Definitions for Robust Self Testing of Unknown Quantum Systems into Any Entangled Two-Qubit States

QAlgorithms.Papers.Independent.Intakethird100.P75.Dichotomic

For a complex inner-product space $H$ with its normed additive group structure and a continuous complex-linear operator $A:H\to H$, this defines the predicate that $A^\dagger=A$ and $A\circ A=I_H$, where $A^\dagger$ denotes the adjoint and $I_H$ the identity operator. Thus the predicate requires both self-adjointness and that applying $A$ twice is the identity; it does not require both $+1$ and $-1$ to occur as eigenvalues.

QAlgorithms.Papers.Independent.Intakethird100.P75.ExtractsWithin

For the underlying complex inner product spaces $H,K$, any finite index type $\iota$ with decidable equality, coefficients $a:\iota\to\mathbb C$, vector $\psi\in H\widehat\otimes K$, and real number $e$, this predicate asserts that there exist bounded complex-linear isometries $F_A:H\to H\widehat\otimes\mathbb C^\iota$ and $F_B:K\to K\widehat\otimes\mathbb C^\iota$, and a vector $j\in H\widehat\otimes K$ with $\|j\|=1$, such that

$$\left\|(F_A\otimes F_B)\psi-\sum_{i\in\iota}a_i(J_i^H\otimes J_i^K)j\right\|\le e.$$

Here all completed tensor products are completions of complex tensor products, $J_i^H(x)=x\otimes e_i$ and $J_i^K(y)=y\otimes e_i$, and $e_i$ is the coordinate vector equal to $1$ at $i$ and $0$ elsewhere; tensor operators act by continuous extension. Neither $a$ nor $\psi$ is assumed normalized. If $\iota$ is empty, the sum is zero, while the isometry and unit-vector requirements remain; a negative $e$ cannot satisfy the norm inequality.

QAlgorithms.Papers.Independent.Intakethird100.P75.IsLocalHilbertUnitary

For a complex inner-product space $H$ with its normed additive group structure and a continuous complex-linear operator $A:H\to H$, this defines the predicate $A^\dagger A=I_H$ and $AA^\dagger=I_H$, with products denoting composition and $A^\dagger$ denoting the adjoint. Both identities are required, including when $H$ is the zero space.

QAlgorithms.Papers.Independent.Intakethird100.P75.ProjectiveFamily

For the underlying complex inner product space $H$, any natural number $d$, and bounded complex-linear operators $P_i:H\to H$ indexed by $0\le i<d$, this predicate means that every $P_i$ is self-adjoint and idempotent, that $P_iP_j=0$ for every distinct pair $i,j$, and that

$$\sum_{i=0}^{d-1}P_i=I_H.$$

Products mean composition. Zero projections are permitted. When $d=0$, the indexed conditions are vacuous and the sum condition is $0=I_H$, which holds only when $H$ is the zero space.

QAlgorithms.Papers.Independent.Intakethird100.P75.alice

For complex inner-product spaces $H,K$ with their normed additive group structures and a continuous complex-linear operator $A:H\to H$, this defines the continuous complex-linear operator $A\widehat\otimes I_K$ on $H\widehat\otimes K$, the completion of $H\otimes_{\mathbb C}K$. It is the continuous extension of $x\otimes y\mapsto Ax\otimes y$, where $I_K$ is the identity on $K$.

QAlgorithms.Papers.Independent.Intakethird100.P75.appendBasisKet

For a complex inner-product space $H$ with its normed additive group structure, a finite index type $\iota$ with decidable equality, and an index $i\in\iota$, this defines the continuous complex-linear map $H\to H\widehat\otimes\mathbb C^\iota$ sending $x$ to $x\otimes e_i$ in the completed tensor product. Here $\mathbb C^\iota$ has its Euclidean inner product, and $e_i$ has coordinate $1$ at $i$ and $0$ elsewhere. The supplied index $i$ rules out an empty $\iota$ for this application.

