Invariants for maximally entangled vectors and unitary bases

Introduction

In Section 2 of the paper, we introduce fan representations of unitary bases. The concepts and results of Section 2 are illustrated via different examples in Section 3. Some of the ideas developed in Section 2 as well as the examples discussed in Section 3 are then applied in Section 4 to discuss the issue of quantum state tomography. Finally, we draw the conclusion in Section 5.

Unitary bases and their fan representation

We begin with some basic material. We shall freely use and in this section. Let $\mathcal{H}$ be a $d$-dimensional Hilbert space with $2\leq d<\infty$ and $X$ a set of $d^{2}$ elements such as $\{1,2,\ldots,d^{2}\}$ or $\{0,1,\ldots,d^{2}-1\}$ etc. .

Let $\mathbf{U}=\{U_{x}:x\in X\}$ be a unitary basis, in short, UB, i.e. a collection $\mathbf{U}$ of unitary operators $U_{x}\in\mathcal{B}(\mathcal{H}),$ the $\ast$-algebra of bounded linear operators on $\mathcal{H}$ to itself, such that $\text{\rm tr}(U_{x}^{\ast}U_{y})=d\delta_{xy}$ for $x,y\in X.$

Rewording a part of the discussion after Proposition 9 , we call two unitary bases $\mathbf{U}$ and $\mathbf{U}^{\prime}$ equivalent if there exist unitaries $V_{1},$ $V_{2}$ in $\mathcal{B}(\mathcal{H})$ and a relabelling $x\rightarrow x^{\prime}$ of $X$ such that $U_{x^{\prime}}^{\prime}=V_{1}U_{x}V_{2}$ for $x$ in $X.$

We recall a one-to-one linear correspondence $\tau$ (in terms of this maximally entangled vector $\Omega$ in $\mathcal{H}\otimes\mathcal{H}$) between $\Psi\in\mathcal{H}\otimes\mathcal{H}$ and $A\in\mathcal{B}(\mathcal{H}),$ as set up in , Lemma 2 or , Proof of Theorem 1, for instance, via $\langle e_{j}|Ae_{k}\rangle=\sqrt{d}\langle e_{j}\otimes e_{k},\Psi\rangle.$ At times we express this as $\Psi\rightarrow A_{\Psi}$ or, even $A\rightarrow\Psi_{A}$ and write $\tau(\Psi)=A_{\Psi}.$

The map $\tau$ takes the set $\mathcal{M}$ of maximally entangled vectors in $\mathcal{H}\otimes\mathcal{H}$ to the set $\mathcal{U}(\mathcal{H})$ of unitaries on $\mathcal{H}.$

The rank of $A$ and the Schmidt rank of $\Psi_{A}$ are the same.

Entropy of $A^{\ast}A$ or certain variants are in vogue as measures of entanglement of $\Psi_{A}.$

A non-empty set $\mathbf{W}=\{W_{y}:y\in Y\}$ of unitaries in $\mathcal{H}$ indexed by $Y$ will be called a unitary system, in short, US, if $\text{\rm tr}\,W_{y}=0$ and $\text{\rm tr}\,(W_{x}^{\ast}W_{y})=d\delta_{xy}$ for $x,$ $y$ in $Y.$ The number $s=\#Y$ will be called the size of $\mathbf{W}.$

An abelian unitary system, in short, AUS, is a unitary system $\mathbf{W}$ with $W_{x}W_{y}=W_{y}W_{x}$ for $x,y$ in $Y.$

A maximal abelian subsystem of a unitary system $\mathbf{W},$ in short, $\mathbf{W}$-MASS, is a subset $\mathbf{V}$ of $\mathbf{W}$ which is an AUS and maximal with this property.

A tagged unitary system, in short, TUS, is a triple $\mathbf{T}=(x_{0},U_{x_{0}},\mathbf{W})$ where $x_{0}\in X,$ $U_{x_{0}}\in\mathcal{U}(\mathcal{H}),$ $\mathbf{W}=\{W_{y}:y\in X,y\neq x_{0}\}$ is a unitary system. We say $\mathbf{T}$ is tagged at $x_{0}.$

Let $\mathbf{T}$ be a tag at $x_{0}.$ Set $W_{x_{0}}=I.$ We set $\mathbf{U}=\{U_{x}=U_{x_{0}}W_{x}:x\in X\}.$ Then $\mathbf{U}$ satisfies $W_{x}^{\ast}W_{y}=U_{x}^{\ast}U_{y}$ for $x,$ $y\in X,$ and, therefore, $\mathbf{U}$ is a UB. We call $\mathbf{U}$ the $\mathbf{T}$-associated UB. On the other hand, given a UB $\mathbf{U},$ for $x_{0}\in X,$ setting $\mathbf{W}=\{W_{x}=U_{x_{0}}^{\ast}U_{x},x\in X,x\neq x_{0}\},$ we have $\mathbf{T}=(x_{0},U_{x_{0}},\mathbf{W})$ is a tag at $x_{0}.$ It satisfies $W_{x}^{\ast}W_{y}=U_{x}^{\ast}U_{y}$ for $x,$ $y\in X,$ where $W_{x_{o}}$ has been taken as $I.$ We call $\mathbf{T}$ the $\mathbf{U}$-associated tag at $x_{0}.$ We note that in both cases, for $x,y\in X,$ $W_{y}W_{x}^{\ast}=U_{x_{0}}^{\ast}U_{y}U_{x}^{\ast}U_{x_{0}}.$ Further, for $x,$ $y\in X,$ $W_{x}W_{y}=W_{y}W_{x}$ if and only if $U_{x}U_{x_{0}}^{\ast}U_{y}=U_{y}U_{x_{0}}^{\ast}U_{x}.$ We denote this condition by $\mathbf{T}(x,x_{0},y)$ and call it by Twill. Really speaking, both the associations are on the left and have obvious right versions as well.

