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Approximate Quasiorthogonality of Operator Algebras and Relative Quantum Privacy

David W. Kribs1,2, Jeremy Levick1,2, Mike Nelson1, Rajesh Pereira1, Mizanur Rahaman2,3 1Department of Mathematics & Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1 2Institute for Quantum Computing, University of Waterloo, Waterloo, ON, Canada N2L 3G1 3Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1
Abstract.

We show that the approximate quasiorthogonality of two operator algebras is equivalent to the algebras being approximately private relative to their conditional expectation quantum channels. Our analysis is based on a characterization of the measure of orthogonality in terms of Choi matrices and Kraus operators for completely positive maps. We present examples drawn from different areas of quantum information.

Key words and phrases:
completely positive map, private quantum channel, conditional expectation, operator algebra, private quantum code, quasiorthogonal algebra.
2010 Mathematics Subject Classification:
47L90, 81P45, 81P94, 94A40

1. Introduction

The notion of quasiorthogonality for operator algebras arose from the study of modified forms of orthogonality for algebras and their relative behaviours in a variety of settings in finite-dimensional quantum information. Primarily motivated by mutually unbiased bases (MUB) constructions [2, 19, 18, 14, 34] and their associated commutative algebras initially, over the past decade the work expanded to the non-commutative setting [29, 28, 31, 27, 32, 30]. Built on these efforts, Weiner [36] opened up the study of approximate quasiorthogonality by introducing a measure of orthogonality between two algebras based on joint properties of their conditional expectation channels, and establishing results on the approximate version for some important special cases.

From a different direction, still with quantum information motivation, it has recently been recognized that there are connections between the study of quasiorthogonal operator algebras and work in quantum privacy; specifically, on the topic of what are variously known as private quantum channels or codes, decoherence-full or private subspaces and subsystems, and private algebras [1, 8, 4, 3, 21, 11, 17, 16, 12]. In particular, for a number of special cases of channels or algebras, quasiorthogonality has been linked with certain quantum privacy properties in those cases [24, 25, 22], all suggesting a deeper more general link between the topics.

In this paper, we establish the first general result that ties together approximate quasiorthogonality of operator algebras with approximate privacy for quantum codes, represented as algebras. This involves identification of an appropriate notion of relative quantum privacy, with natural assumptions on the algebras and private quantum codes considered, and in doing so, we also derive a new approach for computing Weiner’s measure of orthogonality in terms of Choi matrices and Kraus operators for the conditional expectation channels of the associated algebras. We present examples drawn from the framework for hybrid classical-quantum information theory, from studies of private quantum subsystems, and from work on approximate MUB constructions.

The paper is organized as follows. The next section includes preliminary material and the derivation of our approach to compute the orthogonality measure. In the third section we define relative approximate privacy of two algebras and present our main result and its proof. The fourth section contains examples and we conclude with a brief outlook discussion.

2. Measure of Quasiorthogonality and Quantum Privacy

Let Mn​(ℂ)M_{n}(\mathbb{C}) be the set of n×nn\times n complex matrices. In all of what follows, 𝒜,ℬ⊆Mn​(ℂ)\mathcal{A},\mathcal{B}\subseteq M_{n}(\mathbb{C}) are unital ∗*-algebras (or finite-dimensional C∗-algebras [13]); that is, In∈𝒜I_{n}\in\mathcal{A}, if A∈𝒜A\in\mathcal{A} so is A∗A^{*}, and 𝒜\mathcal{A} is closed under linear combinations and matrix multiplication, and the same is true for ℬ\mathcal{B}.

Given such 𝒜\mathcal{A}, ℬ\mathcal{B} we denote by ℰ𝒜\mathcal{E}_{\mathcal{A}}, ℰℬ\mathcal{E}_{\mathcal{B}} the (unique) trace-preserving, unital conditional expectations onto 𝒜\mathcal{A} and ℬ\mathcal{B} respectively. That is, ℰ𝒜:Mn​(ℂ)→Mn​(ℂ)\mathcal{E}_{\mathcal{A}}:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C}) (and similarly for ℰℬ\mathcal{E}_{\mathcal{B}}) is the linear map uniquely characterized by the following conditions:

  1. (1)

    ℰ𝒜​(A)=A\mathcal{E}_{\mathcal{A}}(A)=A for all A∈𝒜A\in\mathcal{A}

  2. (2)

    ℰ𝒜​(X)⪰0\mathcal{E}_{\mathcal{A}}(X)\succeq 0 whenever X⪰0X\succeq 0

  3. (3)

    ℰ𝒜​(A1​X​A2)=A1​ℰ𝒜​(X)​A2\mathcal{E}_{\mathcal{A}}(A_{1}XA_{2})=A_{1}\mathcal{E}_{\mathcal{A}}(X)A_{2} for all A1,A2∈𝒜A_{1},A_{2}\in\mathcal{A} and X∈Mn​(ℂ)X\in M_{n}(\mathbb{C})

  4. (4)

    Tr​(ℰ𝒜​(X))=Tr​(X)\mathrm{Tr}(\mathcal{E}_{\mathcal{A}}(X))=\mathrm{Tr}(X) for all X∈Mn​(ℂ)X\in M_{n}(\mathbb{C}).

We refer to ℰ𝒜\mathcal{E}_{\mathcal{A}} as the conditional expectation channel for 𝒜\mathcal{A}, reflecting standardized use of the term quantum channel to describe completely positive trace-preserving maps.

We now define the key notion of quasiorthogonal algebras, noting the early literature on the subject sometimes referred to the notion as ‘orthogonal’ or ‘complementary’. (We stick to use of the ‘quasi’ prefix as in [36] as it avoids possible confusion with other quantum information notions that use these other terms.)

Definition 2.1.

Two unital ∗*-algebras 𝒜\mathcal{A}, ℬ\mathcal{B} are quasiorthogonal if they satisfy any one of the following equivalent conditions:

  1. (1)

    Tr​((A−Tr​(A)n​In)​(B−Tr​(B)n​In))=0\mathrm{Tr}\bigl{(}(A-\frac{\mathrm{Tr}(A)}{n}I_{n})(B-\frac{\mathrm{Tr}(B)}{n}I_{n})\bigr{)}=0 for all A∈𝒜A\in\mathcal{A}, B∈ℬB\in\mathcal{B}

  2. (2)

    Tr​(A​B)=Tr​(A)​Tr​(B)n\mathrm{Tr}(AB)=\frac{\mathrm{Tr}(A)\mathrm{Tr}(B)}{n} for all A∈𝒜A\in\mathcal{A}, B∈ℬB\in\mathcal{B}

  3. (3)

    ℰ𝒜​(B)=Tr​(B)n​In\mathcal{E}_{\mathcal{A}}(B)=\frac{\mathrm{Tr}(B)}{n}I_{n} for all B∈ℬB\in\mathcal{B} and ℰℬ​(A)=Tr​(A)n​In\mathcal{E}_{\mathcal{B}}(A)=\frac{\mathrm{Tr}(A)}{n}I_{n} for all A∈𝒜A\in\mathcal{A}

The ideal (ϵ=0\epsilon=0 – see next section) notion of quantum privacy we consider here is given as follows.