QAlgorithms.Papers.Independent.Intakethird100.P75.attachSchmidt

For complex inner-product spaces $H,K$ with their normed additive group structures, a finite index type $\iota$ with decidable equality, arbitrary coefficients $a:\iota\to\mathbb C$, and an arbitrary vector $j\in H\widehat\otimes K$, this defines the vector $\sum_{i\in\iota}a_i(E_i^H\widehat\otimes E_i^K)j$ in $(H\widehat\otimes\mathbb C^\iota)\widehat\otimes(K\widehat\otimes\mathbb C^\iota)$, where completed tensor products are completions of the corresponding normed algebraic tensor products, $e_i$ is the Euclidean coordinate vector, and $E_i^H(x)=x\otimes e_i$, $E_i^K(y)=y\otimes e_i$. Each tensor map is extended continuously to the completions. Neither the coefficients nor $j$ are required to be normalized; if $\iota$ is empty, the sum is zero.

QAlgorithms.Papers.Independent.Intakethird100.P75.bellOperator

For complex inner-product spaces $H,K$ with their normed additive group structures, a real number $\alpha$, and arbitrary continuous complex-linear operators $A_0,A_1:H\to H$ and $B_0,B_1:K\to K$, this defines the continuous complex-linear operator $\alpha(A_0\widehat\otimes I_K)+A_0\widehat\otimes(B_0+B_1)+A_1\widehat\otimes(B_0-B_1)$ on $H\widehat\otimes K$. Here the completed tensor product is the completion of $H\otimes_{\mathbb C}K$, and each tensor operator is the continuous extension of its action on elementary tensors. No dichotomic, self-adjointness, or unitarity hypotheses are imposed on these operators.

QAlgorithms.Papers.Independent.Intakethird100.P75.bellValue

For the underlying complex inner product spaces $H,K$, any real number $\alpha$, bounded complex-linear operators $A_0,A_1:H\to H$ and $B_0,B_1:K\to K$, and vector $\psi\in H\widehat\otimes K$, where $\widehat\otimes$ denotes the completion of the complex tensor product, this defines the real number

$$\operatorname{Re}\left\langle\psi,\left[\alpha(A_0\otimes I_K)+A_0\otimes(B_0+B_1)+A_1\otimes(B_0-B_1)\right]\psi\right\rangle.$$

Each tensor operator is the continuous extension of its action on elementary tensors. No normalization of $\psi$, self-adjointness, or other condition on the operators is required.

QAlgorithms.Papers.Independent.Intakethird100.P75.bob

For complex inner-product spaces $H,K$ with their normed additive group structures and a continuous complex-linear operator $B:K\to K$, this defines the continuous complex-linear operator $I_H\widehat\otimes B$ on $H\widehat\otimes K$, the completion of $H\otimes_{\mathbb C}K$. It is the continuous extension of $x\otimes y\mapsto x\otimes By$, where $I_H$ is the identity on $H$.

QAlgorithms.Papers.Independent.Intakethird100.P75.extractionBudget

For arbitrary real numbers $\theta,\varepsilon,\varepsilon_2$, this defines

$$4\varepsilon+\frac{\varepsilon_2}{2|\cos\theta|}.$$

Neither error parameter is required to be nonnegative. Division uses the total real-field convention $x/0=0$, so when $\cos\theta=0$, the value is $4\varepsilon$.

QAlgorithms.Papers.Independent.Intakethird100.P75.idealB0

For every real number $\theta$, set $\varphi=\arctan(\sin(2\theta))$. This defines the bounded complex-linear operator on the Euclidean space of complex functions on the Booleans with matrix

$$\begin{pmatrix} \cos\varphi&\sin\varphi\\ \sin\varphi&-\cos\varphi \end{pmatrix}$$

in the coordinate order $(\mathrm{false},\mathrm{true})$. Equivalently, it is $\cos\varphi$ times the operator that negates the true coordinate, plus $\sin\varphi$ times the operator that exchanges the two coordinates. There is no restriction on $\theta$.

QAlgorithms.Papers.Independent.Intakethird100.P75.idealB1

For every real $\theta$, this defines a continuous complex-linear operator on $\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$ by

$$B_1(\theta)=\cos(\arctan(\sin(2\theta)))Z-\sin(\arctan(\sin(2\theta)))X, \qquad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\quad X=\begin{pmatrix}0&1\\1&0\end{pmatrix},$$

where the basis is ordered as $\mathrm{false},\mathrm{true}$ and the real coefficients are regarded as complex scalars. There is no restriction on $\theta$.