Let $\mathbf{W}$ be a unitary system. Then $\text{\rm tr}\,A=0$ for each $A$ in the linear span $\mathcal{L}$ of $\mathbf{W}.$ So $I\not\in\mathcal{L}.$ Also $\text{\rm tr}\,(W_{x}^{\ast}W_{y})=d\delta_{xy}$ for $x,y$ in $Y$ forces $\{W_{x}:x\in Y\}$ to be linearly independent. So we have $s=\#Y\leq d^{2}-1$ and, thus, we may consider $Y$ as a proper subset of $X,$ if we like.

Given a US $\mathbf{W},$ $\widetilde{\mathbf{W}}=\mathbf{W}\cup\{I\}$ will be called the unitization of $\mathbf{W}.$ We note that $\widetilde{\mathbf{W}}$ is a system of mutually orthogonal unitaries and it is a UB if and only if the size of $\mathbf{W}$ is $d^{2}-1.$

AUS of size $d-1$ have been very well utilized by Wootters and Fields and Bandyopadhyay et. al. to study mutually unbiased bases in $\mathcal{H},$ see also , , , and . (, Lemma 3.1) records the basic fact that the size of an AUS can at most be $d-1.$

In fact, if $\mathbf{W}=\{W_{x}:x\in Y\}$ is an AUS of size $s,$ then there is a unitary $U$ on $\mathcal{H}$ and mutually orthogonal operators represented by mutually orthogonal diagonal matrices $D_{x},$ $x\in Y$ with entries in the unit circle $\mathbf{S}^{1}$ such that $\text{\rm tr}\,D_{x}=0,$ $W_{x}=UD_{x}U^{\ast},$ $x\in Y.$ As a consequence, the $(s+1)\times d$ matrix $H$ formed by the diagonals of $I$ and $D_{x},$ $x\in Y,$ as rows is a partial complex Hadamard matrix in the sense that $HH^{\ast}=dI_{s+1}$ and $|H_{jk}|=1$ for each entry $H_{jk}$ of $H.$ This forces $s\leq d-1.$ History and development of Hadamard matrices is very long and fascinating. We just mention a few sources , , , and , , , which we can directly use.

Obstructions to construction of MUB have occupied many researchers, e.g., see , , , , , . Even when, say for, $d$ a prime power, complete systems of MUB’s exist, some subsystems of certain UB’s may not be extendable to complete systems of MUB’s, this is explained well by Mandayam, Bandyopadhyay, Grassl and Wootters for $d=2^{2},2^{3},\ldots.$

Contents of the next three items are based on the fourth section of the second author’s preprint .

Let $\mathcal{F},$ $\mathcal{G}$ be subsets of $\mathcal{B}(\mathcal{H}).$

We say that $\mathcal{F}$ is collectively unitarily equivalent to $\mathcal{G},$ in short, $\mathcal{F}$CUE$\mathcal{G},$ if there exists a $V\in\mathcal{U}(\mathcal{H})$ such that $\mathcal{G}=\{V^{\ast}AV:A\in\mathcal{F}\}.$ We may say $\mathcal{F}$CUE$\mathcal{G}$ via $V.$

In case $\mathcal{F}$ and $\mathcal{G}$ are decomposed as $\mathcal{F}=\underset{\gamma\in\Gamma}{\cup}\mathcal{F}_{\gamma},$ $\mathcal{G}=\underset{\gamma\in\Gamma}{\cup}\mathcal{G}_{\gamma}$ respectively, then we may require that for $\gamma\in\Gamma,$ $\mathcal{F}_{\gamma}$CUE$\mathcal{G}_{\gamma}$ via $V,$ and say that $\mathcal{F}_{\Gamma}$CUE$\mathcal{G}_{\Gamma},$ or, if no confusion arises $\mathcal{F}$CUE$\mathcal{G}.$

In case $\mathcal{F}$ and $\mathcal{G}$ are indexed by a set $\Lambda$ as $\{A_{\alpha}:\alpha\in\Lambda\}$ and $\{B_{\alpha}:\alpha\in\Lambda\}$ respectively, then (as a special case of (ii) above) we may require that $B_{\alpha}=V^{\ast}A_{\alpha}V,\alpha\in\Lambda$ and say that $\mathcal{F}_{\Lambda}$CUE$\mathcal{G}_{\Lambda},$ or, if no confusion arises, $\mathcal{F}$CUE$\mathcal{G}.$

The operator $V$ in the definition above may not be unique. In fact, if $U^{\ast}\mathcal{G}U=\mathcal{G}$ for some $U\in\mathcal{U}(\mathcal{H})$ then $VU$ works fine too.

