Quantification and manipulation of magic states

TL;DR

This paper investigates the possibility of converting one magic state into another using stabilizer-preserving quantum operations in a single-copy setting, providing a mathematical framework and criteria for such transformations.

Contribution

It formulates the magic state conversion problem as an SDP feasibility problem and derives necessary and sufficient conditions using a variant of the Farkas Lemma.

Findings

Provides a procedure to determine the existence of stabilizer-preserving conversions.

Derives a complete set of magic monotones for state conversion.

Establishes necessary and sufficient conditions for single-shot magic state transformations.

Abstract

Magic states can be used as a resource to circumvent the restrictions due to stabilizer-preserving operations, and magic-state conversion has not been studied in the single-copy regime thus far. Here we solve the question of whether a stabilizer-preserving quantum operation exists that can convert between two given magic states in the single-shot regime. We first phrase this question as a feasibility problem for a semi-definite program (SDP), which provides a procedure for constructing a stabilizer-preserving quantum operation (free channel) if it exists. Then we employ a variant of the Farkas Lemma to derive necessary and sufficient conditions for existence, and this method is used to construct a complete set of magic monotones.

Equations

$$\begin{array}{rl}-1& ⩽x+y+z⩽1\\ -1& ⩽-x+y+z⩽1\\ -1& ⩽x-y+z⩽1\\ -1& ⩽x+y-z⩽1.\end{array}$$

$$\mathrm{tr}{A}_i{A}_j=\{\begin{array}{ll}2& \text{if }i=j\\ 0& \text{if }i\ne j.\end{array}$$

$$H=\sum _{i=1}^4{\alpha }_i{A}_i=\sum _{i=1}^4{\alpha }_i^{\prime }{A}_i^{\prime }$$

$${\alpha }_i=\frac{1}{2}\mathrm{tr}(H{A}_i){\alpha }_i^{\prime }=\frac{1}{2}\mathrm{tr}(H{A}_i^{\prime })$$

$$\begin{array}{rl}\mathrm{tr}(\rho A_i)& ⩾0,\\ \mathrm{tr}(\rho A_i^{\prime })& ⩾0\end{array}$$

$$\mathrm{tr}({\Pi }_{b,v}{\Pi }_{b^{\prime },v^{\prime }})=\{\begin{array}{ll}1& \text{if }b=b^{\prime },v=v^{\prime }\\ 0& \text{if }b=b^{\prime },v\ne v^{\prime }\\ 1/d& \text{if }b\ne b^{\prime }.\end{array}$$

$$W_{p,q}(\rho ):=\mathrm{tr}(\rho A_{p,q}),$$

$$\begin{array}{rl}\mathrm{tr}{A}_{p,q}& =1\\ \mathrm{tr}(A_{p,q}{A}_{p^{\prime },q^{\prime }})& =d{\delta }_{p{p}^{\prime }}{\delta }_{q{q}^{\prime }}.\end{array}$$

$$A_{0,0}:=\frac{1}{d}\sum _{p,q}{D}_{p,q},A_{p,q}:=D_{p,q}{A}_{0,0}{D}_{p,q}^{†}$$

$$D_{p,q}:={\omega }^{-pq/2}{P}^p{S}^q,\omega :=e^{2\pi i/d}$$

$$S|k⟩=|k+1⟩,P|k⟩={\omega }^k|k⟩,k\in {\mathbb{Z}}_d.$$

$$A_{\mathbf{v}}:=\sum _{b=1}^{d+1}{\Pi }_{b,v_b}-\mathbb{I}.$$

$$\mathrm{tr}(\rho A_{\mathbf{v}})⩾0\forall \mathbf{v}\in {\mathbb{Z}}_d^{d+1}.$$

$$\mathcal{E}(\rho )={\mathrm{tr}}_E\left[U\left(\rho \otimes {\rho }_E\right)U^{†}\right],$$

$$J⩾0,$$

$$\mathrm{tr}\left(J({\mathbb{I}}^{\prime }\otimes X)\right)=\mathrm{tr}(X)\forall X\in \mathcal{H},$$

$$\mathrm{tr}\left(J\left(A_{\mathbf{v}}\otimes {\Pi }_{a,u}^T\right)\right)⩾0\forall \mathbf{v},a,u,$$

$$q(X):=\underset{S⩾0}{\min }\{\mathrm{tr}S|S\in {\mathcal{H}}^B,{\mathbb{I}}^A\otimes S⩾X\}.$$

$${\Omega }_{\rho }:=N\left(\sigma \otimes {\rho }^T+t\sum _{\mathbf{v},b,a,u}{p}_{\mathbf{v},a,u}{\Pi }_{b,v_b}^{\prime }\otimes {\Pi }_{a,u}^T\right)$$

$$M_{\sigma ,t}:=\underset{p_{\mathbf{v},a,u}}{\min }q({\Omega }_{\rho })-c_{\sigma ,t}$$