Definition 2.2.

Given a unital quantum channel Φ:Mn​(ℂ)→Mn​(ℂ)\Phi:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C}), we say a unital ∗*-algebra 𝒜\mathcal{A} is private for Φ\Phi whenever

Φ​(A)=Tr​(A)n​In,\Phi(A)=\frac{\mathrm{Tr}(A)}{n}I_{n},

for all A∈𝒜A\in\mathcal{A}.

Remark 2.3.

Notice that Condition 3 from Definition 2.1 asserts that if 𝒜\mathcal{A} and ℬ\mathcal{B} are quasiorthogonal then the conditional expectation onto the one algebra privatizes the other. The simplest example of this phenomena can be seen in the extreme case with 𝒜=Mn​(ℂ)\mathcal{A}=M_{n}(\mathbb{C}) and Φ=𝒟n\Phi=\mathcal{D}_{n} is the ‘complete depolarizing’ channel, 𝒟n​(X)=n−1​Tr​(X)​In\mathcal{D}_{n}(X)=n^{-1}\mathrm{Tr}(X)I_{n}, which is the conditional expectation channel onto the scalar algebra ℬ=ℂ​In\mathcal{B}=\mathbb{C}I_{n}. Observe also in this case that 𝒜\mathcal{A} and ℬ\mathcal{B} are quasiorthogonal, which is a simple special case of our main result below.

More general notions of private algebras have been considered in the literature, often with different nomenclature as well, such as private quantum channels, decoherence-full or private subspaces and subsystems, and private algebras [1, 8, 4, 3, 21, 11, 17, 16, 12, 24, 25]. The distinguished special case we consider here captures many of the most naturally occurring examples from these settings, in addition to, as we shall see, allowing us to establish a tight connection with quasiorthogonality in the approximate case.

In [36], Weiner introduced the following quantitative measure of orthogonality for algebras. We will focus on this notion for the rest of the section.

Definition 2.4.

For 𝒜,ℬ⊆Mn​(ℂ)\mathcal{A},\mathcal{B}\subseteq M_{n}(\mathbb{C}) unital ∗*-algebras, the measure of orthogonality between them is given by

(1) 𝒬​(𝒜,ℬ):=Tr​(T𝒜​Tℬ)\mathcal{Q}(\mathcal{A},\mathcal{B}):=\mathrm{Tr}(T_{\mathcal{A}}T_{\mathcal{B}})

where T𝒜T_{\mathcal{A}} is the (any) matrix representation of ℰ𝒜\mathcal{E}_{\mathcal{A}} acting on the vector space Mn​(ℂ)M_{n}(\mathbb{C}).

We shall make use of the explicit forms of our matrix representations so let us introduce notation {|i⟩:1≤i≤n}\{|i\rangle:1\leq i\leq n\} for a fixed orthonormal (o.n.) basis for ℂn\mathbb{C}^{n}, and then Ei​j=|i⟩​⟨j|E_{ij}=|i\rangle\langle j| for the corresponding set of matrix units of MnM_{n}, which themselves form an orthonormal basis in the trace inner product; <A,B>=Tr⁡(B∗​A)<A,B>=\operatorname{Tr}(B^{*}A), A,B∈MnA,B\in M_{n}. We will then work with the so-called ‘natural representation’ TΦT_{\Phi} [35] for a given linear map Φ:Mn​(ℂ)→Mn​(ℂ)\Phi:M_{n}(\mathbb{C})\rightarrow M_{n}(\mathbb{C}) in the basis {Ei​j⊗Ek​l}\{E_{ij}\otimes E_{kl}\} this set defines for Mn2​(ℂ)≅Mn​(ℂ)⊗Mn​(ℂ)M_{n^{2}}(\mathbb{C})\cong M_{n}(\mathbb{C})\otimes M_{n}(\mathbb{C});

TΦ=∑i,j,k,l<Φ​(Ek​l),Ei​j>Ei​k⊗Ej​l.T_{\Phi}=\sum_{i,j,k,l}<\Phi(E_{kl}),E_{ij}>E_{ik}\otimes E_{jl}.

On the other hand, we can consider the Choi matrix [10] for Φ\Phi, given by

CΦ:=∑i,j=1nEi​j⊗Φ​(Ei​j).C_{\Phi}:=\sum_{i,j=1}^{n}E_{ij}\otimes\Phi(E_{ij}).

Using the expansion Φ​(Ei​j)=∑k,l<Φ​(Ei​j),Ek​l>Ek​l\Phi(E_{ij})=\sum_{k,l}<\Phi(E_{ij}),E_{kl}>E_{kl}, we see that CΦC_{\Phi} and TΦT_{\Phi} have the same matrix coefficients up to the (unitarily implemented) permutation that sends Ei​j⊗Ek​l↦Ei​k⊗Ej​lE_{ij}\otimes E_{kl}\mapsto E_{ik}\otimes E_{jl}.

Hence, if we denote by C𝒜C_{\mathcal{A}} the Choi matrix of ℰ𝒜\mathcal{E}_{\mathcal{A}}, and similarly for CℬC_{\mathcal{B}}, then we have the following observation.

Proposition 2.5.

Given ∗*-algebras 𝒜\mathcal{A} and ℬ\mathcal{B}, we have

(2) 𝒬​(𝒜,ℬ)=Tr​(C𝒜​Cℬ).\mathcal{Q}(\mathcal{A},\mathcal{B})=\mathrm{Tr}(C_{\mathcal{A}}C_{\mathcal{B}}).

We shall make use of the following internal description of the Choi matrix for a conditional expectation. Recall that if AA is a matrix, A¯\overline{A} is its complex conjugate matrix.

Lemma 2.6.

Let 𝒜⊆Mn​(ℂ)\mathcal{A}\subseteq M_{n}(\mathbb{C}) be a ∗*-algebra and let {Ai}i=1d1\{A_{i}\}_{i=1}^{d_{1}} be any o.n. basis for 𝒜\mathcal{A}. Then we have

(3) C𝒜=∑i=1d1Ai¯⊗Ai.C_{\mathcal{A}}=\sum_{i=1}^{d_{1}}\overline{A_{i}}\otimes A_{i}.
Proof.