$\mathcal{F}$CUE$\mathcal{G}$ if and only if $\mathcal{F}_{\Lambda}$CUE$\mathcal{G}_{\Lambda}$ for some indexing $\mathcal{F}_{\Lambda}$ and $\mathcal{G}_{\Lambda}$ of $\mathcal{F}$ and $\mathcal{G}$ respectively by the same index set $\Lambda.$ So we may fix some such indexing, if we like.

The relation CUE (in both the senses in Definition 2.4 above) is an equivalence relation.

It can be readily seen that if $\mathcal{F}$CUE$\mathcal{G}$ and $\mathcal{F}$ is a commuting family then so is $\mathcal{G}.$

If $\mathcal{F}_{\Lambda}$CUE$\mathcal{G}_{\Lambda}$ via $V,$ then for each $\alpha\in\Lambda,$ the spectrum $\sigma(A_{\alpha})=\sigma(B_{\alpha});$ and for $\alpha\in\Lambda,$ for an eigenvalue $\lambda$ of $A_{\alpha},$ $\xi$ is an eigenvector for $B_{\alpha}$ with eigenvalue $\lambda$ if and only if $V\xi$ is an eigenvector for $A_{\alpha}$ with eigenvalue $\lambda.$

If $\mathcal{F}=\{F_{\alpha}:1\leq\alpha\leq n\}$ is a commuting $n$-tuple of normal operators in $\mathcal{B}(\mathcal{H}),$ then there exists a $U\in\mathcal{U}(\mathcal{H})$ and an $n$-tuple $\mathcal{D}$ of operators ($D_{\alpha}:1\leq\alpha\leq n$) represented by diagonal matrices $\{\widetilde{D}_{\alpha}:1\leq\alpha\leq n\}$ with respect to basis $\mathbf{e}$ such that $F_{\alpha}=UD_{\alpha}U^{\ast},$ $1\leq\alpha\leq n.$ In other words, $\mathcal{F}$CUE$\mathcal{D}.$ The converse is also true.

Garcia and Tener (, Theorem 1.1) obtained a canonical decomposition for complex matrices $T$ which are UET, i.e., unitarily equivalent to their transposes $T^{t},$ we may call a tuple $\mathcal{F}=(F_{\alpha}:1\leq\alpha\leq n)$ of $d\times d$ matrices collectively unitarily equivalent to the respective transposes, in short, CUET, if $\mathcal{F}$CUE$\mathcal{F}^{t},$ where $\mathcal{F}^{t}=(F_{\alpha}^{t}:1\leq\alpha\leq n).$

is not CUET where $\lambda$ is a non-real complex number and $\mu$ is a complex number with $|\mu|=(1+|\lambda|^{2})^{\frac{1}{2}}.$

Garcia and Tener [, expressions (1.4), (1.5), (1.6)] note that if T=\left(\begin{array}[]{cc}A&B\\ D&A^{t}\end{array}\right), with $d=2n,$ $A,B,D$ $n\times n$ matrices satisfying $B^{t}=-B,$ $D^{t}=-D$ (to be called skew-Hamiltonian (SHM, in short)) then it is UET simply because $T=JT^{t}J^{\star},$ with J=\left(\begin{array}[]{cc}0&I\\ -I&0\end{array}\right). We can now immediately strengthen this to : for every $d,$ every non-empty collection $\mathcal{F}$ of $d\times d$ SHM’s is CUET via $J.$ For more details one can see , particularly §6 and §7.

(, items 8.3, 8.4 and 8.5) tell us how to construct CUET tuples.

Let $[A_{jk}]$ be a positive block matrix such that $(A_{jk}:1\leq j,k\leq n)$ is CUET. Then $[A_{jk}^{t}]$ is positive. To see this we first note that there is a unitary matrix $U$ such that $A_{jk}=UA_{jk}^{t}U^{\ast}$ for $1\leq j,k\leq n.$ Let $\widetilde{U}$ be the block matrix $[\delta_{jk}U],$ with $\delta_{jk}=0$ for $j\neq k$ and $1$ for $j=k.$ Then $\widetilde{U}$ is unitary and $[A_{jk}^{t}]=\widetilde{U}^{\ast}[A_{jk}]\widetilde{U}.$ So $[A_{jk}^{t}]$ is positive.

Items (i)(b) and (ii) above can be put together to construct matrices with positive partial transposes, in short, PPT matrices.

Let $\mathbf{U},$ $\mathbf{U}^{\prime}$ be unitary bases for $\mathcal{H}.$ Then the following are equivalent.