$$M_{\sigma ,t}(\rho )⩾M_{\sigma ,t}({\rho }^{\prime }).$$

$$\Lambda \left({\Pi }_{b,v_b}^{\prime }\otimes {\Pi }_{a,u}^T\right)$$

$$={\Pi }_{b,v_b}^{\prime }\otimes \sum _{a,u}{p}_{a,u}{\Pi }_{a,u}^T$$

$$M_{\sigma ,t}(\rho )+c_{\sigma ,t}=$$

$$q\left(N\left(\sigma \otimes {\rho }^T+t\sum _{\mathbf{v},b,a,u}{\tilde{p}}_{\mathbf{v},a,u}{\Pi }_{b,v_b}^{\prime }\otimes {\Pi }_{a,u}^T\right)\right)$$

$$⩾q\left(N\left(\sigma \otimes {\rho }^{\prime T}+t\sum _{\mathbf{v},b,a,u}{\bar{p}}_{\mathbf{v},a,u}{\Pi }_{b,v_b}^{\prime }\otimes {\Pi }_{a,u}^T\right)\right)$$

$$⩾M_{\sigma ,t}({\rho }^{\prime })+c_{\sigma ,t}$$

$$\mathrm{tr}\left(J\left(Y\otimes {\rho }^T\right)\right)=\mathrm{tr}({\rho }^{\prime }Y)\forall Y\in {\mathcal{H}}^{\prime }.$$

$$J⩾0$$

$$\mathrm{tr}\left(J({\mathbb{I}}^{\prime }\otimes \tilde{X})\right)=0\forall \tilde{X}\in \mathcal{H}$$

$$\mathrm{tr}\left(J\left(A_{\mathbf{v}}\otimes {\Pi }_{a,u}^T\right)\right)⩾0\forall \mathbf{v},a,u,$$

$$\mathrm{tr}\left(J\left(\tilde{Y}\otimes {\rho }^T-\frac{\mathrm{tr}(\tilde{Y}{\rho }^{\prime })}{d}{\mathbb{I}}^{\prime }\otimes \mathbb{I}\right)\right)=0\forall \tilde{Y}\in {\mathcal{H}}^{\prime },$$

$$\begin{array}{rl}& \mathrm{tr}(JV)=0\text{ for all }V\in \mathcal{V}\\ & \mathrm{tr}(J{W}_i)⩾0\text{ for all }i=1,\dots ,m.\end{array}$$

$$V-(y_1{W}_1+\cdots +y_m{W}_m)\ngtr 0.$$

$${\mathbb{I}}^{\prime }\otimes \tilde{X}+\tilde{Y}\otimes {\rho }^T-\frac{\mathrm{tr}(\tilde{Y}{\rho }^{\prime })}{d}{\mathbb{I}}^{\prime }\otimes \mathbb{I}-\sum _{\mathbf{v},a,u}{y}_{\mathbf{v},a,u}{A}_{\mathbf{v}}\otimes {\Pi }_{a,u}^T\ngtr 0$$

Full text

††preprint: APS/123-QED

I Introduction

Magic states have interesting applications in disparate areas of quantum physics from foundations of quantum mechanics to quantum computation. According to the Gottesman-Knill’s theorem Gottesman (1997); Bartlett et al. (2002); Mari and Eisert (2012), if we restrict the allowed set of states and operations to stabilizer states and operations, then the dynamics and measurements of quantum states can be simulated efficiently on a classical computer. However, it is well-known that universal quantum computing can be achieved by addition of magic states Bravyi and Kitaev (2005); Howard and Campbell (2017). In circuit synthesis Howard and Campbell (2017), the set of Clifford unitaries is supplemented with the $T$-gate in order to achieve universal quantum computation. There, it is favorable to reduce the number of $T$-gates as much as possible, since to implement a $T$-gate, magic states need to be consumed as a resource. Moreover, recently an interesting connection between contextuality and the resource theory of magic has been pointed out Spekkens (2008); Delfosse et al. (2015); Raussendorf et al. (2017); Delfosse et al. (2017). In particular, it was established that quantum contextuality is a resource for quantum speed-up within one of the most successful models for fault-tolerant quantum computation Howard et al. (2014), namely the magic state distillation model.

