The slogan information is physical has been so successful that it led to some excess. Classical and quantum information can be thought of independently of any physical implementation. Pure information tasks can be realized using such abstract c- and qu-bits, but physical tasks require appropriate physical realizations of c- or qu-bits. As illustration we consider the problem of communicating chirality.

Assume that two distance partners like to compare the chiralities of their Cartesian reference frames. This is impossible by exchanging only classical information, i.e. by sending only abstract 0’s and 1’s. This is true in the most general relativistic context Feynman . But for our purpose it suffice to consider a Newtonian physics worldview. It is then quite intuitive to grasp why information about chirality can’t be encoded in classical bits: bits measure the quantity of information, but have per se no meaning, in particular no meaning about geometric and physical concepts. Hence, if our world is invariant under left ↔↔\leftrightarrow↔ right, then mere information is unable to distinguish between left and right. Now, information is physical, as Landauer used to emphasize and as every physicists knows today, hence let’s consider classical bits physically realized in some system. For example the bits 0 and 1 could be realized by right-handed and left-handed gloves, respectively. It is obvious that such physical bits can be used to send chirality information. But bits realized by black. White balls couldn’t do the job.

Let us now consider what quantum information brings to the situation under investigation. Note that recently many authors did consider similar formal problems, like aligning reference frames, without paying too much attention to the deep physical meaning of the problem. At first sight one may quickly conclude that sending abstract superpositions of basic kets |0⟩ket0|0\rangle| 0 ⟩ and |1⟩ket1|1\rangle| 1 ⟩, i.e. qubits c0|0⟩+c1|1⟩subscriptݑ0ket0subscriptݑ1ket1c_0|0\rangle+c_1|1\rangleitalic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | 0 ⟩ + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 1 ⟩ with complex amplitudes cjsubscriptݑݑ—c_jitalic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, doesn’t help for the same reason as classical bits: they have no geometric nor physical meaning. The quantum situation is however more tricky because of the phenomenon of entanglement: Alice and Bob (we adopt the by now traditional labelling of the two partners) may exchange many qubits in such a way as to share many entangled qubit pairs in the singlet state: |0,1⟩-|1,0⟩ket01ket10|0,1\rangle-|1,0\rangle| 0 , 1 ⟩ - | 1 , 0 ⟩. For clarity assume first that Alice. Bob both use the same reference frame. Then, using local operations and classical communication (LOCC), they can measure all the correlations ⟨σj⊗11⟩delimited-⟨⟩tensor-productsubscriptݜŽݑ—11\langle\sigma_j\otimes\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\rangle⟨ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ 1 1 ⟩, ⟨11⊗σk⟩delimited-⟨⟩tensor-product11subscriptݜŽݑ˜\langle\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\otimes\sigma_k\rangle⟨ 1 1 ⊗ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ and ⟨σj⊗σk⟩delimited-⟨⟩tensor-productsubscriptݜŽݑ—subscriptݜŽݑ˜\langle\sigma_j\otimes\sigma_k\rangle⟨ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩, j,k=x,y,zformulae-sequenceݑ—ݑ˜ݑ¥ݑ¦ݑ§j,k=x,y,zitalic_j , italic_k = italic_x , italic_y , italic_z where σxsubscriptݜŽݑ¥\sigma_xitalic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, σysubscriptݜŽݑ¦\sigma_yitalic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT and σzsubscriptݜŽݑ§\sigma_zitalic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT denote the 3 Pauli matrices. From these measured correlations Alice and Bob compute the matrix:

14(11+⟨σ→⊗11⟩σ→⊗11+⟨11⊗σ→⟩11⊗σ→+⟨σj⊗σk⟩σj⊗σk)1411tensor-productdelimited-⟨⟩tensor-product→ݜŽ11→ݜŽ11tensor-productdelimited-⟨⟩tensor-product11→ݜŽ11→ݜŽtensor-productdelimited-⟨⟩tensor-productsubscriptݜŽݑ—subscriptݜŽݑ˜subscriptݜŽݑ—subscriptݜŽݑ˜\frac14\big{(}\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1+\langle% \vec\sigma\otimes\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\rangle% \vec\sigma\otimes\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1+\langle% \leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\otimes\vec\sigma\rangle% \leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\otimes\vec\sigma+\langle% \sigma_j\otimes\sigma_k\rangle\sigma_j\otimes\sigma_k\big{)}divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 1 + ⟨ over→ start_ARG italic_σ end_ARG ⊗ 1 1 ⟩ over→ start_ARG italic_σ end_ARG ⊗ 11 + ⟨ 1 1 ⊗ over→ start_ARG italic_σ end_ARG ⟩ 1 1 ⊗ over→ start_ARG italic_σ end_ARG + ⟨ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) (1)
which is nothing but the density matrix representing the quantum state they share, the projector onto the singlet state in our example. Consequently, the 4 eigenvalues of (1) are positive (≥0absent0\geq 0≥ 0). Next, if Bob’s reference frame is rotated compared to Alice’s one, but still of the same chirality, then the matrix (1) represents the singlet state with one part accordingly rotated by a unitary transformation; the 4 eigenvalues would still be positive. What now if Bob’s axes are all opposite to Alice’s ones: r→Bob=-r→Alicesubscript→ݑŸݐµݑœݑsubscript→ݑŸݐ´ݑ™ݑ–ݑݑ’\vecr_Bob=-\vecr_Aliceover→ start_ARG italic_r end_ARG start_POSTSUBSCRIPT italic_B italic_o italic_b end_POSTSUBSCRIPT = - over→ start_ARG italic_r end_ARG start_POSTSUBSCRIPT italic_A italic_l italic_i italic_c italic_e end_POSTSUBSCRIPT? This corresponds to Bob using a frame with opposite chirality with respect to that of Alice. In such a case it might seem that if Alice and Bob proceed exactly as above, they would end up with the following computed matrix:

14(11+⟨σ→⊗11⟩σ→⊗11-⟨11⊗σ→⟩11⊗σ→-⟨σj⊗σk⟩σj⊗σk)1411tensor-productdelimited-⟨⟩tensor-product→ݜŽ11→ݜŽ11tensor-productdelimited-⟨⟩tensor-product11→ݜŽ11→ݜŽtensor-productdelimited-⟨⟩tensor-productsubscriptݜŽݑ—subscriptݜŽݑ˜subscriptݜŽݑ—subscriptݜŽݑ˜\frac14\big{(}\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1+\langle% \vec\sigma\otimes\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\rangle% \vec\sigma\otimes\leavevmode\hbox\small 1\kern-3.8pt\normalsize 1-\langle% \leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\otimes\vec\sigma\rangle% \leavevmode\hbox\small 1\kern-3.8pt\normalsize 1\otimes\vec\sigma-\langle% \sigma_j\otimes\sigma_k\rangle\sigma_j\otimes\sigma_k\big{)}divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( 1 1 + ⟨ over→ start_ARG italic_σ end_ARG ⊗ 1 1 ⟩ over→ start_ARG italic_σ end_ARG ⊗ 1 1 - ⟨ 1 1 ⊗ over→ start_ARG italic_σ end_ARG ⟩ 1 1 ⊗ over→ start_ARG italic_σ end_ARG - ⟨ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⟩ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) (2)
