$$ \newcommand\ketbra[1]{\lvert #1 \rangle \langle #1 \lvert} \renewcommand\braket[2]{\langle #1 \lvert #2 \rangle} \newcommand\ket[1]{\lvert #1 \rangle} \newcommand\bra[1]{\langle #1 \lvert} \newcommand\Tr{\text{Tr}} $$

Introduction to the structure of free operations in the resource theories of entanglement and magic

The operational and axiomatic definitions have a gap in both entanglement and magic


After three semesters into grad school, I’m realizing that my initial lofty goal of keeping a blog is not going to happen easily. I just prioritize everything else (homework, classes, research, private life) over it. However, I still generate some form of written material - final projects in classes. So, for the next two semesters (of my last three classes), I’m just going to put down here my final projects in the form of blog posts. As final projects are typically done in a rush, I’ll probably clean them up a bit in the process of translating them to markdown from LaTeX.

This particular write-up was for my 2022 Fall semester Quantum Information Theory class taught by Professor Iman Marvian.


Resource theories formalize the notion of valuable resources. A quantum resource theory identifies “free states” that can be created with “free operations” - these are “easy” relative to assumed capabilities. To create “resource states”, one must either have states that contain “some” of the resource or use resource-generating operations. The theory of entanglement (where the resource is entanglement) and the theory of magic (where the resource is “non-stabilizerness”, or “magic”) share a similar structure through the lens of resource theory. Our main focus is a particular hierarchical structure of free operations, namely, the gap between the operational definition and the axiomatic definition of these classes of operations. For entanglement, this is the gap between the classes of Local Operations and Classical Communication (LOCC) and Separable Operators. In the case of magic, the gap is between the classes of Stabilizer Operators and Completely Stabilizer Preserving operators. These gaps have potential implications for protocols related to entanglement and magic state distillation.


The majority of physical theories establish laws based on experimental observations of nature and predict behavior for situations that were not yet verified experimentally. Resource theories in contrast have an inherent process-, or engineering focus as they examine what is possible given a demarcation line in operational capabilities. With roots in Carnot’s explorations of how to exploit thermodynamic non-equilibrium as a resource for heat engines, resource theories provide a framework to organize the concepts and questions of availability and quantification of resources, conversion rates, reversibility, and more. At the core of a quantum resource theory, a set of operations is defined to be the free operations, or resource-non-generating (RNG) operations, and a set of states defined to be the free states. The intuition is that RNG operations and free states are the “easy ones”, i.e. assumed to be available to acquire without incurring high costs. This is in contrast to valuable dynamic resources (resource generating operations), i.e. operations generating static resources (resource states) which are “hard” to acquire. (Chitambar & Gour, 2019).

While there are numerous quantum resource theories, the two we are interested in in this report are the theories of entanglement and magic. It is well known that quantum entanglement can be used as a resource for communication to extend our capabilities and open up use cases like super-dense coding or quantum teleportation (Wilde, 2017). These use cases would be impossible with the “free” separable states, and Local Quantum Operations and Classical Communication (LOCC).

Magic is a resource for universal fault-tolerant quantum computation (FTQC). Under the stabilizer formalism (Gottesman, 1997), large classes of Quantum Error Correction codes can be expressed for both storing and fault tolerantly executing quantum computation in the logical space of these codes. However, stabilizer operations are not universal in the quantum sense, and in-fact are classically efficiently simulatable, as stated by the Gottesman-Knill theorem (missing reference). The term “magic” was introduced in (Bravyi & Kitaev, 2005), where Bravyi and Kitaev showed that for universal quantum computation, one has to inject the valuable resource of magic states or use non-stabilizer operations.

This report focuses on a particular aspect of resource theories of magic and entanglement, namely, the structure of free operations. Free operations are defined either axiomatically or operationally. The operational definition is usually the more natural, stemming from the assumed limitation on capabilities, and having a clear operational interpretation. Axiomatically defined free operations are more mathematically motivated, they have less clear operational meaning; they are the maximal set of operations that map free states to free states. The question we are interested in is whether the axiomatic class is larger or not than the operational one and if yes, what the implications may be.

In entanglement theory, the class of separable channels (SEP) are the axiomatically defined free operations and the operational ones are the LOCC operations. Chitambar et al. (Chitambar et al., 2012) found an operationally interpretable measure of entanglement, that shows a 12.5% gap between SEP and LOCC. In the case of magic theory, a recent counterexample from Heimendahl et al. (Heimendahl et al., 2022) shows that the axiomatically defined class of free operations, Completely Stabilizer Preserving (CSP) is strictly larger than the operationally defined class of stabilizer operations (SO).

The structure of this paper is as follows: in Fundamental concepts we will cover the minimum prerequisite of notations and concepts to understand the separation of these two classes in these two theories. In The structure of free operations we will then get into the details of the separation of these classes in the two theories and finally, in Discussion, we discuss the potential consequences and draw our conclusions.