We first extend {Ai}i=1d1\{A_{i}\}_{i=1}^{d_{1}} to an o.n. basis on the full matrix space by appending the elements of {Aj⟂}j=1n2−d1\{A^{\perp}_{j}\}_{j=1}^{n^{2}-d_{1}}. We claim that

(4) ∑i=1d1Ai¯⊗Ai+∑j=1n2−d1Aj⟂¯⊗Aj⟂=∑i,j=1nEi​j⊗Ei​j.\sum_{i=1}^{d_{1}}\overline{A_{i}}\otimes A_{i}+\sum_{j=1}^{n^{2}-d_{1}}\overline{A_{j}^{\perp}}\otimes A_{j}^{\perp}=\sum_{i,j=1}^{n}E_{ij}\otimes E_{ij}.

To see why this is so, observe that for all X∈Mn​(ℂ)X\in M_{n}(\mathbb{C}),

(Tr⊗id)​((XT⊗In)​(∑i=1d1Ai¯⊗Ai+∑j=1n2−d1Aj⟂¯⊗Aj⟂))(\operatorname{Tr}\otimes\mathrm{id})\biggl{(}(X^{T}\otimes I_{n})(\sum_{i=1}^{d_{1}}\overline{A_{i}}\otimes A_{i}+\sum_{j=1}^{n^{2}-d_{1}}\overline{A_{j}^{\perp}}\otimes A_{j}^{\perp})\biggr{)}
=∑i=1d1Tr​(XT​Ai¯)​Ai+∑j=1n2−d1Tr​(XT​Aj⟂¯)​Aj⟂\displaystyle=\sum_{i=1}^{d_{1}}\mathrm{Tr}(X^{T}\overline{A_{i}})A_{i}+\sum_{j=1}^{n^{2}-d_{1}}\mathrm{Tr}(X^{T}\overline{A_{j}^{\perp}})A_{j}^{\perp}
=∑i=1d1Tr​(X​Ai∗)​Ai+∑j=1n2−d1Tr​(X​Aj⟂∗)​Aj⟂\displaystyle=\sum_{i=1}^{d_{1}}\mathrm{Tr}(XA_{i}^{*})A_{i}+\sum_{j=1}^{n^{2}-d_{1}}\mathrm{Tr}(XA_{j}^{\perp*})A_{j}^{\perp}
=∑i=1d1<X,Ai>Ai+∑j=1n2−d1<X,Aj⟂∗>Aj⟂\displaystyle=\sum_{i=1}^{d_{1}}<X,A_{i}>A_{i}+\sum_{j=1}^{n^{2}-d_{1}}<X,A_{j}^{\perp*}>A_{j}^{\perp}
=X,\displaystyle=X,

since {Ai}∪{Aj⟂}\{A_{i}\}\cup\{A_{j}^{\perp}\} forms an o.n. basis for Mn​(ℂ)M_{n}(\mathbb{C}). If we denote B=∑i=1d1Ai¯⊗Ai+∑j=1n2−d1Aj⟂¯⊗Aj⟂B=\sum_{i=1}^{d_{1}}\overline{A_{i}}\otimes A_{i}+\sum_{j=1}^{n^{2}-d_{1}}\overline{A_{j}^{\perp}}\otimes A_{j}^{\perp}, then we observe that the property encoded above is (Tr⊗id)​((XT⊗In)​B)=X(\operatorname{Tr}\otimes\mathrm{id})\bigl{(}(X^{T}\otimes I_{n})B\bigr{)}=X, which uniquely characterizes BB as the Choi matrix of the identity map [35], and so B=∑i,j=1nEi​j⊗id​(Ei​j)=∑i,j=1nEi​j⊗Ei​jB=\sum_{i,j=1}^{n}E_{ij}\otimes\mathrm{id}(E_{ij})=\sum_{i,j=1}^{n}E_{ij}\otimes E_{ij}, as claimed.

Now, we recall that C𝒜:=(id⊗ℰ𝒜)​(∑i,j=1nEi​j⊗Ei​j)C_{\mathcal{A}}:=(\mathrm{id}\otimes\mathcal{E}_{\mathcal{A}})\bigl{(}\sum_{i,j=1}^{n}E_{ij}\otimes E_{ij}\bigr{)} and so by Eq. (4) we can equivalently say that

C𝒜=(id⊗ℰ𝒜)​(B)=∑i=1d1Ai¯⊗ℰ𝒜​(Ai)+∑j=1n2−d1Aj⟂¯⊗ℰ𝒜​(Aj⟂).C_{\mathcal{A}}=(\mathrm{id}\otimes\mathcal{E}_{\mathcal{A}})\bigl{(}B\bigr{)}=\sum_{i=1}^{d_{1}}\overline{A_{i}}\otimes\mathcal{E}_{\mathcal{A}}(A_{i})+\sum_{j=1}^{n^{2}-d_{1}}\overline{A_{j}^{\perp}}\otimes\mathcal{E}_{\mathcal{A}}(A_{j}^{\perp}).

Since ℰ𝒜​(A)=A\mathcal{E}_{\mathcal{A}}(A)=A for all A∈𝒜A\in\mathcal{A}, and ℰ𝒜​(Aj⟂)=0\mathcal{E}_{\mathcal{A}}(A_{j}^{\perp})=0 for each Aj⟂A_{j}^{\perp}, this simplifies to

C𝒜=∑i=1d1Ai¯⊗Ai,C_{\mathcal{A}}=\sum_{i=1}^{d_{1}}\overline{A_{i}}\otimes A_{i},

completing the proof. ∎

This leads to the following 2-norm type characterization of Q​(𝒜,ℬ)Q(\mathcal{A},\mathcal{B}).

Corollary 2.7.

Let 𝒜\mathcal{A}, ℬ⊆Mn​(ℂ)\mathcal{B}\subseteq M_{n}(\mathbb{C}) be unital ∗*-algebras. Then the measure of orthogonality may equivalently be expressed as:

(5) 𝒬​(𝒜,ℬ)=∑i,j=1d1,d2|Tr​(Ai​Bj)|2,\mathcal{Q}(\mathcal{A},\mathcal{B})=\sum_{i,j=1}^{d_{1},d_{2}}|\mathrm{Tr}(A_{i}B_{j})|^{2},

for any o.n. bases {Ai}i=1d1\{A_{i}\}_{i=1}^{d_{1}}, {Bj}j=1d2\{B_{j}\}_{j=1}^{d_{2}} for 𝒜\mathcal{A} and ℬ\mathcal{B} respectively.