(notice the 2 sign changes!). This amounts to change ⟨σ→⟩delimited-⟨⟩→ݜŽ\langle\vec\sigma\rangle⟨ over→ start_ARG italic_σ end_ARG ⟩ to -⟨σ→⟩delimited-⟨⟩→ݜŽ-\langle\vec\sigma\rangle- ⟨ over→ start_ARG italic_σ end_ARG ⟩ on Bob’s side which is equivalent, up to a unitary transformation, to the well known partial transposition introduced by Peres Peres96 as a criteria for entanglement. And indeed the matrix (2) has a negative eigenvalue whenever the original state (1) is entangled Horodecki96 . This observation led Lajos Diosi to the initial conclusion that pure quantum information. Entanglement allows one to compare chiralities. But, as Diosi himself noticed soon after posting Diosi00 , this is not quite so. Indeed, the Poincaré sphere that we did implicitly use above does not float in our ”real” 3-dimensional space, but is a convenient mathematical construction. Abstract qubits don’t point in any direction. Thus can’t be used to define spatial directions nor chirality. ARG. One may think that now we have spatial directions. Can thus use entanglement to define chirality. One may think that now we have spatial directions. Can thus use entanglement to define chirality. But this is still not quite so Rudolph99 ; CollinsPopescu04 . ARG particle in a typical Stern-Gerlach experiment. Assume it is deflected towards the ceiling. From this elementary fact one can’t conclude that the resulting spin state is ”up”; in fact the relevant direction is that of the gradient of the magnetic field. If a physicist using a right-handed reference frame correctly concludes from the above fact that the spin is ”up”, then a physicist using a left-handed frame would equally correctly conclude that the spin is ”down”, because the latter physicist ”sees” a magnetic field pointing in the opposite direction than that assigned (by convention!) by the first physicist. This is not special to quantum physics, the same holds in classical physics for all axial vectors, like e.g. angular momentum. Hence, spins are not appropriate realizations of qubit if the task is to communicate chirality. Rather than an axial vector one should use a polar vector like the vector joining an electron in an atom to its nucleus. This can be done using the orbital angular momenta YmlsubscriptsuperscriptݑŒݑ™ݑšY^l_mitalic_Y start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. But before elaborating on this, let us look for the closest quantum equivalent to the glove implementation of classical bits discussed in the introduction.