Fundamental concepts

In this section, we review the concepts and mathematical tools in our two theories of interest and establish notation, mostly following Wilde (Wilde, 2017) for the quantum information theoretical and entanglement-related concepts. To introduce the stabilizer formalism, we will follow multiple sources (Gottesman, 1997), (Aaronson & Gottesman, 2004), (Heimendahl et al., 2022).

State vector and density matrix representation of quantum states

Quantum mechanics is inherently probabilistic in the sense that measurement outcomes on a system are defined by the amplitude square of a set of complex numbers (probability amplitudes) that describe a state as defined by the Born rule. If we can describe the probability amplitudes of a quantum mechanical system with certainty, we call that state a pure state. However, this is far from realistic, usually, we don’t know perfectly the description of a system. Thus, on top of the inherent “quantum probability distribution”, we might have an uncertainty of these amplitudes themselves, layering a “classical probability distribution” over the quantum probability, which we then call mixed states (or noisy states).

Pure quantum states are represented as normalized vectors (or we can think of them as rays) in a potentially composite Hilbert space, i.e. one that allows for a structure of H=i=1nHi\mathcal{H}=\otimes_{i=1}^n \mathcal{H}_i with each sub-Hilbert space having dimension did_i. We will only discuss cases where di=dj=d,i,jd_i = d_j = d, \forall i, j, i.e. Hi=Cd\mathcal{H}_i=\mathbb{C}^d, and where dd is prime. When d=2d=2, we talk about a Hilbert space of nn qubits. If d=3d=3, we call each subsystem a qutrit and d>3d\gt3 is the case of qudits. We denote pure states in the Dirac notation as ψH\ket{\psi} \in \mathcal{H}. The members of the computational basis for the nn qudit Hilbert space of dd-dimensional qudits will be labeled by the elements of the vector space Fdn\mathbb{F}_d^n over the finite field Fd\mathbb{F}_d (i.e. a nn length vector with elements 0 to d1d-1, with addition is modulo dd). For example 124=124(C5)3\ket{124} = \ket{1}\otimes \ket{2} \otimes \ket{4} \in (\mathbb{C}^5)^{\otimes 3} is a 3-qudit state of dimension 5.

As for mixed states, we don’t know exactly which pure state a system is in, we can represent this lack of knowledge as a probability distribution PXP_X over possible pure states ψx\ket{\psi_x}. This ensemble of states and their probabilities {(PX(x),ψx)}\{(P_X(x), \ket{\psi_x})\} affords a density operator representation ρxpX(x)ψxψx\rho \equiv \sum_x p_X(x) \ket{\psi_x}\bra{\psi_x}, where ψxψxL(H)\ket{\psi_x}\bra{\psi_x} \in \mathcal{L}(\mathcal{H}) is the density operator for the pure state ψx\ket{\psi_x} defined by the outer product of ψx\ket{\psi_x} and ψxψx\bra{\psi_x} \equiv \ket{\psi_x}^\dagger. We denote the set of square linear operators acting on H\mathcal{H} with L(H)\mathcal{L}(\mathcal{H}). A density operator is always positive semi-definite, denoted as ρ0\rho \geq 0, and has unit trace, Tr{ρ}=1\text{Tr}\{\rho\}=1. The set of density operators is denoted as DO(H)L(H)DO(\mathcal{H}) \subset \mathcal{L}(\mathcal{H}).

Describing change with operators and quantum channels

When the change in a system is noiseless, i.e. the system is closed, it is postulated to be a transformation UL(H)U \in \mathcal{L}(\mathcal{H}) on the system that is unitary, i.e. UU=IUU^\dagger=I. In the vector representation, the effect of UU is ψUψ\ket{\psi} \rightarrow U\ket{\psi}, in the density operator formalism, it is ρUρU\rho \rightarrow U \rho U^\dagger. We will make use of the generalizations of the Pauli group, which is generated by the Pauli-X and Pauli-Z operators for a single qubit, with their action on the computational basis:

Xk=k+1  ,Zk=(1)kk,kF2. X\ket{k}=\ket{k+1} \ \ , Z\ket{k}=(-1)^{k}\ket{k}, k\in \mathbb{F}_2.

The generalized nn-qudit Pauli-group for dd dimension qudits is generated by the generalized nn-qudit X and Z operators, with action:

X(x)k=k+x  ,Z(z)k=ωkzk,x,z,kFdn. X(x)\ket{k}=\ket{k+x} \ \ , Z(z)\ket{k}=\omega^{k \cdot z}\ket{k}, x,z,k\in \mathbb{F}_d^n.

, where ω=ei2π/d\omega = e^{i2\pi/d} is the ddth root of unity and kz=i=1nkizik \cdot z = \sum_{i=1}^{n} k_i z_i is the inner product between kk and zz.