$\mathbf{U}$ is equivalent to $\mathbf{U}^{\prime},$

for some $\mathbf{U}$-associated tag $\mathbf{T}$ and some $\mathbf{U}^{\prime}$-associated tag $\mathbf{T}^{\prime},$ $\mathbf{W}$CUE $\mathbf{W}^{\prime}.$

for each $\mathbf{U}$-associated tag $\mathbf{T},$ there is a $\mathbf{U}^{\prime}$-associated tag $\mathbf{T}^{\prime}$ such that $\mathbf{W}$CUE$\mathbf{W}^{\prime}.$

(i) $\Rightarrow$ (iii), Suppose $\mathbf{U}\sim\mathbf{U}^{\prime}.$ Then there exist $V_{1},V_{2}\in\mathcal{U}(\mathcal{H})$ and a relabelling $x\rightarrow x^{\prime}$ of $X$ such that $U_{x^{\prime}}^{\prime}=V_{1}U_{x}V_{2}$ for $x\in X.$ Consider any $x_{0}\in X$ and let $\mathbf{T}=(x_{0},U_{x_{0}},\mathbf{W})$ be the $\mathbf{U}$-associated tag at $x_{0}$ and $\mathbf{T}^{\prime}=(x_{0}^{\prime},U_{x_{0}^{\prime}}^{\prime},\mathbf{W}^{\prime}),$ the $\mathbf{U}^{\prime}$-associated tag at $x_{0}^{\prime}.$ Set $W_{x^{\prime}_{0}}^{\prime}=W_{x_{0}}=I.$ Then $U_{x_{0}^{\prime}}^{\prime}=V_{1}U_{x_{0}}V_{2}$ and, therefore, for $x\in X,$ $W_{x^{\prime}}^{\prime}=U_{x_{0}^{\prime}}^{\prime^{\ast}}U_{x^{\prime}}^{\prime}=V_{2}^{\ast}U_{x_{0}}^{\ast}V_{1}^{\ast}V_{1}U_{x}V_{2}=V_{2}^{\ast}W_{x}V_{2}.$ So $\mathbf{W}$CUE$\mathbf{W}^{\prime}$ via $V_{2}.$

(iii) $\Rightarrow$ (ii) is trivial. (ii) $\Rightarrow$ (i), Let $\mathbf{T}=(x_{0},U_{x_{0}},\mathbf{W})$ and $\mathbf{T}^{\prime}=(x_{0}^{\prime},U_{x_{0}^{\prime}}^{\prime},\mathbf{W}^{\prime})$ be the $U$-associated tag at $x_{0}$ and $\mathbf{U}^{\prime}$-associated tag at $x_{0}^{\prime}$ respectively with $\mathbf{W}$CUE$\mathbf{W}^{\prime}.$ Then by Remark 2.5(ii), there exist a $V\in\mathcal{U}(\mathcal{H})$ and a bijective function on $X\backslash\{x_{0}\}$ onto $X\backslash\{x_{0}^{\prime}\},$ say $x\rightarrow x^{\prime}$ such that $W_{x^{\prime}}^{\prime}=V^{\ast}W_{x}V$ for $x\in X\backslash\{x_{0}\}.$

Set $V_{1}=U_{x_{0}^{\prime}}^{\prime}V^{\ast}U_{x_{0}}^{\ast}.$ Then $V_{1}\in\mathcal{U}(\mathcal{H}).$ Also $U_{x_{0}^{\prime}}^{\prime}=V_{1}U_{x_{0}}V.$ Further, for $x\in X\backslash\{x_{0}\}$

So $\mathbf{U}$ is equivalent to $\mathbf{U}^{\prime}.$ ∎

Then there is a unique maximal family $\mathcal{V}_{\mathbf{W}}=\{\mathbf{V}_{\alpha}:\alpha\in\Lambda\}$ of $\mathbf{W}$-MASS’s, such that $\mathbf{W}=\underset{\alpha\in\Lambda}{\cup}\mathbf{V}_{\alpha}.$ If each $W_{y}$ in $\mathbf{W}$ has simple eigenvalues, then $\mathbf{V}_{\alpha}$’s are mutually disjoint.

Let $\mathbf{W}^{\prime}$ be a unitary system and $\mathcal{V}_{\mathbf{W}^{\prime}}=\{\mathbf{V}_{\beta}^{\prime}:\beta\in\Lambda^{\prime}\},$ the family of $\mathbf{W}^{\prime}$-MASS’s as in (i). Then $\mathbf{W}$CUE$\mathbf{W}^{\prime}$ via $V$ iff there is a bijective map on $\Lambda$ to $\Lambda^{\prime},$ say, $\alpha\rightarrow\alpha^{\prime},$ such that $\mathbf{V}_{\alpha}$CUE$\mathbf{V}^{\prime}_{\alpha^{\prime}}$ via $V.$

Let $\mathbf{W}=\{W_{y}:y\in Y\}.$ For any $y\in Y,$ $\{W_{y}\}$ is an AUS $\subset\mathbf{W}.$ Let $\mathcal{W}=\{\mathbf{A}:\mathbf{A}\subset\mathbf{W}\,\mbox{and}\,\mathbf{A}\,\mbox{is an AUS}\}$ and order it by inclusion. Then $\mathcal{V}_{\mathbf{W}}$ is made up of maximal elements of $\mathcal{W}.$ The second part follows from elementary Linear Algebra as indicated in Remark 2.9 (iv) below.