In order to proceed, we borrow from Qglove the concept of quantum gloves, that is elementary quantum states that contain nothing but the abstract concept of chirality, let us introduce the following chirality operator Qglove :

χݜ’\displaystyle\chiitalic_χ =\displaystyle== 123(σx⊗σy⊗σz+σy⊗σz⊗σx+σz⊗σx⊗σyfragments123fragments(subscriptݜŽݑ¥tensor-productsubscriptݜŽݑ¦tensor-productsubscriptݜŽݑ§subscriptݜŽݑ¦tensor-productsubscriptݜŽݑ§tensor-productsubscriptݜŽݑ¥subscriptݜŽݑ§tensor-productsubscriptݜŽݑ¥tensor-productsubscriptݜŽݑ¦\displaystyle\frac12\sqrt3(\sigma_x\otimes\sigma_y\otimes\sigma_z+% \sigma_y\otimes\sigma_z\otimes\sigma_x+\sigma_z\otimes\sigma_x% \otimes\sigma_ydivide start_ARG 1 end_ARG start_ARG 2 square-root start_ARG 3 end_ARG end_ARG ( italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT + italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT + italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT (3)

-\displaystyle-- σx⊗σz⊗σy-σz⊗σy⊗σx-σy⊗σx⊗σz)fragmentssubscriptݜŽݑ¥tensor-productsubscriptݜŽݑ§tensor-productsubscriptݜŽݑ¦subscriptݜŽݑ§tensor-productsubscriptݜŽݑ¦tensor-productsubscriptݜŽݑ¥subscriptݜŽݑ¦tensor-productsubscriptݜŽݑ¥tensor-productsubscriptݜŽݑ§)\displaystyle\sigma_x\otimes\sigma_z\otimes\sigma_y-\sigma_z\otimes% \sigma_y\otimes\sigma_x-\sigma_y\otimes\sigma_x\otimes\sigma_z)italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT - italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT - italic_σ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT )
Note that all positive permutations appear with positive signs, while all negative permutations have a negative sign. This chirality operator has 3 eigenvalues: 0 and ±1plus-or-minus1\pm 1± 1, hence χ=χ+-χ-ݜ’subscriptݜ’subscriptݜ’\chi=\chi_{+}-\chi_{-}italic_χ = italic_χ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT - italic_χ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT. If Alice sends the mixed state ρ+=12χ+subscriptݜŒ12subscriptݜ’\rho_{+}=\frac12\chi_{+}italic_ρ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_χ start_POSTSUBSCRIPT + end_POSTSUBSCRIPT corresponding to the projector onto the 2-dimensional eigenspace associated to the eigenvalue +1 and if Bob measures χݜ’\chiitalic_χ, then Bob obtains the results +1 if and only if he uses the same chirality as Alice 111Note that both states ρ±subscriptݜŒplus-or-minus\rho_\pmitalic_ρ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT are invariant under global rotation; they could thus be used for decoherence free decohFreeSupspace quantum communication.. But, if the qubits are realized with axial vectors, like spins, then knowing χݜ’\chiitalic_χ is equivalent to knowing the chirality, hence in such a case Bob can measure χݜ’\chiitalic_χ only if he does already know Alice’s chirality! Accordingly the states ρ±subscriptݜŒplus-or-minus\rho_\pmitalic_ρ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT are not good abstract quantum gloves, for this we need polar vectors, i.e. vectors that really point in some spatial direction. Since we are not interested in any radial variable, we now on use the spherical harmonics YmlsubscriptsuperscriptݑŒݑ™ݑšY^l_mitalic_Y start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. Under reflection about the origin, Pn→=-n→ݑƒ→ݑ›→ݑ›P\vecn=-\vecnitalic_P over→ start_ARG italic_n end_ARG = - over→ start_ARG italic_n end_ARG, they transform as PYml=(-1)lYmlݑƒsubscriptsuperscriptݑŒݑ™ݑšsuperscript1ݑ™subscriptsuperscriptݑŒݑ™ݑšPY^l_m=(-1)^lY^l_mitalic_P italic_Y start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ( - 1 ) start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT italic_Y start_POSTSUPERSCRIPT italic_l end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT. Consider the two following 4-particle (e.g. 3 electrons and one nucleus) rotationally invariant states Qglove :

|S⟩ketݑ†\displaystyle|S\rangle| italic_S ⟩ =\displaystyle== Y00Y00Y00subscriptsuperscriptݑŒ00subscriptsuperscriptݑŒ00subscriptsuperscriptݑŒ00\displaystyle Y^0_0Y^0_0Y^0_0italic_Y start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (4)

|A⟩ketݐ´\displaystyle|A\rangle| italic_A ⟩ =\displaystyle== 16(Y11Y01Y-11+Y01Y-11Y11+Y-11Y11Y01fragments16fragments(subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ10subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ10subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ10\displaystyle\frac1\sqrt6\big{(}Y^1_1Y^1_0Y^1_-1+Y^1_0Y^% 1_-1Y^1_1+Y^1_-1Y^1_1Y^1_0divide start_ARG 1 end_ARG start_ARG square-root start_ARG 6 end_ARG end_ARG ( italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT + italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT (5)

-Y11Y-11Y01-Y-11Y01Y11-Y01Y11Y-11)fragmentssubscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ10subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ10subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ10subscriptsuperscriptݑŒ11subscriptsuperscriptݑŒ11)\displaystyle-Y^1_1Y^1_-1Y^1_0-Y^1_-1Y^1_0Y^1_1-Y^1_% 0Y^1_1Y^1_-1\big{)}- italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_Y start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT )
Clearly P|S⟩=|S⟩ݑƒketݑ†ketݑ†P|S\rangle=|S\rangleitalic_P | italic_S ⟩ = | italic_S ⟩ and P|A⟩=-|A⟩ݑƒketݐ´ketݐ´P|A\rangle=-|A\rangleitalic_P | italic_A ⟩ = - | italic_A ⟩. The following two states can thus be defined as quantum gloves:

|G±⟩=|S⟩±|A⟩2,ketsuperscriptݐºplus-or-minusplus-or-minusketݑ†ketݐ´2|G^\pm\rangle=A\rangle\over\sqrt2,| italic_G start_POSTSUPERSCRIPT ± end_POSTSUPERSCRIPT ⟩ = divide start_ARG | italic_S ⟩ ± | italic_A ⟩ end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG , (6)
they do indeed transform properly: P|G+⟩=|G-⟩ݑƒketsuperscriptݐºketsuperscriptݐºP|G^{+}\rangle=|G^{-}\rangleitalic_P | italic_G start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ = | italic_G start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ⟩ and P|G-⟩=|G+⟩ݑƒketsuperscriptݐºketsuperscriptݐºP|G^{-}\rangle=|G^{+}\rangleitalic_P | italic_G start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ⟩ = | italic_G start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩!

Note that the two states |S⟩ketݑ†|S\rangle| italic_S ⟩ and |A⟩ketݐ´|A\rangle| italic_A ⟩ can be used to define qubits c0|S⟩+c1|A⟩subscriptݑ0ketݑ†subscriptݑ1ketݐ´c_0|S\rangle+c_1|A\rangleitalic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | italic_S ⟩ + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_A ⟩: each qubit is realized with 4 particles. Space reflections act on such qubits as phase-flips: P(c0|S⟩+c1|A⟩)=c0|S⟩-c1|A⟩ݑƒsubscriptݑ0ketݑ†subscriptݑ1ketݐ´subscriptݑ0ketݑ†subscriptݑ1ketݐ´P(c_0|S\rangle+c_1|A\rangle)=c_0|S\rangle-c_1|A\rangleitalic_P ( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | italic_S ⟩ + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_A ⟩ ) = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | italic_S ⟩ - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | italic_A ⟩. The chirality operator χݜ’\chiitalic_χ is then invariant under spatial reflection: P⊗P⊗PχP⊗P⊗P=χtensor-producttensor-productݑƒݑƒݑƒݜ’ݑƒݑƒݑƒݜ’P\otimes P\otimes P\chiP\otimes P\otimes P=\chiitalic_P ⊗ italic_P ⊗ italic_P italic_χ italic_P ⊗ italic_P ⊗ italic_P = italic_χ. Hence, using such realized qubits, Bob could measure χݜ’\chiitalic_χ without knowing Alice’s chirality. But this requires the use of 3x4=12 particles, the states (6) are thus much simpler.

Like classical information, quantum information per se is not physical. As emphasized by Landauer, information - including quantum information - requires a physical implementation. Depending on the task, some implementations are advantageous. The advantage can be so large as to render possible seemingly impossible tasks. In the present context, the determination of chirality is impossible using some implementations of classical bits or of quantum bits, like e.g. optical pulses and spin ½ respectively, but is perfectly possible both using c-bits or qubits provided appropriate physical implementations are used, like e.g. gloves and superpositions of excited atomic states, respectively 222Note that a physicist not polluted by excessive use of quantum information would probably come up with a much simpler solution: send a circularly polarized photon, the momentum provides the polar vector and the circular polarization the axial vector, a seemingly minimal set of 2 vectors..

In conclusion, quantum information can be thought of independently of any implementation, similarly to classical information. This rather trivial remark implies that quantum information can only achieve tasks which are expressed in pure information theoretical terms, like cloning and factoring, but can’t perform physical tasks like aligning reference frames Rudolph99 ; CollinsPopescu04 or defining temperature. This stresses that quantum teleportation is an information concept and does not permit the teleportation of a physical object, including its mass and chirality. This underlines that information is physical, but physics is more than mere information infoVsPhysics . Acknowledgements: This work has been elaborated thanks to -. Much influenced by - numerous discussions with many colleagues. In particular, I enjoyed stimulating debates with Lajos Diosi, Sandu Popescu, Serge Massar and with participants at the Perimeter Institute’s workshop last July on Quantum Reference Frames. Financial supports by the swiss NCCR Quantum Photonics. From the European project RESQ IST-2001-37559 are gratefully acknowledged.

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