However, in a realistic scenario, evolution is not perfectly known, and as such, it can be thought of as a probabilistic mixture of certain operators. Formally, we represent the most general type of evolution an experimenter can impart on a system with the quantum channel formalism. A quantum channel is a map NAB:L(HA)L(HB)\mathcal{N}_{A \rightarrow B}: \mathcal{L}(\mathcal{H}_A) \rightarrow \mathcal{L}(\mathcal{H}_B), that must map density operators to density operators, i.e. we have to make sure that the output operator ρB:=NAB(ρA)\rho_B := \mathcal{N}_{A \rightarrow B}(\rho_A) has trace 1 and is positive. This leads to two mandatory properties. The first property a quantum channel has to satisfy is to be trace-preserving (TP), i.e. Tr{ρA}=Tr{ρB}\text{Tr}\{\rho_A\}=\text{Tr}\{\rho_B\}. The second related to positivity is more subtle, namely that ρB0\rho_B \geq 0 is not enough, because positivity is not closed under the tensor product. Our map has to be completely positive (CP), meaning that for a ρA\rho_A density operator on a system AA, even if it is extended with an arbitrary reference system RR to a density operator ρAB\rho_{AB}, (NABidR)(ρAB)0(\mathcal{N}_{A \rightarrow B} \otimes id_R) (\rho_{AB}) \geq 0, i.e. the total system stays positive semi-definite. Both of these properties can be tested via the Choi-Jamiokołwski representation of a channel E\mathcal{E}:

J(E):=(Eidn)(ϕ+ϕ+), J(\mathcal{E}) := (\mathcal{E} \otimes id_n)(\ket{\phi^+}\bra{\phi^+}),

, where ϕ+:=dnxFdnxx\ket{\phi^+}:=d^{-n} \sum_{x \in \mathbb{F}_d^n} \ket{xx} is the maximally entangled bipartite state on 2n2n dd-qudits in the computational basis, and idnid_n is the identity channel acting on the Hilbert space of nn dd-qudits. Namely, E\mathcal{E} is TP if and only if Tr(J(E))=1Tr(J(\mathcal{E})) = 1, and E\mathcal{E} is CP if and only if J(E)0J(\mathcal{E}) \geq 0.

We will use the Kraus representation of a quantum channel E\mathcal{E}, where, for a set of operators called Kraus operators {Mi}\{M_i\}, such that they are complete, i.e. iMiMi=I\sum_i M_i^\dagger M_i = I, the channel’s effect is expressed as

E(ρ)=iMiρMi. \mathcal{E}(\rho)=\sum_i M_i \rho M_i^\dagger.

Measurement of an observable can be described as a quantum channel. An observable MM is a Hermitian operator, M=MM^\dagger=M, and thus has a spectral decomposition, say M=iλiϕiϕiM=\sum_i \lambda_i \ket{\phi_i}\bra{\phi_i}, where λiR\lambda_i \in \mathbb{R} are the eigenvalues and ϕi\ket{\phi_i} are the corresponding eigenstates. When representing a measurement with this observable, the measurement outcomes are the eigenvalues, and after a certain eigenvalue λi\lambda_i is observed, the system’s state will be projected to the corresponding eigensubspace (of dimension equal to the multiplicity of the eigenvalue), e.g. if λi\lambda_i has multiplicity 1: ρϕiϕiρϕiϕipM(i)\rho \rightarrow \frac{\ket{\phi_i}\bra{\phi_i}\rho\ket{\phi_i}\bra{\phi_i}}{p_M(i)}, where pM(i)=Tr{ϕiϕiρϕiϕi}p_M(i)=\text{Tr}\{\ket{\phi_i}\bra{\phi_i}\rho\ket{\phi_i}\bra{\phi_i}\} is the probability of observing λi\lambda_i.

Finally, we can always find a Stinespring dilation of a channel EA\mathcal{E}_A acting on system AA, which is a unitary UARU_{AR} on a larger bipartite system ARAR such that on the subsystem AA its effect (i.e. after tracing out system RR) is exactly the channel EA\mathcal{E}_A:

EA(ρA)=TrBUAR(ρA00R)UAR \mathcal{E}_A(\rho_A) = Tr_B\\{ U_{AR}(\rho_A \otimes \ket{0}\bra{0}_R) U_{AR}^\dagger \\}

, where 00R\ket{0}\bra{0}_R is an arbitrary pure state picked from the reference subsystem RR.


A pure quantum state ψAB\ket{\psi}_{AB} on a bipartite system HAHB\mathcal{H}_A \otimes \mathcal{H}_B is called a tensor product state (TPS) or product state if it can be expressed as a tensor product of two local states: ψ=ψAψB\ket{\psi} = \ket{\psi}_A \otimes \ket{\psi}_B. When there are no states local to the subsystems that can completely describe the state of a pure state, it is entangled. A prototypical example is the Bell state, 00+112\frac{\ket{00} + \ket{11}}{\sqrt{2}}, the qubit case of ϕ+\ket{\phi^+}.