Proof.

We apply the formula of Eq. (2) to the expression for C𝒜C_{\mathcal{A}} obtained above to obtain:

𝒬​(𝒜,ℬ)\displaystyle\mathcal{Q}(\mathcal{A},\mathcal{B}) =Tr​(C𝒜​Cℬ)\displaystyle=\mathrm{Tr}(C_{\mathcal{A}}C_{\mathcal{B}})
=∑i,j=1d1,d2Tr​((Ai¯⊗Ai)​(Bj¯⊗Bj))\displaystyle=\sum_{i,j=1}^{d_{1},d_{2}}\mathrm{Tr}\bigl{(}(\overline{A_{i}}\otimes A_{i})(\overline{B_{j}}\otimes B_{j})\bigr{)}
=∑i,j=1d1,d2|Tr​(Ai​Bj)|2.\displaystyle=\sum_{i,j=1}^{d_{1},d_{2}}|\mathrm{Tr}(A_{i}B_{j})|^{2}.

∎

We have stated that 𝒬​(𝒜,ℬ)\mathcal{Q}(\mathcal{A},\mathcal{B}) is a measure of orthogonality. This is most explicitly seen through the following elementary observation of Weiner [36]. We give an alternate simple proof based on the descriptions derived here.

Proposition 2.8.

Let 𝒜\mathcal{A}, ℬ⊆Mn​(ℂ)\mathcal{B}\subseteq M_{n}(\mathbb{C}) be unital ∗*-algebras. Then 𝒜\mathcal{A} and ℬ\mathcal{B} are quasiorthogonal to one another if and only if 𝒬​(𝒜,ℬ)=1\mathcal{Q}(\mathcal{A},\mathcal{B})=1.

Proof.

Let {Ai}i=1d1\{A_{i}\}_{i=1}^{d_{1}}, {Bj}j=1d2\{B_{j}\}_{j=1}^{d_{2}} be o.n. bases for 𝒜\mathcal{A}, ℬ\mathcal{B} respectively, and suppose A1=B1=1n​InA_{1}=B_{1}=\frac{1}{\sqrt{n}}I_{n} (which is possible since both algebras are unital), so that Tr​(Ai)=Tr​(Bj)=0\mathrm{Tr}(A_{i})=\mathrm{Tr}(B_{j})=0 for all 1<i≤d11<i\leq d_{1}, 1<j≤d21<j\leq d_{2}.

Then if 𝒜\mathcal{A} and ℬ\mathcal{B} are quasiorthogonal, and using the fact that Ai,BjA_{i},B_{j} are traceless for i,j≠1i,j\neq 1, we have

𝒬​(𝒜,ℬ)\displaystyle\mathcal{Q}(\mathcal{A},\mathcal{B}) =∑i,j=1d1,d2|Tr​(Ai​Bj)|2\displaystyle=\sum_{i,j=1}^{d_{1},d_{2}}|\mathrm{Tr}(A_{i}B_{j})|^{2}
=∑i,j=1d1,d21n2​|Tr​(Ai)|2​|Tr​(Bj)|2\displaystyle=\sum_{i,j=1}^{d_{1},d_{2}}\frac{1}{n^{2}}|\mathrm{Tr}(A_{i})|^{2}|\mathrm{Tr}(B_{j})|^{2}
=1n2​|Tr​(In)n|2​|Tr​(In)n|2\displaystyle=\frac{1}{n^{2}}\bigl{|}\frac{\mathrm{Tr}(I_{n})}{\sqrt{n}}\bigr{|}^{2}\bigl{|}\frac{\mathrm{Tr}(I_{n})}{\sqrt{n}}\bigr{|}^{2}
=1.\displaystyle=1.

Conversely, keeping our o.n. bases for 𝒜\mathcal{A} and ℬ\mathcal{B}, suppose that

𝒬​(𝒜,ℬ)=∑i,j=1d1,d2|Tr​(Ai​Bj)|2=1.\mathcal{Q}(\mathcal{A},\mathcal{B})=\sum_{i,j=1}^{d_{1},d_{2}}|\mathrm{Tr}(A_{i}B_{j})|^{2}=1.

Since A1=B1=1n​InA_{1}=B_{1}=\frac{1}{\sqrt{n}}I_{n}, we have

1+∑(i,j)≠(1,1)|Tr​(Ai​Bj)|2=1,1+\sum_{(i,j)\neq(1,1)}|\mathrm{Tr}(A_{i}B_{j})|^{2}=1,

and so Tr​(Ai​Bj)=0\mathrm{Tr}(A_{i}B_{j})=0 except when (i,j)=(1,1)(i,j)=(1,1).

Then, for any A=Tr​(A)n​In+∑i=2d1ai​Ai∈𝒜A=\frac{\mathrm{Tr}(A)}{n}I_{n}+\sum_{i=2}^{d_{1}}a_{i}A_{i}\in\mathcal{A} and B=Tr​(B)n​In+∑j=1d2bj​Bj∈ℬB=\frac{\mathrm{Tr}(B)}{n}I_{n}+\sum_{j=1}^{d_{2}}b_{j}B_{j}\in\mathcal{B}, we have that

Tr​(A​B)\displaystyle\mathrm{Tr}(AB) =\displaystyle= Tr​(A)​Tr​(B)n2​Tr​(In)+∑(i,j)≠(1,1)ai​bj​Tr​(Ai​Bj)\displaystyle\frac{\mathrm{Tr}(A)\mathrm{Tr}(B)}{n^{2}}\mathrm{Tr}(I_{n})+\sum_{(i,j)\neq(1,1)}a_{i}b_{j}\mathrm{Tr}(A_{i}B_{j})
=\displaystyle= 1n​Tr​(A)​Tr​(B),\displaystyle\frac{1}{n}\mathrm{Tr}(A)\mathrm{Tr}(B),

which is one of the equivalent conditions for quasiorthogonality. ∎

Remark 2.9.

Note that evidently for 𝒜\mathcal{A}, ℬ\mathcal{B} unital ∗*-algebras, 𝒬​(𝒜,ℬ)≥1\mathcal{Q}(\mathcal{A},\mathcal{B})\geq 1 since both algebras contain 1n​In.\frac{1}{\sqrt{n}}I_{n}. So quasiorthogonality corresponds to the case where 𝒬\mathcal{Q} is minimized.

Now we end the section with the definition of ϵ\epsilon-quasiorthogonal subalgebras.

