QQuantifying the Imaginarity of Quantum Mechanics
Alexander Hickey and Gilad Gour Department of Mathematics and Statistics, University of Calgary,2500 University Drive NW, Calgary, Alberta, Canada T2N 1N4 (Dated: August 14, 2018)The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity,where the free states are taken to be those with density matrices that are real with respect to afixed basis. This theory is closely related to the resource theory of coherence, as it is basis depen-dent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlikecoherence however, the set of physically realizable free operations is identical to both completelyresource non-generating operations, and stochastically resource non-generating operations. More-over, the resource theory of imaginarity does not have a self-adjoint resource destroying map. Afterintroducing and characterizing the free operations, we provide several measures of imaginarity, andgive necessary and sufficient conditions for pure state transformations via physically consistent freeoperations in the single shot regime.
I. INTRODUCTION
Quantum Resource Theories (QRTs) revolutionize the way think about familiar topics like entanglement, coherence,and symmetry. They provide an operational way to quantify and manipulate a given property (resource) of a quantummechanical system, and determine the information processing tasks that are attainable by consuming this resource.A well known example of a resource theory is entanglement theory [1], where the consumption of entanglement leadsto protocols such as superdense coding [3] and quantum teleportation [2]. In recent years, there has been interestin developing resource theories for many other non-classical properties, such as resource theories of coherence [4–9],athermality [10–14], asymmetry [15–18], knowledge [19], magic [20–22], steering [33], nonlocality [34], contextuality[35], and superposition [23]. Furthermore, there has been interest in developing mathematical frameworks that applyto large classes of resource theories [24, 26, 27, 36].In general, a resource theory is completely characterized by physical restrictions on the set of possible transfor-mations [26, 27, 36]. These restrictions give rise to a set of free states F , and a set of free operations , O , whichdo not generate a resource. It is therefore a minimal requirement that the free operations acts invariantly on F .Often times the physical restrictions lead to a much smaller set of free operations than just those that are resourcenon-generating (RNG). For example, Local Operations and Classical Communication (LOCC) in the resource theoryof entanglement, do not contain all operations that do not generate entanglement when acting on separable states.The physical restriction to LOCC leads to a much smaller class of operations than the full set of non-entanglingoperations. Yet, due to the notorious complexity of LOCC operations, other subsets of non-entangling operations arefound to be useful, especially those which have a nice mathematical structure, such as PPT operations and separableoperations (SEP) in entanglement theory [5].A key feature of quantum mechanics is the necessity of imaginary numbers to accurately model the dynamics of aphysical system. Although imaginary numbers have long been used in classical physics to simplify models of oscillatorymotion and wave mechanics, it seems that they play a much deeper role in quantum physics as they are intrinsic toany orthodox formulation [37–39]. Consider, for example, the polarization density matrix of a single photon in the {| H (cid:105) , | V (cid:105)} basis. The imaginary numbers in the density matrix gives rise to rotations of the electric field vector,i.e. elliptical or circular polarization. Motivated by this unique characteristic of quantum theory, we propose here aresource theory of imaginarity in which the resource arises from imaginary terms that appear in the density matrixof a state. With such a resource at hand, real quantum mechanics can achieve the same tasks as standard quantummechanics.Such a resource theory is interesting for several reasons. The resource of imaginarity is closely related to quantumcoherence as the imaginary terms of a density matrix always appear as off-diagonal elements. This relaxation ofonly considering the imaginary part of the off-diagonal elements provides a characterization of how relative phasesbetween the measurement basis states effect the underlying dynamics of a system, as opposed to the total interferencebetween basis states as is the case with coherence. From a conceptual standpoint, this theory may help to quantifythe seemingly fundamental role that imaginary numbers play in quantum theory. In addition, a characterization ofthis theory is of interest as it is a part of a large class of QRTs known as affine resource theories, which also includescoherence theory. Recently, work has been done in developing theorems which can be applied to a general affineresource theory, and currently the resource theory of imaginarity is the only known affine resource theory which does a r X i v : . [ qu a n t - ph ] A ug not have a self-adjoint resource destroying map [27].This paper is structured as follows. First we formally define free states and completely characterize the largestpossible class of free operations. Next we discuss notions of physical consistency and show that by further restrictingour free operations, we obtain a class of operations which simultaneously amends multiple physical inconsistencies(something that does not happen in the resource theory of coherence). We will then introduce the notion of ameasure of imaginarity and propose several measures. Finally, we will discuss pure state transformations in thesingle-shot regime via free operations, where we will give necessary and sufficient conditions for the existence of sucha transformation. II. FREE STATES
Analogous to coherence, imaginarity is intrinsically a basis dependent phenomena. Before defining our free states, wemust therefore fix an orthonormal reference basis {| j (cid:105)} d − j =0 for our d -dimensional Hilbert space H . Such a requirementis not a detriment to the theory however, as in general, one fixes an experimental setup corresponding to someprojective basis measurement prescribed by the physics of the system of interest. In the case of a composite statespace H A ⊗H B , we reference the canonical choice of basis {| j (cid:105) A ⊗ | k (cid:105) B } where the k index is iterated first. Throughoutthis paper we will use H to denote a d -dimensional Hilbert space, L ( H ) to denote the set of linear operators on H and D ( H ) to denote the set of density operators acting on H , which we can represent as matrices with respect to thefixed reference basis. Definition 1.