Recently, the resource theory of magic has attracted much attention Veitch et al. (2014); Howard et al. (2014); Delfosse et al. (2015, 2017); Raussendorf et al. (2017). In this framework, free operations are the set of allowed operations, i.e., stabilizer operations. Resource states, namely the magic states, are required in order to achieve some desired task. In a realistic setting wherein the resources are finite, one is ideally interested in answering the single-shot question: Given two resource states $\rho$ and $\rho^{\prime}$, is there a free operation that will convert $\rho$ to $\rho^{\prime}$? In a recent attempt to answer this question, the necessary and sufficient conditions for the possibility of converting a resource state into another resource state has been obtained for a large class of quantum resource theories coined as affine resource theories Gour (2017). In these resource theories, an affine combination of free states is considered to be a free state itself. Resource theories of coherence Baumgratz et al. (2014); Chitambar and Hsieh (2016); Chitambar and Gour (2016); Marvian and Spekkens (2016); Winter and Yang (2016); Napoli et al. (2016), asymmetry Gour and Spekkens (2008); Gour et al. (2009); Marvian and Spekkens (2013); Ahmadi et al. (2013); Marvian and Spekkens (2014a, b), athermality Brandão et al. (2013, 2015); Horodecki and Oppenheim (2013); Faist et al. (2015); Lostaglio et al. (2015a, b); Gour et al. (2015) are examples of affine resource theories, whereas the resource theories of entanglement and magic states Veitch et al. (2014); Howard et al. (2014); Howard and Campbell (2017) are not affine Gour (2017).

In this paper, we study the single-shot conversion of magic states using free operations. We find the set of free operations in Veitch et al. (2014) too restrictive and we extend it to include all the completely positive trace preserving (CPTP) maps that convert stabilizer states into stabilizer states, i.e., stabilizer-preserving operations (SPOs). SPOs form the largest possible set of physical operations that can be considered free. We have numerical evidence that this set is strictly larger than the set of stabilizer operations as defined in Veitch et al. (2014). Furthermore, we construct a complete set of magic monotones based on the conditional min-entropy König et al. (2009). Our set of magic monotones is complete in the sense that a magic state can be converted into another magic state by an SPO if and only if the value of all the monotones does not increase in the process.

This paper is structured as follows: In section II, we define the set of free states, i.e., stabilizer states. In section III, we characterize the set of SPOs. In section IV, we construct a family of magic monotones which quantify the usefulness of magic states. In section V.1, we formulate the necessary and sufficient conditions for single-shot conversion of a magic state to another using a free operation as an SDP. In section V.2, we prove that our set of magic monotones is complete.

Throughout this paper, $d$ denotes the dimension of the Hilbert space. $\mathcal{H}$ is the set of $d\times d$ Hermitian matrices and $\mathcal{D}$ is the set of $d\times d$ density matrices. When a quantum channel is considered, we use no-prime notation for the input of the channel and prime notations for the output (for example, $\mathcal{F}$ denotes the set of free input states in dimension $d$ while $\mathcal{F}^{\prime}$ denotes the set of free output states in dimension $d^{\prime}$). Greek letters are used for density matrices. Capital letters such as $X,Y$ and $Z$ are used for generic Hermitian matrices, with an added tilde if they are traceless.

II Free states

The free states in our resource theory are defined to be stabilizer states. These consist of all pure stabilizer states, which are eigenstates of the generalized Pauli operators, and their convex mixtures. We use the term magic (or non-stabilizer) states to refer to states that are not stabilizer states. We only consider cases where the Hilbert space dimension $d$ is a prime number. We devote a separate analysis for $d=2$ to provide intuition via the visualization of the free states in the Bloch sphere.

II.1 Qubit case

In the Hilbert space of dimension $d=2$, there are 6 pure stabilizer states, namely the set of eigenstates of Pauli operators $\{\ket{0},\ket{1},\ket{+},\ket{-},\ket{i},\ket{-i}\}$. In the Bloch representation, these states correspond to 3 pairs of antipodal points along the 3 principle axes. The full set of free qubit states $\mathcal{F}$ is the convex hull of these extreme points, which is an octahedron embedded in the Bloch sphere. This octahedron consists of 4 pairs of parallel facets, so a point lies inside the octahedron if and only if it is confined to the space between any pair of parallel planes containing the facets. These planes are described by the equations $\pm x\pm y\pm z=1$. Therefore, a state with the Bloch vector $(x,y,z)$ is a free state if and only if the following inequalities hold:

[TABLE]

The inequalities above characterize free qubit states in terms of their Bloch coordinates. However, as it will be more natural for the later consideration of quantum channels, we would like to have an alternative characterization in the space of Hermitian operators. This can be done by noticing that the facets of the octahedron can be divided into 2 groups, each containing 4 facets that extend to a regular tetrahedron. The octahedron can then be described as the intersection of these 2 regular tetrahedra as shown in Figure 1.

The 8 vertices of the 2 tetrahedra have Bloch coordinates $(\pm 1,\pm 1,\pm 1)$, where the first tetrahedron’s vertices consist of coordinates with an odd number of $1$’s such as $(1,1,1)$ and $(-1,-1,1)$, and the second tetrahedron’s vertices have an even number of $1$’s in their coordinates. Let $A_{i}$ (for $i=1,\dots,4$) denote the unit-trace Hermitian operators corresponding to the vertices of the first tetrahedron (similarly $A^{\prime}_{i}$ for the second one). It is straightforward to check that

[TABLE]

The same properties hold for $A^{\prime}_{i}$. Therefore, $\{A_{i}\}$ and $\{A^{\prime}_{i}\}$ form two orthogonal (but not normalized) bases for the space of qubit Hermitian operators. Any unit-trace Hermitian operator $H$ can therefore be written as