Definition 2.10.

Given an ϵ>0\epsilon>0, two unital ∗*-subalgebras 𝒜,ℬ⊂Mn\mathcal{A},\mathcal{B}\subset M_{n} are called ϵ\epsilon- quasiorthogonal if

𝒬​(𝒜,ℬ)≤1+ϵ.\mathcal{Q}(\mathcal{A},\mathcal{B})\leq 1+\epsilon.

3. Approximate Relative Quantum Privacy and Main Result

We shall consider the following notion of approximate privacy in what follows.

We first recall the 2-norm of an operator is ‖A‖2=(Tr⁡(A∗​A))12||A||_{2}=(\operatorname{Tr}(A^{*}A))^{\frac{1}{2}}, and so <A,A>=‖A‖22<A,A>=||A||_{2}^{2}. Also ‖Φ‖2=sup‖X‖2=1‖Φ​(X)‖2||\Phi||_{2}=\sup_{||X||_{2}=1}||\Phi(X)||_{2} for linear maps Φ\Phi.

Definition 3.1.

Let 𝒜\mathcal{A}, ℬ⊆Mn​(ℂ)\mathcal{B}\subseteq M_{n}(\mathbb{C}) be unital ∗*-algebras, and let ϵ>0\epsilon>0. Then we say ℬ\mathcal{B} is ϵ\epsilon-private relative to 𝒜\mathcal{A} if

(6) ‖(ℰ𝒜−𝒟n)∘ℰℬ‖2<ϵ,||(\mathcal{E}_{\mathcal{A}}-\mathcal{D}_{n})\circ\mathcal{E}_{\mathcal{B}}||_{2}<\epsilon,

where 𝒟n\mathcal{D}_{n} is the complete depolarizing channel on Mn​(ℂ)M_{n}(\mathbb{C}), 𝒟n​(X)=Tr⁡(X)n​In\mathcal{D}_{n}(X)=\frac{\operatorname{Tr}(X)}{n}I_{n} for all X∈Mn​(ℂ)X\in M_{n}(\mathbb{C}).

Remark 3.2.

We are motivated to consider the 2-norm here as it is fairly standard in physically motivated quantum information settings, in addition to the description of 𝒬\mathcal{Q} as a particular 2-norm derived in Corollary 2.7. We also note that our ϵ\epsilon-private language is in the spirit of terminology used in the context of approximate privacy previously (e.g. [21]).

We now state and prove our main result.

Theorem 3.3.

Let 𝒜\mathcal{A}, ℬ⊆Mn​(ℂ)\mathcal{B}\subseteq M_{n}(\mathbb{C}) be unital ∗*-algebras, and let d𝒜​ℬmax=max⁡(dim𝒜,dimℬ)d^{\mathrm{max}}_{{\mathcal{A}}{\mathcal{B}}}=\max(\dim{\mathcal{A}},\dim{\mathcal{B}}) and d𝒜​ℬmin=min⁡(dim𝒜,dimℬ)d^{\mathrm{min}}_{{\mathcal{A}}{\mathcal{B}}}=\min(\dim{\mathcal{A}},\dim{\mathcal{B}}). Let ϵ>0\epsilon>0. Suppose that

𝒬​(𝒜,ℬ)≤1+ϵ2d𝒜​ℬmax−1.\mathcal{Q}(\mathcal{A},\mathcal{B})\leq 1+\frac{\epsilon^{2}}{d^{\mathrm{max}}_{\mathcal{A}\mathcal{B}}-1}.

Then ℬ\mathcal{B} (and respectively 𝒜\mathcal{A}) is ϵ\epsilon-private relative to 𝒜\mathcal{A} (respectively to ℬ\mathcal{B}).

Conversely, suppose that ℬ\mathcal{B} (and respectively 𝒜\mathcal{A}) is ϵ\epsilon-private relative to 𝒜\mathcal{A} (respectively to ℬ\mathcal{B}). Then we have

𝒬​(𝒜,ℬ)≤1+(d𝒜​ℬmin−1)​ϵ2.\mathcal{Q}(\mathcal{A},\mathcal{B})\leq 1+(d^{\mathrm{min}}_{\mathcal{A}\mathcal{B}}-1)\epsilon^{2}.
Proof.

As above, we pick o.n. bases {Ai}i=1d1\{A_{i}\}_{i=1}^{d_{1}} and {Bj}j=1d2\{B_{j}\}_{j=1}^{d_{2}} with A1=B1=1n​InA_{1}=B_{1}=\frac{1}{\sqrt{n}}I_{n}. (Note that d𝒜​ℬmin≤dj≤d𝒜​ℬmaxd^{\mathrm{min}}_{\mathcal{A}\mathcal{B}}\leq d_{j}\leq d^{\mathrm{max}}_{\mathcal{A}\mathcal{B}}, j=1,2j=1,2.) We will also choose the basis so that Ai=Ai∗A_{i}=A_{i}^{*} for each ii, and similarly for BjB_{j}.

For the forward direction, let ϵ′=ϵ2​(d𝒜​ℬ−1)−1\epsilon^{\prime}=\epsilon^{2}(d_{\mathcal{A}\mathcal{B}}-1)^{-1}. Since

𝒬​(𝒜,ℬ)=1+∑(i,j)≠(1,1)|Tr​(Ai​Bj)|2≤1+ϵ′,\mathcal{Q}(\mathcal{A},\mathcal{B})=1+\sum_{(i,j)\neq(1,1)}|\mathrm{Tr}(A_{i}B_{j})|^{2}\leq 1+\epsilon^{\prime},

it follows that for each j≠1j\neq 1, we have

∑i=2d1|Tr(AiBj)|2=∑i=2d1|<Ai,Bj>|2≤ϵ′.\sum_{i=2}^{d_{1}}|\mathrm{Tr}(A_{i}B_{j})|^{2}=\sum_{i=2}^{d_{1}}|<A_{i},B_{j}>|^{2}\leq\epsilon^{\prime}.

Next, we extend {Ai}i=1d1\{A_{i}\}_{i=1}^{d_{1}} to an o.n. basis for the full matrix space by appending elements {Aj⟂}j=1n2−d2\{A_{j}^{\perp}\}_{j=1}^{n^{2}-d_{2}} and express each BjB_{j} in terms of this basis:

Bj=Tr​(Bj)n​In+∑i=2d1bj​i​Ai+∑k=1n2−d1cj​k​Ak⟂.B_{j}=\frac{\mathrm{Tr}(B_{j})}{n}I_{n}+\sum_{i=2}^{d_{1}}b_{ji}A_{i}+\sum_{k=1}^{n^{2}-d_{1}}c_{jk}A_{k}^{\perp}.