QAlgorithms.Papers.Independent.Intakethird100.P75.idealX

This defines the bounded complex-linear operator on the Euclidean space of complex functions on the Booleans whose matrix is zero on the diagonal and $1$ off the diagonal. Thus it exchanges the two coordinates:

$$(z_{\mathrm{false}},z_{\mathrm{true}}) \longmapsto(z_{\mathrm{true}},z_{\mathrm{false}}).$$

QAlgorithms.Papers.Independent.Intakethird100.P75.idealZ

This defines the bounded complex-linear operator on the Euclidean space of complex functions on the Booleans whose matrix has diagonal entries $1$ at false and $-1$ at true, and zero off-diagonal entries. Thus, in the coordinate order $(\mathrm{false},\mathrm{true})$, it sends

$$(z_{\mathrm{false}},z_{\mathrm{true}}) \longmapsto(z_{\mathrm{false}},-z_{\mathrm{true}}).$$

QAlgorithms.Papers.Independent.Intakethird100.P75.mapHilbertFactors

For arbitrary complex inner-product spaces $H,K,L,M$, each equipped with its normed additive group structure, and continuous complex-linear maps $A:H\to L$ and $B:K\to M$, this defines a continuous complex-linear map $H\widehat\otimes K\to L\widehat\otimes M$, where $E\widehat\otimes F$ denotes the completion of the normed algebraic tensor product $E\otimes_{\mathbb C}F$. It is constructed by composing the algebraic tensor map $A\otimes B$ with the canonical map into the target completion and extending along the canonical map into the source completion; on elementary tensors it sends $x\otimes y$ to $Ax\otimes By$. The spaces may be zero-dimensional, and no completeness assumption is imposed on the four original spaces.

QAlgorithms.Papers.Independent.Intakethird100.P75.normalizedProjection

For complex inner-product spaces $H,K$ with their normed additive group structures, a real number $\theta$, a continuous complex-linear operator $Z_A:H\to H$, and a vector $\psi\in H\widehat\otimes K$, this defines $\|q\|^{-1}q$, where $q=(2\cos\theta)^{-1}((I_H+Z_A)\widehat\otimes I_K)\psi$. Here the tensor product is the completion of the normed algebraic tensor product and the tensor operator is extended continuously. Both real reciprocals use the convention $0^{-1}=0$: the result is zero if $q=0$, including whenever $\cos\theta=0$, and otherwise has norm one.

QAlgorithms.Papers.Independent.Intakethird100.P75.projectionWitness

For complex inner-product spaces $H,K$ with their normed additive group structures, a real number $\theta$, a continuous complex-linear operator $Z_A:H\to H$, and a vector $\psi\in H\widehat\otimes K$, this defines $q=(2\cos\theta)^{-1}((I_H+Z_A)\widehat\otimes I_K)\psi$. The completed tensor product is the completion of the normed algebraic tensor product, and the tensor operator is extended continuously from elementary tensors. The real reciprocal is interpreted as zero when $\cos\theta=0$, so the defined vector is then zero; no projection property of $Z_A$ is assumed.

QAlgorithms.Papers.Independent.Intakethird100.P75.quantumMaximum

For every real number $\alpha$, this defines the real number

$$\sqrt{8+2\alpha^2},$$

using the nonnegative real square root. The definition imposes no restriction on $\alpha$ and contains no optimization assertion.

QAlgorithms.Papers.Independent.Intakethird100.P75.qubitCoefficients

For every real number $\theta$ and Boolean value $b$, this defines a complex number equal to $\sin\theta$ when $b$ is true and to $\cos\theta$ when $b$ is false, using the usual inclusion of real numbers into $\mathbb C$. There is no restriction on $\theta$.