Let ρ be a density operator on a d -dimensional Hilbert space H . We say that ρ is real or free if ρ = (cid:88) jk ρ jk | j (cid:105)(cid:104) k | where each ρ jk ∈ R . We denote the set of all real density operators by F . Note that since density matrices are Hermitian, conjugation is equivalent to transposition. It therefore follows that ρ ∈ F if and only if ρ is a symmetric matrix, that is, ρ T = ρ . It is clear that F is convex, as any convex combinationof real density operators will remain a real density operator. Moreover, F is also affine [27], that is, any densityoperator that can be expressed as an affine combination of real density operators is itself real. III. FREE OPERATIONS
In some resource theories, the set of free operations does not correspond to physical or experimental restrictions onthe set of possible operations. Instead, one is interested in studying a certain property of a physical system and usesa resource theoretic framework to do so. For example, the resource theory of coherence is a state-centred theory, inwhich the set of free states (incoherent states) is well defined whereas the choice of free operations is not unique. Infact, there is some controversy as to which set of free (incoherent) operations is most physical [5, 6].The minimal condition on any set of free operations is that a free operation can only map a free state to a free state.This ensures that the set of free operations do not generate resources from free states. The largest possible set of freeoperations in any conceivable resource theory is therefore the resource non-generating (RNG) operations. As we willdiscuss below however, RNG operations are typically too large in the sense that they are not physically consistent.Yet in order to derive a more physically consistent QRT of imaginarity, we start with the following characterizationof RNG operations.
Theorem 1.
Let E : L ( H B ) → L ( H A ) be a quantum channel and J = (cid:88) jk E ( | j (cid:105)(cid:104) k | ) ⊗ | j (cid:105)(cid:104) k | the Choi matrix of E acting on H A ⊗ H B . Then E is resource non-generating if and only if J − J Γ A is symmetric.Here Γ A denotes the partial transpose with respect to the Hilbert space H A .Proof. Suppose E is RNG, then for any ρ ∈ F we have that [ E ( ρ )] T = E ( ρ ). Rewriting this condition in the Choirepresentation we get that Tr B (cid:16)(cid:2) J ( I ⊗ ρ T ) (cid:3) Γ A (cid:17) = Tr B (cid:0) J Γ A ( I ⊗ ρ T ) (cid:1) = Tr B (cid:0) J ( I ⊗ ρ T ) (cid:1) . So we have Tr B (cid:2) ( J − J Γ A )( I ⊗ ρ T ) (cid:3) = 0and thus Tr (cid:2) ( J − J Γ A )( X ⊗ ρ ) (cid:3) = 0for any X ∈ L ( H A ) and symmetric ρ . We may view this as an orthogonality condition with respect to the Hilbert-Schmidt inner product, and conclude that J − J Γ A ∈ L ( H A ) ⊗ K B , where K B is the space of antisymmetric operators in L ( H B ). This in particular implies that (cid:0) J − J Γ A (cid:1) Γ B = J Γ A − J , so that (cid:0) J − J Γ A (cid:1) T = (cid:16)(cid:2) J − J Γ A (cid:3) Γ B (cid:17) Γ A = (cid:0) J Γ A − J (cid:1) Γ A = J − J Γ A . For the converse, suppose J − J Γ A is symmetric. Then J − J Γ A = (cid:0) J − J Γ A (cid:1) T = J T − J Γ B and so (cid:0) J − J Γ A (cid:1) Γ B = ( J T − J Γ B ) Γ B = − (cid:0) J − J Γ A (cid:1) . Thus E ( ρ ) = Tr B (cid:2) J ( I ⊗ ρ T ) (cid:3) = Tr B (cid:2) J Γ A ( I ⊗ ρ T ) (cid:3) + Tr B (cid:2)(cid:0) J − J Γ A (cid:1) ( I ⊗ ρ T ) (cid:3) = [ E ( ρ )] T + Tr B (cid:2)(cid:0) J − J Γ A (cid:1) ( I ⊗ ρ T ) (cid:3) . Finally, note that Tr B (cid:2)(cid:0) J − J Γ A (cid:1) ( I ⊗ ρ T ) (cid:3) = Tr B (cid:104)(cid:0) J − J Γ A (cid:1) Γ B ( I ⊗ ρ T ) Γ B (cid:105) = − Tr B (cid:2)(cid:0) J − J Γ A (cid:1) ( I ⊗ ρ ) (cid:3) and so Tr B (cid:2)(cid:0) J − J Γ A (cid:1) ( I ⊗ ρ T ) (cid:3) = 0 whenever ρ is symmetric, therefore E ( ρ ) = [ E ( ρ )] T whenever ρ ∈ F . A. Physical Constraints on the Free Operations