Suppose $\mathbf{W}$CUE$\mathbf{W}^{\prime}$ via $V.$ Then $\{\mathbf{\widehat{V}_{\alpha}}=\{V^{\ast}AV:A\in\mathbf{V}_{\alpha}\},\alpha\in\Lambda\}$ is a maximal family of $\mathbf{W}^{\prime}$-MASS’s with $\mathbf{W}^{\prime}=\underset{\alpha\in\Lambda}{\cup}\widehat{\mathbf{V}}_{\alpha}.$ So, by uniqueness, each $\widehat{\mathbf{V}}_{\alpha}$ is some unique $\mathbf{V}_{\beta}^{\prime}.$ Set $\beta=\alpha^{\prime}.$ On the other hand, each $\mathbf{V}_{\beta}^{\prime}$ is some unique $\widehat{\mathbf{V}}_{\alpha}.$ So the map $\alpha\rightarrow\alpha^{\prime}$ is bijective on $\Lambda$ to $\Lambda^{\prime}.$ The converse part is trivial.

In view of Theorem 2.7, we may add a sixth condition to Theorem 1 of viz., collection of tagged unitary systems as follows: Tagged unitary systems $\mathbf{T},$ i.e., any arbitrarily fixed $x_{0}\in X,$ a unitary $U_{x_{0}}\in\mathcal{U}(\mathcal{H})$ and unitaries $\{W_{x}:x\in X,x\neq x_{0}\}$ such that $\text{\rm tr}\,W_{x}=0=\text{\rm tr}\,(W_{x}^{\ast}W_{y})$ for $x,y,x\neq y$ in $X\backslash\{x_{0}\}.$

Theorem 2.8 says that, to within CUE, we may think of $\mathcal{V}_{\mathbf{W}}$ as an invariant for a unitary system $\mathbf{W}.$

The role of Hadamard matrices in these invariants has already been indicated above. To elaborate a bit, for each $\alpha\in\Lambda,$ there is a unitary $U_{\alpha}$ and a (partial) Hadamard $s_{\alpha}\times d$ matrix $H_{\alpha}$ with $s_{\alpha}=\,\,\mbox{size}\,\,\mathbf{V}_{\alpha}$ such that $\mathbf{V}_{\alpha}$ consists of operators of the type $U_{\alpha}DU_{\alpha}^{\ast},$ $D$ is a diagonal matrix whose diagonal forms a row of $H_{\alpha}.$ The ordering of rows corresponds to that of operators in $\mathbf{V}_{\alpha}.$ To within that $H_{\alpha}$ is unique upto a permutation of columns, and the corresponding $U_{\alpha}$’s will undergo changes accordingly. For each $\alpha\in\Lambda,$ the augmented matrix $\widetilde{H}_{\alpha}$ formed by adding a top row of all $1$’s is also a Hadamard matrix and it arises from the $\widetilde{\mathbf{W}}$-MASS $\widetilde{\mathbf{V}}_{\alpha}=\mathbf{V}_{\alpha}\cup\{I\}$ with same $U_{\alpha}$ in force.

It is now clear from (iii) above that if each $W_{y}$ in $\mathbf{W}$ has simple eigenvalues, then $\mathbf{V}_{\alpha}$’s are mutually disjoint. This happens for the case $d=2$ and $d=3$ but may not be so for larger $d$ because the requirement is that eigenvalues of $W_{y}$ lie on the unit circle and (counted with multiplicities) add to zero. In the last section on examples for $d>3,$ we give concrete situations. It is as if there is a fan of these subsets $\mathbf{V}_{\alpha}$’s (possibly overlapping) hinged at $I$ and, accordingly, a fan of abelian subspaces of $\mathcal{B}(\mathcal{H})$ (possibly overlapping), hinged at the linear span of $I.$

A $\mathbf{W}$-MASS of size $d-1$ together with $I$ generates a maximal abelian subalgebra of $M_{d},$ in short, a MASA in $M_{d}.$ Theory of orthogonal MASA is well developed by , , . In fact, even defines an entropy $h(A/B)$ between a pair $(A,B)$ of MASA’s and proves that $A,B$ are orthogonal if and only if $h(A/B)$ takes the maximum value, and then the value is $\log\,d.$

Definitions and Discussion 2.10. (Fan representation & Hadamard fans).

In view of item 2.9 (iv) above, we call $\mathcal{V}_{\mathbf{W}}$ in Theorem 2.8 (i), the fan representation of $\mathbf{W}.$ One can figure out the $\mathbf{W}$-MASS fan representation through a common eigenvector system approach. It is neat when eigenvalues of each $W_{x}$ in $\mathbf{W}$ are simple and becomes quite involved when some of them are multiple.

The family $H_{\mathbf{W}}=\left\{H_{\alpha}:\alpha\in\Lambda\right\}$ facilitated as in item 2.9 (iii) above, will be called the Hadamard fan of $\mathbf{W}.$ We note that if $\mathbf{W}$ and $\mathbf{W}^{\prime}$ are unitary systems with $\mathbf{W}$CUE$\mathbf{W}^{\prime}$ then their Hadamard fans are the same to within a labelling of $\Lambda$ and permutation of rows and columns of $H_{\alpha}.$