By orthonormality of our basis we have |bj​i|2=|Tr​(Ai∗​Bj)|2=|Tr​(Ai​Bj)|2|b_{ji}|^{2}=|\mathrm{Tr}(A_{i}^{*}B_{j})|^{2}=|\mathrm{Tr}(A_{i}B_{j})|^{2} for all i,ji,j, and hence for all j≠1j\neq 1,

∑i=2d1|bj​i|2≤ϵ′.\sum_{i=2}^{d_{1}}|b_{ji}|^{2}\leq\epsilon^{\prime}.

Now we apply ℰ𝒜\mathcal{E}_{\mathcal{A}} to BjB_{j}, using its decomposition above and noting that <Bj,In>=Tr⁡(Bj)=0<B_{j},I_{n}>=\operatorname{Tr}(B_{j})=0 and ℰ𝒜​(Ak⟂)=0\mathcal{E}_{\mathcal{A}}(A_{k}^{\perp})=0, to obtain ℰ𝒜​(Bj)=∑i=2d1bj​i​Ai.\mathcal{E}_{\mathcal{A}}(B_{j})=\sum_{i=2}^{d_{1}}b_{ji}A_{i}. Hence by orthonormality of the AiA_{i} we have:

‖ℰ𝒜​(Bj)−Tr​(Bj)n​In‖22=‖∑i=2d1bj​i​Ai‖22=∑i=2d1|bj​i|2≤ϵ′.\|\mathcal{E}_{\mathcal{A}}(B_{j})-\frac{\mathrm{Tr}(B_{j})}{n}I_{n}\|_{2}^{2}=\|\sum_{i=2}^{d_{1}}b_{ji}A_{i}\|_{2}^{2}=\sum_{i=2}^{d_{1}}|b_{ji}|^{2}\leq\epsilon^{\prime}.

Finally, we pick an arbitrary B∈ℬB\in\mathcal{B} and decompose it as B=Tr​(B)n​In+∑j=2d2cj​BjB=\frac{\mathrm{Tr}(B)}{n}I_{n}+\sum_{j=2}^{d_{2}}c_{j}B_{j}. Observe that ‖B‖22≥∑j|cj|2\|B\|_{2}^{2}\geq\sum_{j}|c_{j}|^{2}. Then applying ℰ𝒜\mathcal{E}_{\mathcal{A}} we get

ℰ𝒜​(B)−Tr​(B)n​In=∑j=2d2cj​ℰ𝒜​(Bj).\mathcal{E}_{\mathcal{A}}(B)-\frac{\mathrm{Tr}(B)}{n}I_{n}=\sum_{j=2}^{d_{2}}c_{j}\mathcal{E}_{\mathcal{A}}(B_{j}).

Thus we have

(‖ℰ𝒜​(B)−Tr​(B)n​In‖2)2\displaystyle\big{(}\|\mathcal{E}_{\mathcal{A}}(B)-\frac{\mathrm{Tr}(B)}{n}I_{n}\|_{2}\big{)}^{2} =(‖∑j=2d2cj​ℰ𝒜​(Bj)‖2)2\displaystyle=\big{(}\|\sum_{j=2}^{d_{2}}c_{j}\mathcal{E}_{\mathcal{A}}(B_{j})\|_{2}\big{)}^{2}
≤(∑j=2d2|cj|​‖ℰ𝒜​(Bj)‖2)2\displaystyle\leq\big{(}\sum_{j=2}^{d_{2}}|c_{j}|\|\mathcal{E}_{\mathcal{A}}(B_{j})\|_{2}\big{)}^{2}
≤(∑j=2d2|cj|2)​(∑j=2d2‖ℰ𝒜​(Bj)‖22)\displaystyle\leq\big{(}\sum_{j=2}^{d_{2}}|c_{j}|^{2}\big{)}\big{(}\sum_{j=2}^{d_{2}}||\mathcal{E}_{\mathcal{A}}(B_{j})||_{2}^{2}\big{)}
≤‖B‖22​(d2−1)​ϵ′\displaystyle\leq\|B\|_{2}^{2}(d_{2}-1)\epsilon^{\prime}
≤ϵ2​‖B‖22.\displaystyle\leq\epsilon^{2}\|B\|_{2}^{2}.

As B∈ℬB\in\mathcal{B} was arbitrary, it follows that ‖(ℰ𝒜−𝒟n)∘ℰℬ‖2<ϵ||(\mathcal{E}_{\mathcal{A}}-\mathcal{D}_{n})\circ\mathcal{E}_{\mathcal{B}}||_{2}<\epsilon, and this direction of the proof is complete.

For the converse direction, suppose that ℬ\mathcal{B} is ϵ\epsilon-private relative to 𝒜\mathcal{A}, so that

‖ℰ𝒜​(B)−Tr​(B)n​In‖2≤ϵ​‖B‖2∀B∈ℬ.\|\mathcal{E}_{\mathcal{A}}(B)-\frac{\mathrm{Tr}(B)}{n}I_{n}\|_{2}\leq\epsilon\|B\|_{2}\quad\quad\forall B\in\mathcal{B}.

As each BjB_{j} is traceless and has 2-norm equal to 1, we have ‖ℰ𝒜​(Bj)‖2≤ϵ\|\mathcal{E}_{\mathcal{A}}(B_{j})\|_{2}\leq\epsilon. So if we write (again using <Bj,In>=0<B_{j},I_{n}>=0),

Bj=∑i=2d1<Bj,Ai>Ai+∑k<Bj,Ai⟂>Ai⟂,B_{j}=\sum_{i=2}^{d_{1}}<B_{j},A_{i}>A_{i}+\sum_{k}<B_{j},A_{i}^{\perp}>A_{i}^{\perp},

then we have ℰ𝒜​(Bj)=∑i=2d1<Bj,Ai>Ai\mathcal{E}_{\mathcal{A}}(B_{j})=\sum_{i=2}^{d_{1}}<B_{j},A_{i}>A_{i} and so

∥ℰ𝒜(Bj)∥22=∑i=2d1|<Bj,Ai>|2≤ϵ2.\|\mathcal{E}_{\mathcal{A}}(B_{j})\|_{2}^{2}=\sum_{i=2}^{d_{1}}|<B_{j},A_{i}>|^{2}\leq\epsilon^{2}.