[TABLE]

where the real coefficients $\alpha_{i}$ and $\alpha_{i}^{\prime}$ are determined by

[TABLE]

and they satisfy $\sum_{i}\alpha_{i}=\sum_{i}\alpha_{i}^{\prime}=1$ because $H$ has unit trace. In Bloch space, $H$ lies inside the tetrahedron $\{A_{i}\}$ if and only if it can be written as a convex combination of $A_{i}$, meaning that the coefficients $\alpha_{i}$ are all non-negative. A similar statement holds for the tetrahedron $\{A_{i}^{\prime}\}$. This results in the following characterization of free states: a state $\rho$ is free if and only if

[TABLE]

for all $i=1,\dots,4$.

Note that the number of inequalities in (5) is the same as that in (1), but here we can directly characterize the set of free states $\mathcal{F}$ using the Hilbert-Schmidt inner product in the space of operators. It is also worth noting that $\{\alpha_{i}\}$ and $\{\alpha_{i}\}$ are sets of values for the Wigner functions associated with the phase-point operators $\{A_{i}\}$ and $\{A_{i}^{\prime}\}$.

II.2 Qudit case

Next, we define and characterize free states for the case of a single qudit of dimension $d$, where $d$ is an odd prime number. Following the same procedure for the qubit case, we will start with the definition of pure stabilizer states, and then take the convex hull to obtain the set of all free states $\mathcal{F}$. Similar to the qubit case, we can characterize $\mathcal{F}$ by the simultaneous positivity of a finite number of Wigner functions, as shown below.

Pure stabilizer states can be obtained by applying Clifford unitaries to the state $\ket{0}$ Veitch et al. (2014). In prime dimensions, they form a complete set of Mutually Unbiased Bases (MUBs) Dang (2015). In other words, they can be grouped into $d-1$ orthonormal bases so that states from different bases all have the same overlap. If we denote the density operators of these states by $\Pi_{b,v}$, where $b\in\{1,\dots,d+1\}$ labels the basis and $v\in\{0,\dots,d-1\}$ labels the basis vector, then they form a complete set of MUBs if and only if they are rank-1 projectors that satisfy

[TABLE]

The set of all (pure and mixed) stabilizer states is then defined as the convex hull of the pure stabilizer states, which is denoted by $\mathcal{F}:=\operatorname{conv}\{\Pi_{b,v}\}$.

The standard discrete Wigner function of a quantum state $\rho\in\mathcal{D}$ is a quasi-probability distribution over the discrete phase space $\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ given by

[TABLE]

where $(p,q)\in\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ and $A_{p,q}$ are $d^{2}$ Hermitian operators that satisfy

[TABLE]

$A_{p,q}$ are usually referred to as phase-point operators. For odd prime dimensions $d$, they are defined as

[TABLE]

where

[TABLE]

and the shift operator $S$ and phase operator $P$ are defined by their actions on the standard basis as

[TABLE]

The discrete Hudson’s theorem proves that a pure state is a stabilizer state if and only if its standard Wigner function is non-negative for all $(p,q)\in\mathbb{Z}_{d}\times\mathbb{Z}_{d}$ Gross (2006). This is not true for mixed states: there are mixed states with non-negative standard Wigner function that are not stabilizer states (i.e. not in $\mathcal{F}$) Gross (2006). However, if we consider a larger family of $d^{d-1}$ Wigner functions, it was conjectured Galvão (2005) and later proved Cormick et al. (2006) that $\mathcal{F}$ is characterized by the simultaneous non-negativity of these Wigner functions. For the purpose of characterizing $\mathcal{F}$, we do not need to go into the details of how to define them, and will instead only specify their associated $d^{d+1}$ generalized phase-point operators.

Let $\bm{v}=(v_{1},v_{2},\dots,v_{d+1})$ be a vector with $d+1$ components $v_{i}\in\mathbb{Z}_{d}$. For each $\bm{v}$ we define a generalized phase-point operator $A_{\bm{v}}$ as

[TABLE]

There are $d^{d+1}$ such generalized phase-point operators. It is straightforward to check from their definition that $\operatorname{tr}(A_{\bm{v}})=1$. Using the properties of MUB projectors in (6), one can show that $\operatorname{tr}(A_{\bm{v}}A_{\bm{u}})=d$ if and only if $\bm{v}$ and $\bm{u}$ agree at exactly one component. One can group these operators into $d^{d-1}$ groups of size $d^{2}$ in such a way that for any $A_{\bm{v}}$ and $A_{\bm{u}}$ in the same group, $\bm{v}$ and $\bm{u}$ agree at exactly one component, thus forming $d^{d-1}$ Wigner functions (this is a non-trivial combinatoric problem, see section 4 in Bengtsson (2005)). Here, we focus on the fact that these operators are stabilizer witnesses and we can use them to characterize $\mathcal{F}$: a density operator $\rho$ is a free state if and only if

[TABLE]

A geometric interpretation of the inequalities in (13) is that $\mathcal{F}$ is a polytope with $d^{d+1}$ facets in the space of Hermitian operators.