Let $\mathbf{U}$ be a UB and $\mathcal{W}=\{\mathbf{W}:\mathbf{T}=(x_{0},\mathbf{U}_{x_{0}},\mathbf{W})$ is the tag of $\mathbf{U}\,\,\mbox{at}\,\,x_{0}\}$ and $\mathcal{V}=\left\{\mathcal{V}_{\mathbf{W}}:\mathbf{W\in\mathcal{W}}\right\}.$ Then $\mathcal{V}$ will be called the fan system of $\mathbf{U}.$ We note that it follows from Theorem 2.7 and Theorem 2.8 that $\mathcal{V}$ is an invariant for $\mathbf{U}$ in the sense that if $\mathbf{U}^{\prime}$ is a UB and $\mathcal{V}^{\prime}$ is its fan system then $\mathbf{U}\sim\mathbf{U}^{\prime}$ if and only if to within CUE $\mathcal{V}\cap\mathcal{V}^{\prime}\neq\phi$ if and only if to within CUE $\mathcal{V}=\mathcal{V}^{\prime}.$

Let $\widetilde{H}_{\mathbf{W}}=\{\widetilde{H}_{\alpha}:\alpha\in\Lambda\}.$ We call $\widetilde{\mathbf{H}}=\{\widetilde{H}_{\mathbf{W}}:\mathbf{T}=(x_{0},U_{x_{0}},\mathbf{W})$ is the tag of $\mathbf{U}\,\,\mbox{at}\,\,x_{0}\}$ the Hadamard fan system of $\mathbf{U}.$ We note that if $\mathbf{U}$ and $\mathbf{U}^{\prime}$ are UB’s with Hadamard fan systems $\mathbf{H}$ and $\mathbf{H}^{\prime}$ respectively, and if $\mathbf{U}\sim\mathbf{U}^{\prime}$ then $\mathbf{H}=\mathbf{H}^{\prime}.$ The converse does not hold.

2.11. Maximally entangled state bases: The question that triggered this paper, in fact, is the following one in the context of maximally entangled states (MES) with phases. How to distinguish pairwise orthogonal systems of MES using local qantum operations supplemented by classical communication?

If one can figure out sets of pairwise orthogonal MES, locally unitarily connected up to global phases to the Bell basis then the task of distinguishing the states from the aforesaid sets is equivalent to that of distinguishing locally the Bell states.

(a) We now put the question in the language used in the beginning of this section. Let $\{|\Psi_{x}\rangle:x\in X\}$ be an orthonormal basis in $\mathcal{H}\otimes\mathcal{H}$ consisting of MES only. Do there exist unitaries $V_{1},V_{2}\in\mathcal{U}(\mathcal{H}),$ a bijective function $g$ on $X$ to itself and a function $f$ on $X$ to $S^{1}$ such that $|\psi_{g(x)}\rangle=f(x)(V_{1}\otimes V_{2})(U_{x}\otimes I)\Omega,$ $x\in X$ where $\{U_{x}:x\in X\}$ is the system $\{U_{mn}:m,n\in Y_{d}\}$ as explained in Example 3.1(vii) in the next section.

(b) In view of the item 2.1 (v)(a), there exists a system $\{V_{x}:x\in X\}$ of mutually orthogonal unitaries in $\mathcal{U}(\mathcal{H})$ such that $|\psi_{x}\rangle=(V_{x}\otimes I)\Omega$ for $x\in X$ and further, by item 2.1(vii), for $x\in X,$ $(V_{1}\otimes V_{2})(U_{x}\otimes I)\Omega$

Now $\widetilde{V}_{2}=V_{2}^{t}$ is a unitary if $V_{2}$ is so. So the question reduces to: Do there exist unitaries $V_{1},\widetilde{V}_{2}\in\mathcal{U}(\mathcal{H})$ and a function $f:X\rightarrow S^{1}$ such that $V_{g(x)}=f(x)V_{1}U_{x}\widetilde{V}_{2},$ $x\in X.$

In the terminology of item 2.1(ii): Does there exist a function $\widetilde{f}$ on $X$ to $S^{1}$ such that $\{\widetilde{f}(x)V_{x}:x\in X\}$ is equivalent to $\{U_{x}:x\in X\}.$ We shall utilise the results and methods given above to answer this.

Definition 2.12. We call two unitary bases $\mathbf{U}$ and $\mathbf{U}^{\prime}$ phase-equivalent if there exists a function $\widetilde{f}$ on $X$ to $S^{1}$ such that $\{\widetilde{f}(x)U_{x}^{\prime}:x\in X\}$ is equivalent to $\{U_{x}:x\in X\}.$

Definition 2.13. For subsets $\mathcal{F}$ and $\mathcal{G}$ of $\mathcal{B}(\mathcal{H})$ we say $\mathcal{F}$ is phase-collectively-unitarily equivalent to $\mathcal{G}$ and write $\mathcal{F}$PCUE$\mathcal{G}$ if there exists a function $f$ on $\mathcal{F}$ to $S^{1}$ such that $f\mathcal{F}$ CUE $\mathcal{G}$ where, $f\mathcal{F}=\{f(A)A:A\in\mathcal{F}\}.$

Remark 2.14. Let $\mathbf{W}=\{W_{y}:y\in Y\}$ be a unitary system and $h:Y\rightarrow S^{1}$ be a function. Then

$h\mathbf{W}=\{h(y)W_{y}:y\in W\}$ is a unitary system,

$\mathbf{W}$ is abelian if and only if $h\mathbf{W}$ is abelian,

$\mathbf{V}=\{W_{y}:y\in Z\}$ with $Z\subset Y$ is a $\mathbf{W}$-MASS if and only if $(h|Z)\mathbf{V}$ is a $\mathbf{W}$-MASS, and

$h\mathcal{V}_{\mathbf{W}}=\mathcal{V}_{h\mathbf{W}}.$

We can now have the obvious generalizations of Theorem 2.7, Theorem 2.8, item 2.9 and item 2.10 with obvious modifications of the corresponding proofs. Here is an illustration which will be strengthened further by examples in the next section.