The classification of the complete set of RNG operations is of interest as it was shown in [24] that a QRT isasymptotically reversible if its set of free operations is maximal. We will however, concern ourselves with a restrictedclass of free operations based on the ground of physical consistency. In what follows, we define three types of such freeoperations and then show that they are all identical. We start by showing that there are RNG operations as definedin Theorem 1 which can generate resources when applied to part of a larger composite system.Theorem 1 classifies an arbitrary RNG operation with the condition that the Choi matrix must satisfy a sym-metry constraint upon taking a partial transpose. It is straightforward to construct (even in the qubit case) aChoi matrix J which satisfies the condition in Theorem 1 but is not itself symmetric. On the other hand, we have d J = E ⊗ id d ( | φ + (cid:105)(cid:104) φ + | ) where | φ + (cid:105) = √ d (cid:80) j | jj (cid:105) is the normalized maximally entangled state. It therefore followsthat | φ + (cid:105)(cid:104) φ + | ∈ F but E ⊗ id d (cid:0)(cid:12)(cid:12) φ + (cid:11)(cid:10) φ + (cid:12)(cid:12)(cid:1) = 1 d J (cid:54)∈ F ;i.e. E is a RNG operation that generates a resource when applied to part of a maximally entangled state. Thisinconsistency can be rectified by requiring the free operations to be closed under tensor products, a property whichwe will call completely resource non-generating . Definition 2.
A RNG operation, E , is said to be completely resource non-generating if E ⊗ id k is resource non-generating for all k ∈ N . This definition is analogous to the class of completely non-entangling operations introduced in [25], and ensuresthat resources can never be generated by applying a free operation to part of a larger composite system.One may also impose the requirement that resources cannot be generated stochastically, i.e. with probability lessthan one. This leads to an alternate class of free operations which we call stochastically resource non-generatingoperations.
Definition 3.
A quantum operation E is said to be stochastically RNG if it has an operator sum representation withKraus operators { K j } mj =1 such that K j ρK † j Tr( K j ρK † j ) ∈ F for all j whenever ρ ∈ F . This class of free operations ensures that resources cannot be generated probabilistically, and is precisely howincoherent operations (IO) are defined in the resource theory of coherence [4].Furthermore, it is argued in [5, 6] that for a resource theory to be physically realizable, the set of free operationsshould admit a free dilation. This meaning that the operation can be implemented via free unitaries acting on somelarger composite system.
Definition 4.
A unitary U is said to be a free unitary if U ρU † ∈ F whenever ρ ∈ F . Definition 5.
A RNG operation E is said to be physically realizable if it admits a free dilation. That is, there existsa Hilbert space H E , a free state | (cid:105)(cid:104) | E ∈ H E , and a free unitary U AE acting on the joint state space H A ⊗ H E suchthat E ( ρ ) = Tr E (cid:104) U AE ( ρ ⊗ | (cid:105)(cid:104) | E ) U † AE (cid:105) for any density operator ρ ∈ H . In general, the class of physically realizable operations can be much smaller than the class of all RNG operations(see Fig. 3). However, recall that a CPTP map (and a generalized measurement) provides only an effective descriptionof the evolution of a physical system. Basically, the evolution of a closed system is always unitary. Therefore, inorder to implement a free CPTP map, one has to first implement a free unitary on the system+ancilla and then traceout the ancillary system. This means that many CPTP maps that are RNG, and even completely RNG, cannot beimplemented freely, and in this sense are not really free.It follows from definition 5 that the class of free unitaries must be characterized in order to implement a physicallyrealizable operations. This is done in the following lemma.