III Free operations

In magic-state distillation protocols, one asks whether there exists a procedure, which can transform any mixed magic state to a more resourceful (more magic) state while solely employing free operations. In Veitch et al. (2014), stabilizer protocols and magic monotones such as sum negativity, mana and relative entropy of magic were defined to answer this question quantitatively. In this section, for comparison purposes, we first recall the definition of stabilizer protocols and then present our definition of free operations (SPOs).

A stabilizer protocol may consist of the following types of quantum operations: Clifford unitaries, composition with stabilizer states, partial trace, and measurements in the computational basis. Moreover, these quantum operations can be conditioned on measurement results and classical randomness. Using the Stinespring dilation theorem, any quantum protocol composed of these quantum operations can be written as

[TABLE]

where $U$ is a Clifford unitary111The Clifford group is defined as the normalizer of the generalized Pauli group, i.e., the collection of unitary operators $U_{C}$ that map the generalized Pauli group to itself under conjugation. and the ancilla $\rho_{E}$ is a free state, i.e., a pure or mixed stabilizer state. Note that for a given dimension of the ancillary system there are finitely many Clifford unitaries which means that the number of stabilizer protocols (14) is also finite.

In our definition of SPOs as free operations, we do not put any operational restriction on $U$ and $\rho_{E}$ but instead only require that free operations ${\cal O}_{f}$ do not generate resources. Specifically, we define SPOs as linear maps that satisfy the following conditions:

They are completely positive (CP), 2. 2.

They are trace preserving (TP), 3. 3.

They transform stabilizer states to stabilizer states; i.e., if $\rho\in\mathcal{F}$, then ${\cal E}(\rho)\in\mathcal{F}^{\prime}$.

We would like to point out that in the resource theory framework the set of SPOs as defined above is the maximal set of resource non-generating operations. Therefore, it includes the stabilizer protocols. In fact, we have numerically checked that the set of SPOs strictly contains the set of stabilizer protocols, as described in the appendix.

We write the above conditions (1-3) in the Choi representation which later enables us to phrase our problem as an SDP. The Choi matrix $J$ corresponding to the quantum channel ${\cal E}:\mathcal{H}\to\mathcal{H}^{\prime}$ is given as $J={\cal E}\otimes\mathbb{I}(\ket{\phi^{+}}\bra{\phi^{+}})$ where $\ket{\phi^{+}}=\sum_{i=1}^{d}\ket{ii}$ which is the state vector of the maximally entangled state up to a normalization, and $J$ is a positive semi-definite matrix in $\mathcal{H}\otimes\mathcal{H}^{\prime}$. Note that the TP condition can be phrased in terms of the Choi matrix as $\operatorname{tr}_{B}(J)=\mathbb{I}$ which is equivalent to $\operatorname{tr}\left(J(\mathbb{I}^{\prime}\otimes X)\right)=\operatorname{tr}(X)$ for all $X\in\mathcal{H}$. Finally, the Choi matrix of a free operation ${\cal E}\in{\cal O}_{f}$ must satisfy the conditions

[TABLE]

where $\bm{v}\in\mathbb{Z}_{d^{\prime}}^{d^{\prime}+1}$, $a=1,2,\ldots,(d+1)$ and $u\in\mathbb{Z}_{d}$.

Note that to write the third condition, given in Eq. (17), we have used the fact that, if a CPTP map takes the extreme points of the stabilizer polytope to a free state, it will also take any convex combination of them to a free state. Hence, it is sufficient to demand that the quantum channel takes all the elements of the set of MUB projectors $\{\Pi_{a,u}\}$ to a free state.

IV Quantifying magic

Monotones are used to quantify the resourcefulness of resource states. By definition, they are real-valued functions that are non-increasing under the action of free operations. In other words, for $f$ to be a monotone, it has to satisfy $f(\rho)\geqslant f({\cal E}(\rho))$ for all states $\rho$ and free operations ${\cal E}$. In this section, we construct a family of magic monotones to quantify magic states and prove that they are indeed non-increasing under SPO.

**Definition: ** For any non-zero $X\in\mathcal{H}^{A}\otimes\mathcal{H}^{B}$, $X\geqslant 0$, we define $q(X)$ as

[TABLE]

Note that $q(X)$ is positive and closely related to the conditional min-entropy by $q(X)=2^{-H_{\text{min}}(A|B)_{X}}$ König et al. (2009).