Finally, it follows that

𝒬​(𝒜,ℬ)\displaystyle\mathcal{Q}(\mathcal{A},\mathcal{B}) =\displaystyle= 1+∑i≠1,j≠1|<Bj,Ai>|2\displaystyle 1+\sum_{i\neq 1,j\neq 1}|<B_{j},A_{i}>|^{2}
=\displaystyle= 1+∑j=2d2(∑i=2d1|<Bj,Ai>|2)\displaystyle 1+\sum_{j=2}^{d_{2}}\big{(}\sum_{i=2}^{d_{1}}|<B_{j},A_{i}>|^{2}\big{)}
≤\displaystyle\leq 1+(d2−1)​ϵ2.\displaystyle 1+(d_{2}-1)\epsilon^{2}.

Similarly, 𝒬​(𝒜,ℬ)≤1+(d1−1)​ϵ2\mathcal{Q}(\mathcal{A},\mathcal{B})\leq 1+(d_{1}-1)\epsilon^{2}, and the result follows. ∎

4. Examples

In this section we apply Theorem 3.3 to examples drawn from a number of different quantum information settings.

Example 4.1.

We first present an example fashioned for illustrative purposes, one that also arises in the context of hybrid quantum information memories, processing, and error correction [23, 5, 6, 7, 9, 15, 26].

Consider the algebra 𝒜=M2​(ℂ)⊕M2​(ℂ)⊆M4​(ℂ)\mathcal{A}=M_{2}(\mathbb{C})\oplus M_{2}(\mathbb{C})\subseteq M_{4}(\mathbb{C}) of matrices of the form

A=(A100A2),A=\begin{pmatrix}A_{1}&0\\ 0&A_{2}\\ \end{pmatrix},

with A1,A2∈M2​(ℂ)A_{1},A_{2}\in M_{2}(\mathbb{C}). From a hybrid classical-quantum information perspective, 𝒜\mathcal{A} can encode two separate qubits, each with its own classical address.

Additionally, take ℬ\mathcal{B} to be the unital ∗*-algebra of matrices inside M4​(ℂ)M_{4}(\mathbb{C}) of the form

B=k1​I4+k2​(0UU∗0)=(k1​I2k2​Uk2​U∗k1​I2)B=k_{1}I_{4}+k_{2}\begin{pmatrix}0&U\\ U^{*}&0\\ \end{pmatrix}=\begin{pmatrix}k_{1}I_{2}&k_{2}U\\ k_{2}U^{*}&k_{1}I_{2}\\ \end{pmatrix}

for complex numbers k1,k2k_{1},k_{2} and some fixed unitary U∈M2​(ℂ)U\in M_{2}(\mathbb{C}). One can verify that 𝒜\mathcal{A} is quasiorthogonal to ℬ\mathcal{B}, equivalently, Q​(𝒜,ℬ)=1Q(\mathcal{A},\mathcal{B})=1.

In the specific case that U=σxU=\sigma_{x}, the Pauli bit-flip matrix, we now obtain an algebra 𝒞\mathcal{C} from ℬ\mathcal{B} by assuming ℬ\mathcal{B} is exposed to some unitary noise VV, implemented by the conjugation C=V​B​V∗C=VBV^{*}, B∈ℬB\in\mathcal{B} and C∈𝒞C\in\mathcal{C}, so 𝒞=V​ℬ​V∗\mathcal{C}=V{\mathcal{B}}V^{*}. The unitary is chosen to reflect minimal noise exposure, in that the unitary is a small perturbation of the identity, V=eTV=e^{T}, where

T=(δ0000−δ0000000000)T=\begin{pmatrix}\delta&0&0&0\\ 0&-\delta&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{pmatrix}

for some fixed 0<δ<<10<\delta<<1.

Then computing Q​(𝒜,𝒞)Q(\mathcal{A},\mathcal{C}) using our characterization from Corollary 2.7 and Mathematica software, we find that

Q​(𝒜,𝒞)=12​(cosh⁡4​δ+cosh⁡2​δ).Q(\mathcal{A},\mathcal{C})=\frac{1}{2}(\cosh{4\delta}+\cosh{2\delta}).

Thus we may apply the theorem above to quantify the approximate privacy of 𝒜\mathcal{A} and 𝒞\mathcal{C} in the following way: for any ϵ>0\epsilon>0 satisfying,

ϵ2>7​(12​(cosh⁡4​δ+cosh⁡2​δ)−1),\epsilon^{2}>7\left(\frac{1}{2}(\cosh{4\delta}+\cosh{2\delta})-1\right),

we have Q​(𝒜,𝒞)≤1+ϵ2d𝒜​ℬm​a​x−1Q(\mathcal{A},\mathcal{C})\leq 1+\frac{\epsilon^{2}}{d^{max}_{\mathcal{A}\mathcal{B}}-1}, yielding ϵ\epsilon-privacy as in the theorem. Note that we have used d𝒜​𝒞max=8d^{\mathrm{max}}_{\mathcal{A}\mathcal{C}}=8, since 𝒜\mathcal{A} and 𝒞\mathcal{C} have dimensions 88 and 22 respectively.

Example 4.2.

In [16] the first example of a private quantum subsystem [1, 8, 4, 3] was discovered such that no private subspaces existed for the given channel, and error-correction complementarity [21, 17, 12] failed. This example motivated further work and generalizations, including a framing of it in terms of operator algebra language [24, 25]. With the algebra perspective we can apply the theorem above to that example.

Here we take 𝒜=△4\mathcal{A}=\triangle_{4} to be the algebra of diagonal matrices inside M4​(ℂ)M_{4}(\mathbb{C}) with respect to a given basis. Then let ℬ=U∗​(I2⊗M2​(ℂ))​U\mathcal{B}=U^{*}(I_{2}\otimes M_{2}(\mathbb{C}))U, where UU is the unitary

U=12​(10−i0010i010−i10i0).U=\frac{1}{\sqrt{2}}\begin{pmatrix}1&0&-i&0\\ 0&1&0&i\\ 0&1&0&-i\\ 1&0&i&0\\ \end{pmatrix}.

Then one can check that we have 𝒬​(𝒜,ℬ)=1\mathcal{Q}(\mathcal{A},\mathcal{B})=1; indeed, this can be seen directly through an application of Corollary 2.7 or as a consequence of the results from [17]. Now consider the subalgebra 𝒞=V∗​(I2⊗M2​(ℂ))​V\mathcal{C}=V^{*}(I_{2}\otimes M_{2}(\mathbb{C}))V defined with the modified unitary V=U​eTV=Ue^{T}, where

T=(δ0000−δ0000000000),T=\begin{pmatrix}\delta&0&0&0\\ 0&-\delta&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{pmatrix},

for some 0<δ<<10<\delta<<1. Hence, 𝒞=(eT)∗​ℬ​(eT)\mathcal{C}=(e^{T})^{*}\mathcal{B}(e^{T}).