In the case of mixed states, a density operator ρ\rho is called separable if it is a convex combination of pure product states, i.e. ρ=xpX(x)ψxψxAψxψxB\rho = \sum_x p_X(x) \ket{\psi_x}\bra{\psi_x}_{A} \otimes \ket{\psi_x}\bra{\psi_x}_{B}. We say that a mixed state is entangled when it is not separable. The set of separable states is convex, i.e. ρ,σ\forall \rho, \sigma separable, for all λ[0,1]:λρ+(1λ)σ\lambda \in [0,1] : \lambda \rho + (1-\lambda) \sigma is also separable.

Stabilizer formalism and magic

The stabilizer formalism is an efficient way to represent certain subspaces and states in a Hilbert space. For the class of states called stabilizer states, which we will define precisely in a moment, simulation and computation with stabilizer operations (SO) become classically simulatable. This result is the Gottesman-Knill theorem (Gottesman, 1997), and one of its implications is that it identifies a large subset of quantum computing states and operations that are not classical but can be simulated efficiently classically. If one could simulate everything that a quantum computer can do, the field of quantum computing would be in trouble justifying their efforts. Fortunately, this is not the case, as in order to achieve universality, one has to use extra resources: either use operations outside of the stabilizer operations or use a generous amount of special states called magic states. While we will not need magic states in the later exposition, we remark that Bravyi and Kitaev (Bravyi & Kitaev, 2005) defined magic states as states that are any Clifford transformations of one of these two types of pure magic states:

TT=12(I+13(X+Z+Y))HH=12(I+12(X+Z)). \ket{T}\bra{T} = \frac{1}{2}(I + \frac{1}{\sqrt{3}}(X+Z+Y)) \\ \ket{H}\bra{H} = \frac{1}{2}(I + \frac{1}{\sqrt{2}}(X+Z)).

A state ψH\ket{\psi} \in \mathcal{H} is stabilized by an nn qudit Hermitian Pauli operator PPn(d)P \in \mathcal{P}_n(d) (Hermiticity is a requirement for it to be an observable) if the state is an eigenstate of it with +1 eigenvalue, i.e Pψ=ψP\ket{\psi} = \ket{\psi}. For example 0\ket{0} is stabilized by ZZ and +\ket{+} is stabilized by XX.

For two qubits, the situation changes, one might think that for a 00\ket{00} state, S1:=Z1Z2S_1:=Z_1Z_2 would be sufficient, however, the +1 eigenspace of Z1Z2Z_1Z_2 is two dimensional, both 00\ket{00} and 11\ket{11} have +1 eigenvalues. We need then another operator to “filter out” the extra dimension but has 00\ket{00} as its +1 eigenstate. We find that S2:=Z1I2S_2:=Z_1I_2 is a good example. We can also notice that S1S2=I1Z2S_1S_2 = I_1Z_2, which is also a stabilizer for 00\ket{00}, and naturally I1I2I_1I_2 is also a stabilizer. The set {S1,S2,S1S2,I}\{S_1, S_2, S_1S_2, I\} is a group, that is closed under operator multiplication. Also, all of its members commute with each other (i.e. it’s Abelian), thus they can be diagonalized simultaneously. Their shared +1 eigenspace is the stabilized subspace.

In general, we can represent an Abelian group of size 2k2^k with only kk independent elements, the generators. An Abelian subgroup of the nn-qudit Pauli group is called a stabilizer group if it does not contain the element I- \mathbb{I} (to ensure that all stabilizers are Hermitian). As each generator “halves” the Hilbert space, we can see that a kk dimensional stabilizer group stabilizes an nkn-k dimensional subspace. When n=kn=k, we have a stabilizer state. Single qubit stabilizer states are:

Z:0 Z:1X:+ X:Y:i Y:i Z: \ket{0} \ -Z: \ket{1} \\ X: \ket{+} \ -X: \ket{-} \\ Y: \ket{i} \ -Y: \ket{-i}

, also II stabilizes all states, and I-I stabilizes no states. For two-qubit states, XX,ZZXX,ZZ stabilizes the Bell state, and in general, ϕ+\ket{\phi^+} is a stabilizer state. The set of nqubitn-qubit stabilizer states is denoted as STABn(2)STAB_n(2). Convex combinations of stabilizer states are mixed stabilizer states, and they form stabilizer polytope, denoted SPn(2):=convex(STABn(2))SP_n(2):=convex(STAB_n(2)). In the single qubit case, this is a simple octahedron on the Bloch sphere as displayed in the figure below.

The stabilizer polytope for single qubit stabilizer states. In higher dimensions, the polytope is more complex.