Given a density matrix $\sigma\in\mathcal{D}^{\prime}$, a real number $t\geqslant 0$ and probabilities $p_{\bm{v},a,u}$, we define the density matrix $\Omega_{\rho}$ as a function of input state $\rho\in\mathcal{D}$:

[TABLE]

where $N:=1/(td^{\prime}+t+1)$ is the normalization factor and the sum is over all $\bm{v}\in\mathbb{Z}_{d^{\prime}}^{d^{\prime}+1}$, $b=1,\dots,d^{\prime}+1$, $u\in\mathbb{Z}_{d}$ and $a=1,\dots,d+1$.

Definition (monotones): For each $\sigma\in\mathcal{D}^{\prime}$ and a real number $t\geqslant 0$, we define the function $M_{\sigma,t}:\mathcal{D}\to\mathbb{R}$ as

[TABLE]

where $c_{\sigma,t}:=\min_{p_{\bm{v},a,u}}q(\Omega_{\mathbb{I}/d})$.

Note that $\Omega_{\rho}$ depends on $\sigma$, $t$ and probabilities $p_{\bm{v},a,u}$, but $M_{\sigma,t}$ only depends on parameters $\sigma$ and $t$ because of the minimization over $p_{\bm{v},a,u}$. The constant $c_{\sigma,t}$ is used to offset the value of $M_{\sigma,t}$ to zero on stabilizer states.

Theorem 1: For all $\sigma\in\mathcal{D}^{\prime}$ and $t\geqslant 0$, $M_{\sigma,t}(\rho)$ are magic monotones. That is, for an arbitrary state $\rho$, if $\rho^{\prime}=\cal{E}(\rho)$ for a stabilizer-preserving quantum channel $\cal{E}$, then

[TABLE]

Proof: We first recall a fact about the conditional min-entropy that for a channel $\cal{E}$, $q(X)\geqslant q(\Lambda(X))$, where $\Lambda:=\mathbb{I}\;\otimes\;T\circ{\cal E}\circ T$ and $T$ is the transposition map Gour (2017). Also, note that because $\cal{E}$ is a stabilizer-preserving channel,

[TABLE]

for some probabilities $p_{a,u}$. Denoting the set of probabilities that minimize the function $q\left(\Omega_{\rho}\right)$ by $\tilde{p}_{\bm{v},a,u}$, we then have

[TABLE]

where $N:=1/(td^{\prime}+t+1)$. The first inequality follows from $q(X)\geqslant q(\Lambda(X))$, $\bar{p}_{\bm{v},a,u}$ are probabilities derived from $\tilde{p}_{\bm{v},a,u}$ and $p_{a,u}$, and the second inequality follows from the definition of $M_{\sigma,t}$. $\hfill\square$

Remark 1. Our monotones are also monotones under the set of operations considered in Veitch et al. (2014), since it is a strict subset of our set of SPO.

Remark 2. It is worth pointing out that the double minimization in the definition of $M_{\sigma,t}$ can be computed in whole as an SDP. As the number of probabilities $p_{\bm{v},a,u}$ grows as $d(d+1)d^{\prime d^{\prime}+1}$, in high dimensions it is not practical to compute $M_{\sigma,t}$. The computation can be easily carried out in low dimensions, as described in the following numerical example.

Example (qubit case): Here we consider the case $d=d^{\prime}=2$. Computing $M_{\sigma,t}$ involves a minimization over 48 real parameters and a $2\times 2$ Hermitian matrix. We used CVX in Matlab to evaluate $M_{\sigma,t}(\rho)$ numerically at various input states $\rho$, for several values of parameters $\sigma$ and $t$. With $\sigma$ fixed at $|0\rangle\langle 0|$, for $t=0$ we get the trivial zero monotone. For $t=1$ the monotone is faithful, meaning that $M_{\sigma,t}(\rho)=0$ if and only if $\rho$ is a stabilizer state (see Figure 2). For $0\leqslant t<1$ or $t>1$, the monotone is not faithful: there are magic states $\rho$ at which $M_{\sigma,t}(\rho)=0$ (see Figure 3). For some other choices of $\sigma$, we observed that the monotone became faithful at some particular value of $t$ that depends on $\sigma$.

V Manipulating magic states

Resource state manipulation, i.e., using free operations to convert one resource state to another, is among the most fundamental aspects in studying a quantum resource theory. Here we consider exact manipulation in the single-shot regime, that is, exact conversion of a single copy of one state to that of another by an SPO. We first formulate the conversion question as a semi-definite program and use Farkas’ lemma to derive necessary and sufficient conditions for conversion. We then show that the monotones constructed in the previous section can alternatively be used to determine convertibility.