With a choice of basis for the subalgebras and making use of Corollary 2.7 again, we can compute

𝒬​(𝒜,𝒞)=14​e−4​δ​(e4​δ+1)2.\mathcal{Q}(\mathcal{A},\mathcal{C})=\frac{1}{4}e^{-4\delta}\left(e^{4\delta}+1\right)^{2}.

Then, for some suitable choice of ϵ​(δ)\epsilon(\delta) we have

𝒬​(𝒜,𝒞)≤1+ϵ​(δ)23,\mathcal{Q}(\mathcal{A},\mathcal{C})\leq 1+\frac{\epsilon(\delta)^{2}}{3},

and by Theorem 3.3 we say that 𝒜\mathcal{A} is ϵ\epsilon-private relative to 𝒞\mathcal{C} (and vice-versa). Note that in this case, d𝒜​ℬmax=4d^{\mathrm{max}}_{\mathcal{A}\mathcal{B}}=4.

Example 4.3.

Another example of quasiorthogonal subalgebras for which we can study the approximate case comes from the study of mutually unbiased bases (MUB) [2, 19, 18, 14, 34].

MUB are useful in many quantum information protocols because of their defining property. Specifically, two orthonormal basis {|ϕi⟩}\{|\phi_{i}\rangle\} and {|ψk⟩}\{|\psi_{k}\rangle\} of ℂn\mathbb{C}^{n} are mutually unbiased if for all i,k=1,…,ni,k=1,\dots,n,

|⟨ϕi|ψk⟩|=1n.|\langle\phi_{i}|\psi_{k}\rangle|=\frac{1}{\sqrt{n}}.

There is a maximal abelian subalgebra (MASA) [13] denoted 𝒜ψ\mathcal{A}_{\psi} (and similarly 𝒜ϕ\mathcal{A}_{\phi}) in Mn​(ℂ)M_{n}(\mathbb{C}) associated with each basis in the following way: 𝒜ψ=span​{Pψi:1≤i≤n}\mathcal{A}_{\psi}=\mathrm{span}\{P_{\psi_{i}}:1\leq i\leq n\} is the linear span of the orthonomal projectors Pψi=|ψi⟩​⟨ψi|∈Mn​(ℂ)P_{\psi_{i}}=|\psi_{i}\rangle\!\langle\psi_{i}|\in M_{n}(\mathbb{C}) onto the one-dimensional vector subspaces span​{|ψi⟩}=ℂ​|ψi⟩\mathrm{span}\{|\psi_{i}\rangle\}=\mathbb{C}|\psi_{i}\rangle. The subalgebras 𝒜ψ\mathcal{A}_{\psi}, 𝒜ϕ\mathcal{A}_{\phi} are quasiorthogonal if and only if the bases are mutually unbiased. This can be checked using the fact that

Tr​(Pψi​Pϕk)=|⟨ψi|ϕk⟩|2\mathrm{Tr}(P_{\psi_{i}}P_{\phi_{k}})=|\langle\psi_{i}|\phi_{k}\rangle|^{2}

and the criterion for quasiorthogonality, 𝒬​(𝒜ψ,𝒜ϕ)=1\mathcal{Q}(\mathcal{A}_{\psi},\mathcal{A}_{\phi})=1 using Corollary 2.7 for instance.

The maximum number of MUB in an arbitrary dimension is not known in general (they are known for cases where the dimension is a power of a prime). Here we exploit the concept of ”approximate” mutually unbiased bases as discussed in [20, 33].

Given an ϵ>0\epsilon>0, we call a system of n2+nn^{2}+n vectors in ℂn\mathbb{C}^{n} which are the elements of n+1n+1 orthonormal bases ℬk={ψk,1,⋯,ψk,n}\mathcal{B}_{k}=\{\psi_{k,1},\cdots,\psi_{k,n}\} of ℂn\mathbb{C}^{n} where k=0,1,2,⋯,nk=0,1,2,\cdots,n ϵ\epsilon-approximately mutually unbiased bases if

|⟨ψk,i,ψj,l⟩|2≤1+ϵn,|\langle\psi_{k,i},\psi_{j,l}\rangle|^{2}\leq\frac{1+\epsilon}{n},

for every 0≤k,l≤n,k≠l,1≤i,j≤n.0\leq k,l\leq n,k\neq l,1\leq i,j\leq n.

One such construction, from [20], asserts the existence of a system of approximately MUBs with an inequality of the following type:

|⟨ϕi|ψk⟩|≤(2+O​(n−110))​n−12.|\langle\phi_{i}|\psi_{k}\rangle|\leq\left(2+O(n^{-\frac{1}{10}})\right)n^{-\frac{1}{2}}.

Using the above expression and the measure of orthogonality given in Corollary 2.7, one sees that the associated MASAs are approximately private and quasiorthogonal for some ϵ\epsilon. In the 4-dimensional (two-qubit) case, for instance, one has

|⟨ϕi|ψk⟩|≤1+λ​(4),|\langle\phi_{i}|\psi_{k}\rangle|\leq 1+\lambda(4),

where λ​(4)\lambda(4) is some O​(4−35)O(4^{-\frac{3}{5}}) expression. Thus each pair of MASAs is ϵ\epsilon-private whenever 3​λ​(4)<ϵ\sqrt{3\lambda(4)}<\epsilon .

5. Outlook

We have explicitly linked approximate quasiorthogonality of operator algebras with an appropriate notion of approximate relative privacy for the algebras, determined by the actions of their conditional expectation channels. We focussed on unital algebras and the notion of quantum privacy defined by privatizing to the identity operator as this includes many natural examples and it kept the technical issues manageable. That said, we expect it should be possible to extend this result to more general algebras and more general notions of privacy, for instance as has been accomplished for quantum error correction [7] and private quantum codes [12]. It would also be interesting to see if this work could help to generate new constructions of approximate MUB or be applied to the study of SIC-POVM’s [20, 33] through focus on the commutative algebra case of our result. We leave these and other investigations to be pursued elsewhere.

Acknowledgements. D.W.K. was partly supported by NSERC and a University Research Chair at Guelph. J.L. holds a University of Guelph - Institute for Quantum Computing postdoctoral fellowship. M.N. was partly supported by Mitacs and the African Institute for Mathematical Sciences. R.P. was partly supported by NSERC. M.R. holds a postdoctoral fellowship in the Department of Pure Mathematics, University of Waterloo.

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