Lemma 1.
A unitary matrix U is a free unitary if and only if U = e iθ Q for some θ ∈ [0 , π ) and real orthogonalmatrix Q .Proof. Let U be a free unitary, then for any ρ ∈ F we have that (cid:2) U ρU † (cid:3) T = U ρU † and so ρ = ( U T U ) ρ ( U T U ) † where U T U is also unitary. This equation must hold for all symmetric ρ so for each j = 0 , , ..., d − U T U | j (cid:105)(cid:104) j | ( U T U ) † = ( U T U | j (cid:105) )( U T U | j (cid:105) ) † = | j (cid:105)(cid:104) j | and thus U T U | j (cid:105) = e iθ j | j (cid:105) for some θ j ∈ [0 , π ). Similarly, for each k = 1 , , ..., d − j < k : | j (cid:105)(cid:104) k | + | k (cid:105)(cid:104) j | = U T U ( | j (cid:105)(cid:104) k | + | k (cid:105)(cid:104) j | )( U T U ) † = e i ( θ j − θ k ) | j (cid:105)(cid:104) k | + e i ( θ k − θ j ) | k (cid:105)(cid:104) j | FIG. 1: A Venn diagram showing some possible choices of free operations. Resource non-generating (RNG)operations form the largest possible set of free operations within the set of all CPTP maps. Completely RNGoperations ensure that the operations are closed under tensor products. Stochastically RNG operations ensure thatresources cannot be generated probabilistically. Physically realizable operations are those which can be implementedat no cost.therefore θ j = θ k for each j, k and thus θ = θ = ... = θ d − =: θ . It follows that U T U = e iθ d − (cid:88) j =0 | j (cid:105)(cid:104) j | = e iθ I. Define Q := e − iθ/ U , then Q T Q = I and Q † Q = I so Q is real and orthogonal. The converse is trivial as clearly[ U ρU † ] T = Qρ T Q T ∈ F whenever ρ ∈ F .It follows from Lemma 1 that the set of free unitaries in the resource theory of imaginarity differs quite significantlyfrom the free unitaries in coherence theory. In particular, the most general unitary which does not generate coherencetakes the form U = (cid:80) j e iθ j | π ( j ) (cid:105)(cid:104) j | where π is some permutation on the set of indices { , , ..., d − } [5, 6]. So whilethe resource of imaginarity is preserved under any orthogonal basis transformation, coherence is only preserved underpermutations of the basis elements. In this sense, the resource theory of imaginarity contrasts from coherence theoryin the sense that some of the basis dependence has been alleviated.It turns out that each of the possible physical constraints that were discussed corresponds to the same set of freeoperations, forming an equivalence class of operations that we call physically consistent. We present this fact in themain result of this section. Theorem 2.
Let E be a quantum channel, then the following are equivalent:1. E is completely resource non-generating.2. The Choi matrix of E is real.3. E is stochastically resource non-generating.4. E is physically realizable.Proof. For (1) = ⇒ (2), suppose E ⊗ id k is RNG for all k ∈ N . Fix k = d and let | φ + (cid:105) = √ d (cid:80) j | jj (cid:105) , then | φ + (cid:105)(cid:104) φ + | ∈ F and so E ⊗ id d ( (cid:12)(cid:12) φ + (cid:11)(cid:10) φ + (cid:12)(cid:12) ) ∈ F . Thus the Choi matrix is real.FIG. 2: Each of the physical constraints leads to an equivalent class of free operations in the resource theory ofimaginarity.For (2) = ⇒ (3), suppose that the Choi matrix J is real, then J has a spectral decomposition J = (cid:88) j λ j | v j (cid:105)(cid:104) v j | where {| v j (cid:105)} is a set of real orthonormal eigenvectors of J . A set of Kraus operators { K j } can then be obtained byinverting the vectorization operation on each of the eigenvectorsvec( K j ) = (cid:112) λ j | v j (cid:105) [29]where the vectorization operator is an isomorphism defined in the usual way: vec( | i (cid:105)(cid:104) j | ) := | j (cid:105)⊗| i (cid:105) . Since J is positive,each (cid:112) λ j is real and thus { K j } is a set of real Kraus operators.For (3) = ⇒ (4), suppose { K j } Nj =1 is a real set of Kraus operators, then