V.1 An SDP formulation

The characterization of free operations in (15)-(17) is readily in the form of SDP constraints. The only missing ingredient is the condition for a quantum channel ${\cal E}$ to convert state $\rho$ to state $\rho^{\prime}$. We begin this section by expressing the state conversion statement ${\cal E}(\rho)=\rho^{\prime}$ in the Choi matrix representation as $\operatorname{tr}_{B}\left(J\left(\mathbb{I}\otimes\rho^{T}\right)\right)=\rho^{\prime}$. After transforming the partial-trace to a full trace we get

[TABLE]

Following Gour (2017), we can phrase the existence question of a stabilizer-preserving channel ${\cal E}$ converting density matrices $\rho$ to $\rho^{\prime}$ as the existence question of a non-zero matrix $J\in\mathcal{H}\otimes\mathcal{H}^{\prime}$ that satisfies the conditions:

[TABLE]

for all traceless matrices $\tilde{X}$ and $\tilde{Y}$, $\bm{v}\in\mathbb{Z}_{d^{\prime}}^{d^{\prime}+1}$, $u\in\mathbb{Z}_{d}$ and $a=1,\dots,d+1$. Conditions (25) on the Choi matrix $J$ correspond to the conditions for ${\cal E}$ to be completely-positive, trace-preserving, stabilizer-preserving, and converting $\rho$ to $\rho^{\prime}$. To turn the existence question into necessary and sufficient conditions that are verifiable, we make use of the following lemma.

**Lemma 1: ** Let $\mathcal{V}$ be a subspace of $\mathcal{H}$ and let $W_{1},\dots,W_{m}$ be matrices in $\mathcal{H}$. The following are equivalent.

There exists a non-zero positive semidefinite matrix $J$ such that

[TABLE] 2. 2.

For all $V\in\mathcal{V}$ and non-negative numbers $y_{1},\dots,y_{m}$

[TABLE]

This lemma is a variant of Farkas’ Lemma and its proof is based on the separating hyperplane theorem Boyd and Vandenberghe (2004). Applying this lemma to conditions (25a)-(25d), with $\mathcal{V}$ being the subspace spanned by $\mathbb{I}^{\prime}\otimes\tilde{X}$ and $\tilde{Y}\otimes\rho^{T}-\frac{\operatorname{tr}(\tilde{Y}\rho^{\prime})}{d}\mathbb{I}^{\prime}\otimes\mathbb{I}$, and $W_{i}$ being $A_{\bm{v}}\otimes\Pi^{T}_{a,u}$, we obtain the following equivalent conditions:

[TABLE]

for all traceless $\tilde{X}\in\mathcal{H}$, traceless $\tilde{Y}\in\mathcal{H}^{\prime}$, and $y_{\bm{v},a,u}\geqslant 0$.

To summarize, we have cast the state conversion question in the form of an SDP feasibility problem (25a)-(25d) and alternatively provided a set of necessary and sufficient conditions. The SDP form is more useful for practical purposes. It is also constructive: when the conversion is possible, there are known SDP algorithms that can find an SPO that does the state conversion. The necessary and sufficient conditions in (28) consist of infinitely many conditions. On the other hand, they provide the analytical base for the proof of the completeness of our magic monotones in the next section.

Remark 3. Using CVX in Matlab, we found a numerical example of qutrit-to-qutrit SPO that can increase the Wigner sum negativity222The sum negativity of a state is the sum of the negative elements of the Wigner function Veitch et al. (2014). (a monotone for stabilizer protocols studied in Veitch et al. (2014)), thus confirming that the set of SPOs is strictly larger than the previously studied set of stabilizer protocols. See the appendix for this specific example.

V.2 Completeness of monotones

An alternative way of checking if a resource can be converted to another by a free operation is to use a complete set of monotones. In this section, we show that our set of magic monotones, $M_{\sigma,t}(\rho)$ defined in (20), is complete in the sense that a resource state can be converted to another using a stabilizer-preserving operation if and only if it is more resourceful with respect to every monotone. We begin with the lemma below, which we then use to prove the completeness of magic monotones as stated in Theorem 2.

Lemma 2: Given density operators $\rho\in\mathcal{D}$ and $\rho^{\prime}\in\mathcal{D}^{\prime}$, there exists a stabilizer-preserving quantum channel $\mathcal{E}$ such that $\mathcal{E}(\rho)=\rho^{\prime}$ if and only if for all density operators $\sigma\in\mathcal{D}^{\prime}$, probabilities $p_{\bm{v},a,u}$ and $t\geqslant 0$ we have

[TABLE]

Proof: The necessary and sufficient conversion conditions were cast in (28). Notice that these conditions remain unchanged if we divide the left hand side by an arbitrarily large positive number. Therefore, we can assume $Y<\mathbb{I}^{\prime}/d^{\prime}$ such that $Y=\mathbb{I}^{\prime}/d^{\prime}-\sigma$, where $\sigma$ is a density operator. After making this substitution into (28), diving both sides by $t(d^{\prime}+1)+1$ and re-arranging it so that all the terms of the form $\mathbb{I}^{\prime}\otimes\ldots$ are grouped to the left-hand side of the inequality, we obtain

[TABLE]

where $\Omega_{\rho}$ is defined in (19) and

[TABLE]

for all traceless $\tilde{X}\in\mathcal{H}$, $\sigma\in\mathcal{D}^{\prime}$ , probabilities $p_{a,u}$ and $t\geqslant 0$. This is equivalent to

[TABLE]

for all density operators $\sigma\in\mathcal{D}^{\prime}$ and $t\geqslant 0$. $\hfill\square$

Theorem 2: Given density operators $\rho\in\mathcal{D}$ and $\rho^{\prime}\in\mathcal{D}^{\prime}$, if for all density operators $\sigma\in\mathcal{D}^{\prime}$ and $t\geqslant 0$ it holds that $M_{\sigma,t}(\rho)\geqslant M_{\sigma,t}(\rho^{\prime})$, then there exists a stabilizer-preserving quantum channel $\mathcal{E}$ such that $\mathcal{E}(\rho)=\rho^{\prime}$.