An nn-qubit Pauli operator can be represented by 2n+12n+1 bits, with 2 bits for each qubit for the choice of X=10,Z=01,Y=11,I=00X=10, Z=01, Y=11, I=00 and 1 bit for the global phase, the sign ±\pm. Thus a stabilizer state can be described by nn nn-qubit Pauli operators, and as such it requires only 2n2+n=O(n2)2n^2 + n=O(n^2) bits. This polynomial cost is in sharp contrast with the 2n2^n complex numbers required to represent a general state.

Beyond the efficient representation of states, it is possible to efficiently update the state if we limit the set of operations to those that map stabilizer states to stabilizer states. As an nn-qubit stabilizer state can be represented by nn nn-qubit Pauli operators, we need these operators to map Pauli operators to Pauli operators. The largest set of unitary operators that satisfy this criterion is the Clifford group, Cln(2)Cl_n(2), generated by the HH, CNOTCNOT, and the S=ZS=\sqrt{Z} gates. Gottesman and Knill showed that a polynomial number of operations is sufficient to simulate Clifford operations or Pauli measurements.

The structure of free operations

In this section, we will review how the structure of free operations has a gap between axiomatic and operational definitions. In the case of entanglement, after a quick descriptive example for separation from Bennett et al. (Bennett et al., 1999), we follow Chitambar et al. (Chitambar et al., 2012) and for magic, the work by Heimendahl et al. (Heimendahl et al., 2022).

Entanglement: LOCCSEPLOCC \subset SEP

As mentioned before, in entanglement theory, the operational and axiomatic classes of RNG operations are the LOCC and the Separable (SEP) channels.

DefinitionA channel E\mathcal{E} on a Hilbert space of NN qudits is a separable channel if it has a Kraus decomposition with separable Kraus operators, i.e.:

E(ρ)=lAlρAl,Al=M1,iM2,iMN,i. \mathcal{E}(\rho) = \sum_l A_l \rho A_l^\dagger, A_l = M_{1,i} \otimes M_{2,i} \otimes \ldots \otimes M_{N,i}.

The set of separable channels is denoted SEP.

An early example by Bennett et al. (Bennett et al., 1999) shows the separation between these two classes, which shows “non-locality without entanglement”. In the figure below, we can see 9 states in a domino notation. Alice and Bob both have a qutrit and they can be in one of the 9 states. They are each a bipartite, separable state and mutually orthogonal to each other. However, in order to distinguish them, one has to do a non-local measurement. To create the states we have to use Separable channels. Let ψi\ket{\psi_i} denote the 9 states. Then, the measurement that can distinguish these states can be represented by the Kraus representation S(ρ)=iSiρSiS(\rho)=\sum_i S_i \rho S_i^\dagger, where:

Si=iAiBψi. S_i = \ket{i}_A \ket{i}_B\bra{\psi_i}.

The 9 orthogonal states on the two-qutrit Hilbert space are all separable but only distinguishable by non-local measurements, and indistinguishable locally.

An intuition around how local measurements would act here is that they would always “cut” at least one of the dominoes “in half”, which would make those states indistinguishable. It is laboriously derived and carefully argued in (Bennett et al., 1999) that this measurement operator cannot be implemented by LOCC operations (however, it can be prepared locally!), on the other hand, it is clearly a separable channel. Thus, LOCCSEPLOCC \subset SEP.

The seminal work from Bennett et al. also quantified the gap, but only to the order of 10610^{-6}. A central objective of the approach by (Chitambar et al., 2012) was to quantify this gap and define a measure of entanglement that can have a significant difference between the two classes. They managed to show a 12.5% gap using a measure that also has an operational interpretation.

Definition Suppose we have an arbitrary state ρDO(H)\rho \in DO(\mathcal{H}), and a set of LOCC transformations, each executed with probability pip_i, converting ρ\rho into ρi\rho_i. Then, we call a function μ:DO(H)R\mu: DO(\mathcal{H}) \rightarrow \mathbb{R} an entanglement monotone, when it satisfies the following inequality:

μ(ρ)ipiμ(ρi) \mu(\rho) \leq \sum_i p_i \mu(\rho_i)

Chitambar et al. defined a monotone that measures the success rate of a particular task: random-EPR pair distillation from W-class states. W-class states are any state that is obtainable from the W=13(100+010+001)\ket{W}=\frac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001}) state with LOCC operations and can be parameterized as x0000+xA100+xB010+xC001\sqrt{x_0}\lvert 000\rangle + \sqrt{x_A}\lvert 100\rangle + \sqrt{x_B}\lvert 010\rangle + \sqrt{x_C}\lvert 001\rangle. Due to normalization x0x_0 is fully determined by 1=x0+xA+xB+xC1=x_0+x_A+x_B+x_C, thus we will label W-class states as x={xA,xB,xC}\vec{x}=\{x_A,x_B,x_C\}. In random-EPR pair distillation, one uses a protocol to transform an initial W-class state to random singlet states ψij=12(01ij+10ij)\ket{\psi}_ij=\frac{1}{\sqrt{2}}(\lvert 01\rangle_{ij} + \lvert 10\rangle_{ij}) between any two of the 3 parties as shown in the figure below.