we may construct a unitary U AE actingon the state space along with some ancillary system (just as in definition 5) where U AE satisfies K j = (cid:104) j | E U AE | (cid:105) E for each j . Since each K j is real, we may restrict the column space of U AE to R N and pick the remaining columns ofthe matrix to be orthogonal. The matrix U AE will then be real and orthogonal, and E ( ρ ) = (cid:88) j K j ρK † j = (cid:88) j (cid:104) j | E U AE (cid:104) | (cid:105) E ρ (cid:104) | E (cid:105) U † AE | j (cid:105) E = (cid:88) j (cid:104) j | E U AE (cid:104) ρ ⊗ | (cid:105)(cid:104) | E (cid:105) U † AE | j (cid:105) E = Tr E (cid:104) U AE ( ρ ⊗ | (cid:105)(cid:104) | E ) U † AE (cid:105) Therefore E admits a free dilation and thus E ∈ O .For (4) = ⇒ (1), suppose E ( ρ ) = Tr E (cid:104) U AE ( ρ ⊗ | (cid:105)(cid:104) | E ) U † AE (cid:105) for some real orthogonal matrix U AE and consider a composite state space H ⊗ H B where dim H B = k and some jointfree state σ ∈ D ( H ⊗ H B ) ∩ F . Then σ = (cid:80) j(cid:96) r j(cid:96) A j ⊗ B (cid:96) for some r j(cid:96) ∈ R and real matrices A j and B (cid:96) . Thus E ⊗ id k ( σ ) = E ⊗ id k (cid:88) j(cid:96) r j(cid:96) A j ⊗ B (cid:96) = (cid:88) j(cid:96) r j(cid:96) E ( A j ) ⊗ B (cid:96) = (cid:88) j(cid:96) r j(cid:96) Tr E (cid:104) U AE ( A j ⊗ | (cid:105)(cid:104) | E ) U † AE (cid:105) ⊗ B (cid:96) where U AE is a real matrix. Therefore E ⊗ id k is resource non-generating for any k ∈ N .Taking our set of free operations to be those with real Choi matrices fixes each of the previously discussed physicalinconsistencies simultaneously. This is in contrast to coherence theory, where the physically realizable operations(PIO) are a strict subset of those which are stochastically resource non-generating (IO) [5]. We note that the varioussets of RNG operations listed in Theorem 2 are analogous to the usual representations of quantum channels. Inparticular, we see that if the set of all CPTP maps are restricted to just the completely RNG operations, that theseoperations admit a Stinespring dilation, as well as operator-sum and Choi representations which are valid when onerestricts the underlying Hilbert space to the real numbers. The set of physically consistent free operations thereforeprovide a characterization of channels in real quantum mechanics which largely resemble the standard theory of CPTPmaps.It is clear that the set of physically consistent operations is convex. Furthermore, this class of operations admit anice algebraic structure, which we call transposition covariance . Corollary 1.
A quantum channel E is physically consistent if and only if it commutes with the transposition map.That is, E ( ρ ) T = E ( ρ T ) for all ρ ∈ D ( H ) .Proof. Suppose E is physically consistent, then by theorem 2, the Choi matrix J of E is real and so J T = J . Then E ( ρ ) T = Tr B (cid:2) J Γ A ( I ⊗ ρ T ) (cid:3) = Tr B (cid:2) J T ( I ⊗ ρ ) (cid:3) = Tr B [ J ( I ⊗ ρ )]= E ( ρ T )where the second equality comes from the fact that the partial transpose is self-adjoint under the trace. For theconverse, note that E ( ρ ) T = Tr B [ J Γ A ( I ⊗ ρ T )] = Tr B [ J T ( I ⊗ ρ )]and so 0 = E ( ρ T ) − E ( ρ ) T = Tr B (cid:2) ( J − J T )( I ⊗ ρ ) (cid:3) ∀ ρ ∈ D ( H ) , so that J = J T .The definition of Transposition Covariant Operations is analogous to the set of Dephasing-Covariant IncoherentOperations (DIO) in coherence theory [5], as both the transposition and completely dephasing maps act as theidentity on the set of real and incoherent states respectively. Motivated by the physical consistency of this versatileclass of operations, we henceforth restrict our definition of free operations to those which are physically consistent, ascharacterized in theorem 2. F D ( H ) ρ M ( ρ ) FIG. 3: The measure introduced in theorem 3 is interpreted as the trace distance to the set of real states.