Proof: We begin by noticing that the identity channel is a free channel, and therefore we know that $\rho^{\prime}$ can be converted to $\rho^{\prime}$. Then using lemma (2) and the optimal set of probabilities $\tilde{p}_{\bm{v},a,u}$, for all density operators $\sigma\in\mathcal{D}^{\prime}$ and $t\geqslant 0$ we can write

[TABLE]

and hence there exists a free channel which converts $\rho$ to $\rho^{\prime}$. $\hfill\square$

The following corollary is the result of combining theorems (1) and (2).

Corollary: Given density operators $\rho\in\mathcal{D}$ and $\rho^{\prime}\in\mathcal{D}^{\prime}$, the following statements are equivalent:

There exists a stabilizer-preserving quantum channel $\mathcal{E}$ such that $\mathcal{E}(\rho)=\rho^{\prime}$ 2. 2.

For all density operators $\sigma\in\mathcal{D}^{\prime}$ and $t\geqslant 0$ we have $M_{\sigma,t}(\rho)\geqslant M_{\sigma,t}(\rho^{\prime})$.

To summarize, we have found a complete set of magic monotones which can be used to determine the single-shot convertibility between two states.

VI Conclusions

In this paper, we answered the question: Given two magic states $\rho$ and $\rho^{\prime}$, is there an SPO that can convert $\rho$ to $\rho^{\prime}$? We cast this question as an SDP feasibility problem and we employed a variant of Farkas’ Lemma to find the necessary and sufficient conditions for the existence of such an operation. Then, we provided a set of magic monotones which we proved to be complete. In other words, the answer to the single-shot question is positive in our case if and only if the density matrix $\rho\in\mathcal{D}$ is more resourceful than (or at least as resourceful as) the density matrix $\rho^{\prime}\in\mathcal{D}^{\prime}$ according to all the magic monotones (20). Note that we answered this question for the exact conversion of a resource state to another resource state. As a future line of research, the possibility of approximate conversion of a magic state to another using SPOs is a natural next step.

The usefulness of the tools developed here may extend beyond the study of magic states. As a future line of research, it would be interesting to see if the methods applied in this paper (such as SDP and Farkas’ Lemma) to the resource theory of magic can be applied to the more general case of convex resource theories.

Another possible direction for future work is to extend our results to the case of infinite-dimensional systems, namely the resource theory of non-Gaussianity. It is known that stabilizer states are the discrete analogues of Gaussian states Gross (2006) and that non-Gaussianity is a resource for tasks such as distillation of Gaussian entanglement, violation of continuous-variable Bell inequalities and continuous-variable quantum computation Genoni and Paris (2010). However, it is still unknown what can be achieved using Gaussian operations given that one has access to a non-Gaussian state.

Acknowledgements.

We thank Mercedes Gimeno-Segovia, Mark Girard and David Jennings for useful discussions and comments. The authors were supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Pacific Institute for the Mathematical Sciences (PIMS). BCS appreciates financial support from Alberta Innovates Technology Futures and China’s 1000 Talent Plan.

Appendix A Numerical examples

Here, we present an example for the qutrit case where the state $\rho$ cannot be converted using the stabilizer operations as was defined in Veitch et al. (2014), while this conversion is possible using SPO. To find such examples, we generated random density matrices $\rho$ and $\rho^{\prime}$ and ordered them such that $\rho^{\prime}$ has a higher sum negativity in comparison to $\rho$. This means that the state $\rho$ cannot be converted to $\rho^{\prime}$ using the stabilizer operations defined in Veitch et al. (2014). Then, using the CVX package for Matlab Grant and Boyd (2014) (a package for specifying and solving convex programs), we checked whether there exists an SPO with the Choi matrix $J$ which satisfies conditions (25).333https://github.com/ahmadimehdi/Quantification-and-manipulation-of-magic-states.

As an example for the qutrit case, consider the following density matrices:

[TABLE]

and

[TABLE]

The sum-negativities of these two states are [math] and $0.0074$ respectively, while there exists an SPO that converts $\rho$ to $\rho^{\prime}$, whose Choi matrix has the real part

[TABLE]

and imaginary part

[TABLE]

corresponds to an SPO that converts $\rho$ to $\rho^{\prime}$.