Random-EPR pair distillation from a W-class state.

For x={xA,xB,xC}\vec{x}=\{x_A,x_B,x_C\}, let {n1,n2,n3}\{n_1,n_2,n_3\} be a labeling of {A,B,C}\{A,B,C\} so that xn1xn2xn3x_{n_1} \geq x_{n_2} \geq x_{n_3}. Then, the function

κ(x):=2(xn2+xn3)xn2xn3xn1 \kappa(\vec{x}) := 2(x_{n_2} + x_{n_3}) - \frac{x_{n_2}x_{n_3}}{x_{n_1}}

is an entanglement monotone that strictly decreases on average when a party with maximum component value makes a measurement by Theorem 1 in (Chitambar et al., 2012).

The function κ(x)\kappa(\vec{x}) can reach 1 if and only if xn1=xn2x_{n_1}=x_{n_2} and x0=0x_0=0, and thus pAB+pBC+pACκ(x)p_{AB}+p_{BC}+p_{AC}\leq \kappa(\vec{x}) for any x\vec{x} - thus we have an upper bound on the total probability of the success rate for the random-EPR distillation of x\vec{x}. Interestingly, by Theorem 2. in (missing reference), for a class of xn3>x0=0x_{n_3} \gt x_0=0 states, the success rate of LOCC operations can get arbitrarily close to this bound, for any ϵ>0\epsilon\gt0, pAB+pBC+pAC>κ(x)ϵp_{AB}+p_{BC}+p_{AC}\gt\kappa(\vec{x})-\epsilon, but no LOCC operation can exactly reach κ(x)\kappa(\vec{x}).

The situation is different for SEP channels as there exists a set of separable measurement operators that can randomly distill a three-qubit W-class state with x0=0,1/2xAxBxCx_0=0, 1/2 \geq x_A \geq x_B \geq x_C with probability one:

MAB=(λCB00+11)(λCA00+11)00MAC=(λBC00+11)00(λBA00+11)MBC=00(λAC00+11)(λAB00+11)M000=(1λCA2λCB2λBA2λBC2λAB2λAC2)1/2000000M111=111111 M_{AB} = (\lambda_{CB} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes (\lambda_{CA} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes \ket{0}\bra{0} \\ M_{AC} = (\lambda_{BC} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes \ket{0}\bra{0} \otimes (\lambda_{BA} \ket{0}\bra{0} + \ket{1}\bra{1}) \\ M_{BC} = \ket{0}\bra{0} \otimes (\lambda_{AC} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes (\lambda_{AB} \ket{0}\bra{0} + \ket{1}\bra{1}) \\ M_{000} = (1-\lambda_{CA}^2\lambda_{CB}^2-\lambda_{BA}^2\lambda_{BC}^2-\lambda_{AB}^2\lambda_{AC}^2)^{1/2}\ket{000}\bra{000} \\ M_{111} = \ket{111}\bra{111}

, where λij=12xi2xj\lambda_{ij}=\sqrt{\frac{1-2x_i}{2x_j}}.

The example where the 12.5% difference shows up is the following family of states parameterized by 13s12\frac{1}{3} \leq s \leq \frac{1}{2}:

s100+1s2(010+001) \sqrt{s} \ket{100} + \sqrt{\frac{1-s}{2}}(\ket{010}+\ket{001})

While the SEP probability is 1, the LOCC probability is bound by κ(x)=2(1s)(1s)24\kappa(\vec{x}) = 2(1-s)-\frac{(1-s)^2}{4}. We plot the two functions in the figure below, where the separation is clearly visible.

Maximum probability of generating EPR pairs from a parameterized W-class state.

Stabilizer formalism: SOn(d)CSPSO_n(d) \subset CSP

The operationally defined set of free operations in the stabilizer formalism is the Stabilizer Operators. We will formally define them here. All the above concepts in Fundamental concepts including the Pauli group Pn(d)\mathcal{P}_n(d), the stabilizer group SS, the Clifford group Cln(d)Cl_n(d), and stabilizer states STAB(d,nd,n) readily generalize from qubits to qudits as described in (Heimendahl et al., 2022). Our formal definitions then will be stated using qudits:

Definition A quantum channel mapping nn dd-qudits to mm dd-qudits, for prime dimension dd, is defined as a Stabilizer operation when it is a concatenation of operations that fall into one of the following categories:

Beyond the above, arbitrary random functions of previous measurement results can act as classical logic to decide which quantum operation to do in each step.