IV. MEASURES OF IMAGINARITY
Next we discuss the quantification of imaginarity, which is necessary in determining the degree of resourcefulnessof a given state. This will act as a first step in understanding the extent to which a resource may be used to simulatenon-free operations. Such an understanding is of particular interest in the resource theory of imaginarity, as it willallow us to gauge how well real quantum mechanics approximates the standard theory.
Definition 6.
A measure of imaginarity is a function M : D ( H ) → [0 , ∞ ) such that1. M ( ρ ) = 0 if ρ ∈ F M ( E ( ρ )) (cid:54) M ( ρ ) whenever E is a free operation. Furthermore, we say that a measure of Imaginarity M is faithful if M ( ρ ) = 0 = ⇒ ρ ∈ F , and that M is an imaginarity monotone if it does not increase on average under free operations. Condition 1 requires that any realstate will have no imaginarity. Condition 2 follows from the initial assumption that our free operations should notincrease the resource content of a state. In general, any measure of a resource should be monotonically non-increasingunder the physically free operations, but if the measure is monotonic under RNG, then the measure remains valid forany resource theory with the same set of free states [24].As discussed in [24], there are many choices of valid measures for an arbitrary convex resource theory, which areoften given by minimizing some quantity over a subset of linear operators. A couple of examples that will satisfydefinition 6 include:(i) The robustness of imaginarity R ( ρ ) = min π ∈D ( H ) (cid:26) s (cid:62) sπ + ρ s ∈ F (cid:27) . (ii) A distance measure M ( ρ ) = min σ ∈F C ( ρ, σ )where C is any contractive metric.We will focus our attention to a special case of example (ii), where we consider the contractive metric to be thetrace distance. Theorem 3.
Let ρ be a density matrix, then M ( ρ ) := min σ ∈ F (cid:107) ρ − σ (cid:107) = 12 (cid:13)(cid:13) ρ − ρ T (cid:13)(cid:13) is a faithful measure of imaginarity. Proof.
We will show that the trace distance is minimized by choosing σ = (cid:0) ρ + ρ T (cid:1) by showing that (cid:13)(cid:13) ρ − ρ T (cid:13)(cid:13) satisfies Definition 6. It is clear that M ( ρ ) = 0 if and only if ρ = ρ T since the trace norm is a norm on L ( H ). Toshow the monotonicity condition, recall that the trace norm is contractive under CPTP maps, so (cid:107)E ( ρ ) − E ( σ ) (cid:107) (cid:54) (cid:107) ρ − σ (cid:107) for any CPTP map E and ρ, σ ∈ D ( H ). Since ρ is Hermitian, we can decompose ρ = ρ R + iρ I where ρ R = ( ρ + ρ T )is real and symmetric and ρ I = i ( ρ − ρ T ) is real and antisymmetric. Next note thatTr ρ R = Tr ρ = 1and (cid:104) ψ | ρ R | ψ (cid:105) = 12 (cid:104) ψ | ρ | ψ (cid:105) + 12 (cid:104) ψ | ρ T | ψ (cid:105) (cid:62) | ψ (cid:105) ∈ H , therefore ρ R ∈ D ( H ) ∩ F . We can therefore pick σ = ρ R to get (cid:107) i E ( ρ I ) (cid:107) (cid:54) (cid:107) iρ I (cid:107) . Finally, note that if E is a free operation then E ( ρ I ) = 12 i (cid:2) E ( ρ ) − E ( ρ ) T (cid:3) = 12 i (cid:2) E ( ρ ) − E ( ρ T ) (cid:3) and so (cid:107)E ( ρ I ) (cid:107) = M ( E ( ρ )). Therefore M ( E ( ρ )) (cid:54) M ( ρ )As an example, consider the case where ρ is a qubit density matrix. Then ρ can be expanded in the Pauli basis ρ = 12 ( I + xσ x + yσ y + zσ z )where x, y, z ∈ R are the components of the Bloch vector of ρ [28]. Then ( ρ − ρ T ) = yσ y and thus M ( ρ ) = 12 (cid:13)(cid:13) ρ − ρ T (cid:13)(cid:13) = (cid:107) yσ y (cid:107) = 2 | y | = 2 | Tr( ρσ y ) | since the eigenvalues of σ y are ±
1. Therefore in the qubit case, the measure defined in theorem 3 reduces to twicethe y − component of the Bloch vector. V. STATE TRANSFORMATIONS
Next we will consider the problem of transforming resourceful states via free operations. That is, given tworesourceful states ρ and ρ (cid:48) , is there a free operation E ∈ O such that E ( ρ ) = ρ (cid:48) ? This is an important question in anyresource theory as it induces a hierarchy of resourcefulness to be used in information processing tasks. In particular,we will consider transformations between single copies of a pure state. To answer this question, it will be useful firstto determine a standard way to represent the resource of a given state. By manipulating resourceful states with freeunitary operations, one can ensure that the resource content does not change as the transformation is reversible andits inverse is also free. The following lemma presents a standardized form for any pure state. Lemma 2.