QAlgorithms.Papers.Independent.Intakethird100.P75.quditCoefficients

For every natural number $d$, every family $\delta:\{0,\ldots,d-1\}\to\mathbb R$, and every index $i<d$, this defines the complex number obtained by embedding the real number

$$a_i=\frac{r_i}{\sqrt{\sum_{j=0}^{d-1}r_j^2}}, \qquad r_0=1,\quad r_j=\tan(\delta_j)\ \text{for }j\ne0,$$

into $\mathbb C$. The angles are unrestricted, and the coefficients may be negative. For $d>0$, the denominator is at least $1$ because $r_0=1$; for $d=0$, the sum is empty and equals $0$, but there is no index $i$ at which a coefficient can be evaluated. Tangent and division use their total real-valued definitions, including division by zero having value $0$.

QAlgorithms.Papers.Independent.Intakethird100.P75.quditLocal

For the underlying complex inner product space $H$, any natural number $d$, and arbitrary families of bounded complex-linear operators $P_i,X_i:H\to H$, indexed by $0\le i<d$, this defines the bounded complex-linear map into the completed complex tensor product $H\widehat\otimes\mathbb C^d$ given by

$$x\longmapsto\sum_{i=0}^{d-1}(X_i(P_i x))\otimes e_i,$$

where $e_i$ is the $i$-th coordinate vector. No projection, orthogonality, or unitarity assumptions are imposed on either family. For $d=0$, the map is zero.

QAlgorithms.Papers.Independent.Intakethird100.P75.quditRatio

For every natural number $d$, every real-valued family $\delta=(\delta_0,\ldots,\delta_{d-1})$, and every index $i\in\{0,\ldots,d-1\}$, this defines the real number

$$r_i=\begin{cases}1,&i=0,\\ \tan(\delta_i),&i\ne0.\end{cases}$$

Thus $\delta_0$ is ignored. No restriction on the angles is imposed; tangent uses total real division, so $\tan t=\sin t/\cos t$ also has its defined value $0$ when $\cos t=0$. When $d=0$, there is no index $i$ at which this function can be evaluated.

QAlgorithms.Papers.Independent.Intakethird100.P75.schmidtTarget

For any finite index type $\iota$ with decidable equality and arbitrary coefficients $a:\iota\to\mathbb C$, this defines the vector

$$\sum_{i\in\iota}a_i\,e_i\otimes e_i \quad\text{in}\quad \mathbb C^\iota\widehat\otimes\mathbb C^\iota,$$

where $\mathbb C^\iota$ has its Euclidean inner product, $e_i$ equals $1$ at $i$ and $0$ elsewhere, and the tensor product is completed. No normalization or positivity condition is imposed on the coefficients. For empty $\iota$, the vector is zero.

QAlgorithms.Papers.Independent.Intakethird100.P75.swapLocal

For a complex inner-product space $H$ with its normed additive group structure and arbitrary continuous complex-linear operators $X,Z:H\to H$, this defines the continuous complex-linear map $H\to H\widehat\otimes\mathbb C^{\{\mathrm{false},\mathrm{true}\}}$ given by $v\mapsto\frac12\bigl((I_H+Z)v\otimes e_{\mathrm{false}}+X(I_H-Z)v\otimes e_{\mathrm{true}}\bigr)$. Here the tensor product is completed and the two $e_b$ are Euclidean coordinate vectors. No self-adjointness, involution, or unitarity hypothesis on $X,Z$ is part of this definition.

QAlgorithms.Papers.Independent.Intakethird100.P75.tiltParameter

For every real number $\theta$, this defines

$$\frac{2\cos(2\theta)}{\sqrt{1+\sin^2(2\theta)}}.$$

There is no restriction on $\theta$; the denominator is always positive.

QAlgorithms.Papers.Independent.Intakethird100.P75.transferError

For complex inner-product spaces $H,K$ with their normed additive group structures, a real number $\theta$, continuous complex-linear operators $X_A,Z_A:H\to H$ and $X_B,Z_B:K\to K$, and a vector $\psi\in H\widehat\otimes K$, this defines the vector $\sin\theta\,(X_A\widehat\otimes I_K)(I_H\widehat\otimes(I_K+Z_B))\psi-\cos\theta\,(I_H\widehat\otimes X_B)((I_H-Z_A)\widehat\otimes I_K)\psi$. Here $H\widehat\otimes K$ is the completion of the normed algebraic tensor product, and each displayed tensor operator is its continuous extension from elementary tensors. All operators and $\psi$ are arbitrary.