We are taking the stabilizer polytope for nn-qudits of dd dimension, SPn(d)SP_n(d) to be the set of free states. Then, the axiomatically defined RNG operations are the ones that map the stabilizer-polytope to itself. Similar to positivity, in some resource theories, these channels are not closed under tensor product. In order to enforce closeness under tensor product, the set of completely-RNG channels are of interest - in our case the Completely Stabilizer Preserving (CSP) channels, which - for any arbitrary kk dimensional extension, maps the n+kn+k dimensional stabilizer polytope to the m+km+k dimensional stabilizer polytope:

Definition A linear superoperator E\mathcal{E} mapping nn and mm qudit spaces of dd dimension is Completely Stabilizer Preserving if and only if Eidk(SPn+k(d))SPm+k(d)\mathcal{E} \otimes id_k (SP_{n+k}(d)) \subset SP_{m+k}(d) for all kNk \in \mathbb{N}.

A CSP operator is also CPTP (Heimendahl et al., 2022), thus they are legitimate quantum channels. How can we determine whether a stabilizer-preserving channel E\mathcal{E} is also CSP? An interesting result (Seddon & Campbell, 2019) utilizes the Choi operator to prove that if the Choi operator (which can be looked at as a state) of E\mathcal{E} is stabilizer-preserving and trace-preserving, then E\mathcal{E} is CSP.

With the definitions out of the way, let’s look at the main result by Heimendahl et al. For the single qudit (qubit) case, SO1(d)=CSP1(d)SO_1(d)= CSP_1(d). This is intuitively reasonable, as However, starting from two qubits, CSP is strictly larger than SO. Heimendahl et al used the following channel as a counterexample to prove the separation between SO2(2)SO_2(2) and CSP2(2)CSP_2(2):

Λ(ρ)=ρ00+++++x01,10,11ρx,xxx+12x,y01,10,11xyρx,yxy \Lambda(\rho) = \rho_{00} \ket{++}\bra{++} + \sum_{x \in \\{01, 10, 11\\}} \rho_{x,x} \ket{x}\bra{x} + \frac{1}{2} \sum_{\substack{x,y \in \\{01, 10, 11\\} \\ x\neq y}} \rho_{x,y}\ket{x}\bra{y}

, where ρx,y=xρy\rho_{x,y}=\bra{x} \rho \ket{y}.

This channel can be written as the following three operations:

  1. measure a projective measurement that distinguishes between 00\ket{00} and the subspace spanned by 01,10,11\ket{01},\ket{10},\ket{11}, i.e. 00\ket{00}^\perp. This operation cannot be implemented using stabilizer operations. This makes the density matrix block-diagonal:
  2. (ρ000000ρ01,01ρ01,10ρ01,110ρ10,01ρ10,10ρ10,110ρ11,01ρ11,10ρ11,11) \begin{pmatrix} \rho_{00} & 0 & 0 & 0 \\ 0 & \rho_{01,01} & \rho_{01,10} & \rho_{01,11} \\ 0 & \rho_{10,01} & \rho_{10,10} & \rho_{10,11} \\ 0 & \rho_{11,01} & \rho_{11,10} & \rho_{11,11} \end{pmatrix}
  3. Apply partial dephasing with probability 1/2. Off-diagonal terms become half their size:
  4. (ρ000000ρ01,01ρ01,10/2ρ01,11/20ρ10,01/2ρ10,10ρ10,11/20ρ11,01/2ρ11,10/2ρ11,11) \begin{pmatrix} \rho_{00} & 0 & 0 & 0 \\ 0 & \rho_{01,01} & \rho_{01,10}/2 & \rho_{01,11}/2 \\ 0 & \rho_{10,01}/2 & \rho_{10,10} & \rho_{10,11}/2 \\ 0 & \rho_{11,01}/2 & \rho_{11,10}/2 & \rho_{11,11} \end{pmatrix}
  5. Depending on the outcome of the measurement, apply Hadamard on both qubits if 00\ket{00} was observed. At this point, we get the expression for Λ(ρ)\Lambda(\rho).

It is possible to show that all three steps are necessary for the channel to be CSP but not SO. In a bit more intuitive explanation, in the geometry of the space of channels, after the first step, we are outside of the CSP class (which is also a polytope). The dephasing step removes sufficient “magic” from the channel to make it CSP and the Hadamard at the end sets the direction the CSP polytope is being approached from to make the channel a vertex.

We will introduce the notion of a Clifford dilation for the sketch of the proof. A superoperator E\mathcal{E} mapping nn and mm qudit spaces of dd dimension has a Clifford dilation if there exists a number kk, a stabilizer state sSPk(d)\ket{s} \in SP_k(d) and a Clifford unitary UU on n+kn+k qudits such that: E(ρ)=Trm+1,...n+kU(ρss)U\mathcal{E}(\rho) = Tr_{m+1,...n+k}\\{U(\rho \otimes \ket{s}\bra{s})U^\dagger\\}

The channels that have a Clifford dilation are the ones that have no measurements or classical randomness.