Let | ψ (cid:105) be a pure state in a d − dimensional Hilbert space with reference basis {| j (cid:105)} d − j =0 . Then there existsa free unitary U such that U | ψ (cid:105) ∈ span {| (cid:105) , | (cid:105)} .Proof. Decompose | ψ (cid:105) = a | ψ R (cid:105) + ib | ψ I (cid:105) , with | ψ R (cid:105) and | ψ I (cid:105) being two real normalized vectors, and a, b two positivereal numbers satisfying a + b = 1. Let O be an orthogonal real matrix satisfying O | ψ R (cid:105) = | (cid:105) . It follows that | ψ (cid:105) is resource equivalent to O | ψ (cid:105) = a | (cid:105) + ib | φ (cid:105) , where | φ (cid:105) ≡ O | ψ I (cid:105) is some real normalized vector. Next, denote | φ (cid:105) = cos( θ ) | (cid:105) + sin( θ ) | χ (cid:105) , where | χ (cid:105) is a real vector in the span of {| j (cid:105)} d − j =1 , and apply another orthogonal matrix0in the form O (cid:48) ≡ | (cid:105)(cid:104) | ⊕ ˜ O , where ˜ O is an orthogonal matrix in the d − {| j (cid:105)} d − j =1 ,satisfying ˜ O | χ (cid:105) = | (cid:105) . We therefore get that O (cid:48) O | ψ (cid:105) = ( a + ib cos( θ )) | (cid:105) + ib sin( θ ) | (cid:105) . Hence, any state | ψ (cid:105) can be converted reversibly by free operations into a pure qubit state in the span of {| (cid:105) , | (cid:105)} .We can thus represent the resource of an arbitrary pure state with a vector in the two dimensional subspace spannedby the first two reference basis vectors. However, there is still more that we can do: Lemma 3.
Let ρ be a qubit mixed state. Then there exists a × orthogonal matrix O such that OρO T = (cid:18) / x − iyx + iy / (cid:19) , where x and y are non-negative real numbers such that x + y (cid:54) / .Proof. Denote ρ = (cid:18) a c ¯ c − a (cid:19) with 0 (cid:54) a (cid:54) c a complex number with | c | (cid:54) a (1 − a ). Take O = (cid:18) cos α sin α − sin α cos α (cid:19) with tan(2 α ) = 2 a − c + ¯ c . Then, it is straight forward to check that
OρO T has 1/2 along the diagonal.Note that if ρ is a pure state then x + y = 1 /
4. We therefore conclude that any d -dimensional pure state | ψ (cid:105) canbe converted reversibly by free operations to a state of the form | θ (cid:105) ≡ √ (cid:0) | (cid:105) + e iθ | (cid:105) (cid:1) where x + iy = 12 e iθ . Note that since x, y (cid:62) (cid:54) θ (cid:54) π . Hence, the resourcefulness of pure states is determinedcompletely by one parameter θ . Moreover, note that we have M ( | ψ (cid:105)(cid:104) ψ | ) = M ( | θ (cid:105)(cid:104) θ | ) = 2 sin( θ ) . Theorem 4.
Let | ψ (cid:105) and | φ (cid:105) be two pure states in a d -dimensional Hilbert space H . Then there exists a free operation E such that E ( | ψ (cid:105)(cid:104) ψ | ) = | φ (cid:105)(cid:104) φ | if and only if M ( | ψ (cid:105)(cid:104) ψ | ) (cid:62) M ( | φ (cid:105)(cid:104) φ | ) . Proof.