Theorem 2.15. Let $\mathbf{U},\mathbf{U}^{\prime}$ be unitary bases for $\mathcal{H}.$ Then the following are equivalent.

$\mathbf{U}$ is phase equivalent to $\mathbf{U}^{\prime}.$

For some $\mathbf{U}$-associated tag $\mathbf{T}$ and some $\mathbf{U}^{\prime}$-associated tag $\mathbf{T}^{\prime},$ $\mathbf{W}$PCUE$\mathbf{W}^{\prime}.$

For each $\mathbf{U}$-associated tag $\mathbf{T},$ there is a $\mathbf{U}^{\prime}$-associated tag $\mathbf{T}^{\prime}$ such that $\mathbf{W}$ PCUE $\mathbf{W}^{\prime}.$

$\mathbf{U}$ and $\mathbf{U}^{\prime}$ have the same fan systems to within PCUE.

Examples

For $m,n\in Y_{d},$ let $U_{mn}$ (or, at times, also written as $U_{m,n}$ or $U_{(m,n)}$) be the operator which takes $|k\rangle$ to $H_{mk}^{n}$ $|\lambda(n,k)\rangle,$ $k\in Y_{d}.$

Then $\mathbf{U}=\{U_{m,n}:m,n\in Y_{d}\}$ is a UB. We note that its indexing set is $X=Y_{d}\times Y_{d}.$ Further, $U_{m,n}=I$ if and only if $\lambda(n,k)=k$ and $H_{mk}^{n}=1$ for each $k\in Y_{d}.$ In this case $\mathbf{W}=\mathbf{U}\backslash\{I\}$ is a unitary system.

For $(m,n),(m^{\prime},n^{\prime})\in X$ we say $(m,n)$ commutes with $(m^{\prime},n^{\prime})$ and write it as $(m,n)\Delta(m^{\prime},n^{\prime})$ if $U_{m,n}$ commutes with $U_{m^{\prime},n^{\prime}}.$ We now proceed to obtain maximal commuting subsets of $\mathbf{U}$ (to be called $\mathbf{U}$-MASS’s) or, equivalently, of $X.$ Let $(m,n),(m^{\prime},n^{\prime})\in X.$ Then $U_{mn}U_{m^{\prime}n^{\prime}}=U_{m^{\prime}n^{\prime}}U_{mn}$ if and only if for $k\in Y_{d},$ $U_{mn}\left(H_{m^{\prime}k}^{n^{\prime}}|\lambda(n^{\prime},k)\rangle\right)=U_{m^{\prime}n^{\prime}}\left(H_{mk}^{n}|\lambda(n,k)\rangle\right)$ if and only if for $k\in Y_{d},$ $H_{m,\lambda(n^{\prime},k)}^{n}H_{m^{\prime}k}^{n^{\prime}}|\lambda(n,\lambda(n^{\prime},k))\rangle\!\!=\!\!H_{m^{\prime},\lambda(n,k)}^{n^{\prime}}$ $H_{mk}^{n}|\lambda(n^{\prime},\lambda(n,k))\rangle$ if and only if

and $H_{m,\lambda(n^{\prime},k)}^{n}H_{m^{\prime}k}^{n^{\prime}}=H_{m^{\prime},\lambda(n,k)}^{n^{\prime}}H_{mk}^{n},$ $k\in Y_{d}\,\,\,\,\,\left[H_{(m,n),(m^{\prime},n^{\prime})}\right].$ We call these conditions Latin criss-cross and Hadamard criss-cross respectively.