The sketch of the proof is the following:

  1. it is shown that Λ\Lambda is an almost-diagonal channel, meaning that it belongs to AD2AD_2 that is defined by its effect on the computational basis states:
  2. E(0000)=++++,E(xx)=xx,x00 \mathcal{E}(\ket{00}\bra{00})=\ket{++}\bra{++}, \mathcal{E}(\ket{x}\bra{x})=\ket{x}\bra{x}, x \in \ket{00}^\perp
  3. This property is then used to show that Λ\Lambda is extremal in CS2CS_2. From here, if Λ\Lambda is a stabilizer operation, then it would follow that it is an extremal point in SO2SO_2 as well. This is due to convex subsets inheriting the extremality of points from their supersets.
  4. It is shown that extremal points in SO2SO_2 do not have a Clifford dilation.
  5. However, it can be proved that Λ\Lambda cannot have a Clifford dilation - thus it must be that ΛCSP2\Lambda \in CSP_2 but ΛSO2\Lambda \notin SO_2

For larger qubit and qudit dimensions we refer the reader to (Heimendahl et al., 2022). For the full picture though, it is worth mentioning that for larger dimensional qudit states even CSPn(d)CSP_n(d) might not be maximal in terms of free operations. For dd odd dimensions, resource theories of magic use the Wigner function representation of a function. This is a quasi-probability distribution in phase-space and its total negativity is a magic monotone called mana. Thus, the set of free states can be enlarged to include those with positive Wigner functions, and channels that preserve the positivity of the Wigner function and do so closed under the tensor product are called the completely-Wigner-positivity-preserving (CWPPn,m(d)CWPP_n,m(d)) channels. Thus in case of odd d>2d\gt2 dimensions, the full hierarchy is SOn,m(d)CSPn,mCWPPn,m(d)SO_{n,m}(d) \subset CSP_{n,m} \subseteq CWPP_{n,m}(d), where the last containment is conjectured to be proper by Heimendahl et al.


Starting from the basics, we reviewed how the structure of the free operations in the theories of entanglement and magic are structured. To explore entanglement, we looked at an early example by Bennett et al. that demonstrated measurement that is separable but not LOCC. Following Chitambar et al., we explored an entanglement monotone specifically designed to measure the maximum success of random-EPR distillation from W-class states, and how it can show a 12.5% gap between LOCC and SEP.

For the theory of magic, Heimendahl et al. did not work with magic monotones, instead, for 2-qubit states, they showed through a specific counter-example that is a completely stabilizer preserving operation but not a stabilizer operation. One could say that their result is similar in spirit to the Bennett et al. example, where careful arguments around a specific set of counterexamples demonstrate the separation of the two classes. Future work might explore actually quantifying the gap in this theory as well for magic distillation tasks. Further implications of the CSP / SO separation could be that the classically simulatable class of channels will become larger, beyond the Gottesman-Knill theorem.


Comments are provided by giscus and Github Discussions. You will have to login with your Github account to comment.

  1. Chitambar, E., & Gour, G. (2019). Quantum Resource Theories. Reviews of Modern Physics, 91(2), 025001. https://doi.org/10.1103/RevModPhys.91.025001
  2. Wilde, M. (2017). Quantum information theory (Second edition). Cambridge University Press.
  3. Gottesman, D. (1997). Stabilizer Codes and Quantum Error Correction. ArXiv:Quant-Ph/9705052. http://arxiv.org/abs/quant-ph/9705052
  4. Bravyi, S., & Kitaev, A. (2005). Universal Quantum Computation with ideal Clifford gates and noisy ancillas. Physical Review A, 71(2), 022316. https://doi.org/10.1103/PhysRevA.71.022316
  5. Chitambar, E., Cui, W., & Lo, H.-K. (2012). Increasing Entanglement Monotones by Separable Operations. Physical Review Letters, 108(24), 240504. https://doi.org/10.1103/PhysRevLett.108.240504
  6. Heimendahl, A., Heinrich, M., & Gross, D. (2022). The axiomatic and the operational approaches to resource theories of magic do not coincide. Journal of Mathematical Physics, 63(11), 112201. https://doi.org/10.1063/5.0085774
  7. Aaronson, S., & Gottesman, D. (2004). Improved simulation of stabilizer circuits. Physical Review A, 70(5), 052328. https://doi.org/10.1103/PhysRevA.70.052328
  8. Bennett, C. H., DiVincenzo, D. P., Fuchs, C. A., Mor, T., Rains, E., Shor, P. W., Smolin, J. A., & Wootters, W. K. (1999). Quantum nonlocality without entanglement. Physical Review A, 59(2), 1070–1091. https://doi.org/10.1103/PhysRevA.59.1070
  9. Seddon, J. R., & Campbell, E. T. (2019). Quantifying magic for multi-qubit operations. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2227), 20190251. https://doi.org/10.1098/rspa.2019.0251