The necessity of this condition follows from the fact that the measure M is a monotone. It is therefore left toshow that if M ( | ψ (cid:105)(cid:104) ψ | ) (cid:62) M ( | φ (cid:105)(cid:104) φ | ) then there exists a free operation E such that E ( | ψ (cid:105)(cid:104) ψ | ) = | φ (cid:105)(cid:104) φ | .Recall that every qubit state ρ can be expanded in the Pauli basis ρ = 12 ( I + (cid:126) r · (cid:126)σ )where (cid:126) r : (cid:107) (cid:126) r (cid:107) (cid:54) ρ and (cid:126)σ = ( X Y Z ) T is a vector of Paulimatrices. Next, every quantum operation E on a qubit has a geometric interpretation as an affine transformationwhich maps the Bloch sphere onto itself [32]. The action of the operation E on ρ will thus produce a new state ρ (cid:48) with corresponding Bloch vector given by (cid:126) r (cid:48) = T(cid:126) r + (cid:126) t (1)1where (cid:126) t is a 3-component real vector and T is a real 3x3 matrix. In our case, w.l.o.g. we assume that the input state ρ is the pure state | ψ (cid:105) = | θ (cid:105) and the output state | φ (cid:105) is the pure state | θ (cid:48) (cid:105) with 0 (cid:54) θ (cid:48) (cid:54) θ (cid:54)
1. Consequently, wehave (cid:126) r = cos( θ )sin( θ )0 and (cid:126) r (cid:48) = cos( θ (cid:48) )sin( θ (cid:48) )0 Clearly, Eq. (1) holds for T = sin ( θ (cid:48) ) sin( θ )
00 0 0 and (cid:126) t = cos( θ (cid:48) ) − cos( θ )00 Moreover, note that sin ( θ (cid:48) ) sin( θ ) (cid:54) θ (cid:48) ) − cos( θ ) (cid:62)
0. It is straight forward to check that these choices lead to aChoi matrix that is real and positive semi-definite.It follows from the theorem above, that the state | + i (cid:105) := √ ( | (cid:105) + i | (cid:105) ), can be transformed to all other d -dimensional pure states via free operations. This, in turn, implies that | + i (cid:105) can also be converted to any mixedstate, since | + i (cid:105) can be converted to any state | φ j (cid:105) with probability p j , achieving the transformation | ψ (cid:105) → ρ ≡ (cid:80) j p j | φ j (cid:105)(cid:104) φ j | . This means that the resource theory of imaginarity has a unique maximally resourceful state, | + i (cid:105) ,that can be converted to any other state, which we call the maximally imaginary state. Note that |− i (cid:105) and | + i (cid:105) arerelated by a free orthogonal transformation and therefore do not correspond to two distinct maximally imaginarystates.The characterization of the transformations attainable by physically consistent free operations in Theorem 4 allowsone to achieve the same tasks as standard quantum mechanics with the maximally imaginary state. For example,one may use the | + i (cid:105) state to generate the product state | + i (cid:105) | + i (cid:105) and subsequently transform a single copy of themaximally imaginary state to any other state deterministically.The fact that the existence of a process which takes one pure state to another requires the comparison of only asingle parameter is not too surprising, as Lemma 2 shows that we can equivalently represent the resourcefulness ofany pure state by a qubit. This is in contrast to many other resource theories, for example, in entanglement theorywith LOCC, one must evaluate d − Z -superselection rule[15], where one can reversibly map any pure state to a two-dimensional subspace and the condition for the existenceof a state transformation also depends only on a single parameter. VI. CONCLUSION
We introduced a framework for the resource theory of imaginarity, which was first mentioned in [27] where it wasshown to be the only known affine resource theory that does not have a self-adjoint resource destroying map. Upondefining real states, we completely characterized two sets of real operations; those which are resource non-generatingand those which we call physically consistent. We showed that not only do the physically consistent free operationsprevent resources from being generated probabilistically, but they coincide with all completely resource non-generatingoperations as well as admit a free dilation. This provides a level of consistency which is not present in coherence theory[5], and further suggests that imaginarity is a physically significant property of a quantum system. We also introducedseveral measures of imaginarity and provided a closed form measure in terms of the trace distance. Furthermore,Theorem 4 provides necessary and sufficient conditions for the the existence of a physically consistent channel betweentwo pure states, and shows that there is an equivalence class of states in any dimension from which any pure statecan be generated.The results presented in this paper may act as a starting point for the analysis of other imaginarity measures, mixedstate transformations, transformations in the asymptotic limit of many copies of states and the possible existenceof catalysts [40–42] under the restriction to free operations. An interesting direction for future work would beto investigate deviations from the complex numbers, in a resource theoretic framework, with reference to numbersystems other than the reals. This could be a restriction to some smaller field such as the rationals, or an extensionof the complex numbers such as the quaternions. The next steps in this theory could include extensions to infinitedimensional Hilbert spaces as well as analyzing the connection of this theory to entanglement theory [31] and otherconvex resource theories [27].2
Acknowledgments
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