Latin squares. A latin square $\lambda$ may be called a quasigroup $L$ in the sense that the binary operation ‘.’ on $L$ given by $a.b=\lambda(a,b)$ satisfies the condition that, given $s,t\in L,$ the equations $x.s=t$ and $s.y=t$ have unique solutions in $L;$ one may see, for instance, the book by Smith for more details. Keeping this in mind we introduce a few notions for $\lambda.$ (a) An element $a$ of $L$ will be called a left identity if $a.b=b$ for $b$ in $L.$ We note that a left identity, if it exists, is unique. Similar remarks apply to the notion and uniqueness of right identity. (b) $\lambda$ is called associative if ‘$\cdot$’ is associative. (c) Elements $a,b$ in $L$ will be said to be commuting if $a.b=b.a.$ (d) The centre $Z(L)$ of $L$ is $\{a\in L:a.b=b.a\,\,\mbox{for each}\,\,b\,\,\mbox{in}\,\,L\}.$ We shall mainly consider latin squares arising from a group $G$ (with multiplication written as juxtaposition and identity written as $e$) or right divisors or left divisors in the group $G$ as follows: (e) $a.b=ab,$ (f) $a.b=ab^{-1},$ (g) $a.b=a^{-1}b,$ for $a,b$ in $G.$ Direct computations give the following. (h) A right (respectively, left) divisor latin square has $e$ as right (respectively, left) identity. Further, any such latin square has both right and left identity if and only if $a^{2}=e$ for each $a$ in $G$ if and only if $L$ is the same as $G.$ (i) Any such $\lambda$ is associative if and only if $G$ and $L$ coincide. (j) Elements $a,b$ in any such $L$ commute if and only if $(ab^{-1})^{2}=e.$ (k) In particular, if the number of elements in $G,$ $|G|$ is an odd number $\geq 3$ then no two distinct elements in any such $L$ commute. We may have twisted version of (e), (f) and (g) as follows and then draw the same conclusions as above for them. (l) $a.b=ba,$ (m) $a.b=b^{-1}a,$ (n) $a.b=ba^{-1},$ for $a,b$ in $G.$ Let $\lambda^{-1}$ be the latin square $\mu$ defined by $\mu(a,\lambda(a,b))=b$ for $a,b$ in $L.$ (o) Direct computations give that $\mu^{-1}=\lambda.$ We may say that $(\lambda,\mu)$ is an inverse-pair. We note that latin squares listed above may then be inverse-paired as $((e),(g)),$ $((f),(m))$ and $((l),(n)).$

Hadamard criss-cross. (a) We first consider the case $n=n^{\prime}.$ Hadamard criss-cross becomes

Tags and Twills Contents of Section 2 tell us that it is $\mathbf{W}$-MASS’s for tags of $\mathbf{U}$ that really help us. And $\mathbf{U}$-MASS’s help directly if (a scalar multiple of) $I$ is in $\mathbf{U}$ simply because then apart from (the scalar multiple of) $I$ occurring in all $\mathbf{U}$-MASS’s, $\mathbf{W}$-MASS’s and $\mathbf{U}$-MASS’s are same. Let $\mathbf{T}=(x_{0},U_{x_{0}},\mathbf{W})$ be a tag of $\mathbf{U}.$ As noted in Remark 2.2, for $x,y\in X\backslash\{x_{0}\},$ $W_{x}W_{y}=W_{y}W_{x}$ if and only if Twill $\mathbf{T}(x,x_{0},y)$ viz., $U_{x}U_{x_{0}}^{\ast}U_{y}=U_{y}U_{x_{0}}^{\ast}U_{x}$ is satisfied. We now figure these out for some of the cases considered above. (a) Let $x_{0}=(m_{0},n_{0}).$ Then for $k\in Y_{d},$

So the Twill is equivalent to $\left[\lambda(n,n_{0},n^{\prime})\right]$ and $\left[H(m,n),(m_{0},n_{0}),(m^{\prime},n^{\prime})\right]$, given by

respectively. We call them Latin twill and Hadamard twill respectively. We may re-write Hadamard twill in another form as

(b) For latin squares coming from group $G$ as in (iii) (e) above. Latin twill reduces to

For the inverse latin square arising as in (iii)(g) above, it is $n^{-1}n_{0}n^{\prime-1}=n^{\prime-1}n_{0}n^{-1},$ which on taking inverses, becomes

This gives us exactly $4$ $\mathbf{W}$-MASS’s $\{(m_{0},k):k\neq n_{0}\},$ $\{(j,n_{0}):j\neq m_{0}\},$ $\{(m_{0}+1,n_{0}+1),(m_{0}+2,n_{0}+2)\}$ and $\{(m_{0}+1,n_{0}+2),(m_{0}+2,n_{0}+1)\}.$ They are all full-size and mutually disjoint.

One may see more details in the papers referred to, particularly for $d=p^{r},$ where $p$ is a prime and $d=6\,\,\mbox{or}\,\,10.$

Each of the sets $Y_{j}$ is a maximal commuting set in $Y$ and accordingly $\{U_{x}:x\in Y_{j}\}$ is a $\mathbf{W}$-MASS. Their number is 3 for $d=2$ and $2(d-1)$ for $d>2.$ Each of the sets $Y_{j}$ has cardinality $d-1.$ So for $d>3,$ they do overlap for some combinations. Sets $Y_{-1},$ $Y_{0},$ and $Y_{1}$ are pairwise disjoint. Some combinations of sets may have non-empty pairwise intersection. For instance, for $2<d=2r,$ $(r,r)\in Y_{1}\cap Y_{d-1}$ and $(r,0)\in Y_{0}\cap Y_{2},$ $(2,0)\in Y_{0}\cap Y_{r},$ $(2,2)\in Y_{1}\cap Y_{r+1}.$ However, for odd $d,$ $Y_{-1},Y_{0},Y_{1},Y_{d-1}$ are all pairwise disjoint, whereas, for even $d>2,$ they are all pairwise disjoint except for $Y_{1},$ $Y_{d-1}$ containing one element, viz., ($\frac{1}{2}d,$ $\frac{1}{2}d$) in common. Finally, for $j$ co-prime with $d,$ we have unique $\mathbf{W}$-MASS’s $Y_{-1},Y_{0},Y_{1},Y_{d-1}$ for $(0,j),(j,0),(j,j)$ and $(j,d-j)$ respectively.