(PPT, SEP, global) POVM, $\left\{\ M_{j1}\right\}_{j=1,2,...,J}$ is a $LOCC_{1}$(PPT, SEP, global) POVM of $\otimes_{s=1}^{K}C^{d_{s}}$, by the above Lemma.

Note that $Tr(M_{j1}{\rho_{i}})\neq 0$ implies that $Tr(M_{j}\widetilde{\rho_{i}})\neq 0$, since that they are equal. Hence, at most one $\rho_{i}$ satisfies $Tr(M_{j1}{\rho_{i}})\neq 0$, which means that the states are $LOCC_{1}$(PPT, SEP, global) distinguishable via POVM $\left\{\ M_{j1}\right\}_{j=1,2,...,J}$ in $\otimes_{s=1}^{K}C^{d_{s}}$. $\hfill\blacksquare$

The result of other indistinguishabilities may also hold. However, the method in this paper may not work. For example, the up-left block of a LOCC(LPCC) POVM in $\otimes_{k=1}^{K}C^{d_{k}+h_{k}}$ may not be a LOCC(LPCC) POVM in $\otimes_{k=1}^{K}C^{d_{k}}$. The reason may relate to that measuring a state in a larger dimensional system, the collapsing state may not in the ordinary lower dimensional system. We construct a projective POVM in $C^{4}\otimes C^{4}$, which is not projective when only looking at the up-left $3\times 3$ block as follow.

However, it will not be surprising if the result can be extended to other cases. Note that for a completed or unextendible product basis, the result holds for LOCC distinguishability19 (19, 36). Hence, we have the following definitions and conjecture:

This gives a framework of local-global indistinguishability, which states the independence of state distinguishability and the dimension of the system. Now, Theorem 1 can be restated as: For M=$LOCC_{1}$ (unambiguous LOCC, (perfect or unambiguous) PPT, SEP, global) distinguishability, the conjecture is true.

In this paper, we consider the nature problem that whether the indistinguishability of states depends on the dimension of the system. We demonstrate that LOCC₁, PPT and SEP indistinguishabilities, both perfect and unambiguous, are properties of states themselves and independent of the dimensional choice. More exactly, we show that if states are LOCC₁ (or unambiguous LOCC, (perfect or unambiguous) PPT, SEP) indistinguishable in a lower dimensional system, then they are LOCC₁ (or unambiguous LOCC, (perfect or unambiguous) PPT, SEP) indistinguishable in a dimensional extended system. The result is true for both bipartite and multipartite systems and for both pure and mixed states.

Assisted with previous results, Theorem 1 gives the maximal numbers of local distinguishable states and can be employed to construct a LOCC indistinguishable orthogonal product basis in general systems, except for one or two small dimensional ones. Note that the corollaries are even suitable for multipartite systems.

For further discussions, we define the local-global indistinguishable property and present a conjecture. Both proving the validity and searching counter-examples for it could be interesting.

We wish to acknowledge professor Zhu-Jun Zheng who gave advices and helped to check the paper.