Quantum Music, Quantum Arts and Their Perception

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Similarly, assorted trans- G (ˇ (ˇ ˇ formations process quantum art. Finally, the process of (ˇ (ˇ (ˇ (ˇ information transmission requires instruments of percep- |Ψci |Ψdi |Ψei |Ψf i |Ψgi |Ψai |- Ψbi tion or “receivers” [22, Fig. 1]. Let us mention typical components and theoretical entities as example transformations. For instance, FIG. 1. (Color online) Temporal succession of quantum tones Hadamard transformations produce perfect “mixtures” |Ψci, |Ψdi, . . ., |Ψbi in the C major scale forming the octave of classically mutually exclusive signals. Quantum basis B. Fourier transforms produce generalized mixtures. All of them have to be uniformly unitary—that is, in terms (i) bundling octaves by coherent their superposition of the various equivalent formal definitions, they have to (aka simultaneous linear combination), as well as transform orthonormal basis into orthonormal ones, they have to preserve scalar products or norms, and their in- (ii) considering pseudo-field theoretic models treating verse is the adjoint. One of the physical realizations is in notes as field modes that are either bosonic or terms of generalized beam splitters [23, 24]. fermionic. Depending on whether we are willing to contemplate genuine quantum receivers or merely classical ones we The seven tones c, d, e, f, g, a, and b of the octave end up with either a quantum cognition or with merely a can be considered as belonging to disjoint events (maybe classical cognition of this quantum art; and, in particular, together with the null event 0) whose probabilities should of quantum music. In the first, radical deviation from add up to unity. This essentially suggests a formalization C7 classical music, we would have to accept the possibility of by a seven (or eight) dimensional Hilbert space or C8 human or sentient consciousness or audience to perceive ) with the standard Euclidean scalar product. The quantum impressions. respective Hilbert space represents a full octave. C7 This is ultimately a neurophysiologic question. It We shall study the seven-dimensional case . The might well be that the processing of signals exterior to the seven tones forming one octave can then be represented B C7 receiving and perceiving “somewhere along those chan- as an orthonormal basis of by forming the set the- nels” requires a breakdown to classicality; most likely oretical union of the mutually orthogonal unit vectors; that is, B = Ψc , Ψd ,... Ψb , where the basis ele- through the introduction of stochasticity [25]. This is {| i | i | i} very much in the spirit of Schrödinger’s cat [8] and (later) ments are the Cartesian basis tuples quantum jellyfish [26] metaphors based on the assump- Ψc = 0, 0, 0, 0, 0, 0, 1 , tion that, ultimately, even if decoherence by environmen- | i Ψd = 0, 0, 0, 0, 0, 1, 0 , tal intake can be controlled, there cannot be any simul- | i taneous co-experience of being both dead and alive, just ... as there might not be any co-experience of passing into Ψb = 1, 0, 0, 0, 0, 0, 0 a room by two separate doors simultaneously. | i On the other hand, nesting of the Wigner’s friend of C7. Fig. 1 depicts the basis B by its elements, drawn type [27–30], suggests that there might be substance to in different colors. a sort of mindful co-experience of two classical distinct experiences. Whether such experiences remain on the subconscious primordial level of perception, or whether A. Bundling octaves into single tones this can be levied to a full cognitive level is a fascinating question on its own that exceeds the limited scope of this Pure quantum musical states could be represented as article. 7 unit vectors ψ C which are linear combinations of the basis B;| thati ∈ is,

II. QUANTUM MUSICAL TONES ψ = αc Ψc + αd Ψd + + αb Ψb , (1) | i | i | i ··· | i with coeﬃcients α satisfying In what follows we closely follow our nomenclature and i presentation of quantum music [2]. Those formal choices 2 2 2 αc + αd + + αb =1. (2) are neither unique nor comprehensive. Alternatives are | | | | ··· | | mentioned. Equivalent representations of ψ are in terms of the one- We consider a quantum octave in the C major scale, dimensional subspace φ φ| i= α ψ , α C spanned {| i | | i | i ∈ } which classically consists of the tones c, d, e, f, g, a, and by ψ , or by the projector Eψ = ψ ψ . b, represented by eight consecutive white keys on a piano. A| musicali “composition”—indeed,| ih and| any succession (Other scales are straightforward.) At least three ways of quantized tones forming a “melody”—would be ob- to quantize this situation can be given: tained by successive unitary permutations of the state 3

2 G (ˇ G 4 ˇ ˇ |Φg i ˇ ˇ

FIG. 2. Representation of a 50:50 quantum tone |Φg i = FIG. 3. (Color online) A two-note quantum musical 1 √2 (|0g i − |1gi) in gray (without indicating phase factors). composition—a natural ﬁfth.

ψ . The realm of such compositions would be spanned by irreducably stochastic [32] quantum-to-classical transla- | i tion [33] represents an “irreducible” [34–42] stochastic the succession of all unitary transformations U : B B′ mapping some orthonormal basis B into another7→ or- measurement. This can never render a unique classical listening experience, as the probability to hear the tone thonormal basis B′; that is [31], U = Ψ′i Ψi . i | ih | i is α 2. Therefore, partitions of the audience will hear P i different| | manifestations of the quantum musical composition made up of all varieties of successions of tones. B. Coherent superposition of tones as a new form of musical parallelism These experiences multiply and diverge as more tones are perceived. For the sake of a demonstration, let us try a two- One of the mind-boggling quantum features of this note quantum composition. We start with a pure quan- “bundling” is the possibility of the simultaneous “co- tum mechanical state in the two-dimensional subspace existence” of classically excluding musical states, such as spanned by Ψc and Ψg , specified by a 50:50 quantum g in the C major scale obtained by send- | i | i 1 1 1 ing 0g through the Hadamard gate H = , 4 3 1 4 | i √2 1 1 ψ1 = Ψc + Ψg = . (3) 1 − | i 5| i 5| i 5 3 resulting in ( 0g 1g ), and depicted in Fig. 2 by a √2 | i − | i 50 white 50 black; that is, gray, tone (though without the ψ1 would be detected by the listener as c in 64% of all relative “ ” phase). |measurementsi (listenings), and as g in 36% of all listen- − This novel form of musical expression might contribute 0 1 ings. Using the unitary transformation X = , the to novel musical experiences; in particular, if any such 1 0 coherent superposition can be perceived by the audience next quantum tone would be in full quantum uniformity. This would require the cognition of the recipient to experience quantum coherent 3 4 1 3 ψ2 = X ψ1 = Ψc + Ψg = . (4) superpositions—a capacity that is highly speculative. It | i | i 5| i 5| i 5 4 has been mentioned earlier that any such capacity is re- lated to Schrödinger’s cat [8] and quantum jellyfish [26] This means for the quantum melody of both quan- metaphors, as well as to nestings of the Wigner’s friend tum tones ψ1 and ψ2 in succession—for the score, type [27–30]. see Fig. 3—that| i in repeated| i measurements, in 0.642 = 40.96% of all cases c g is heard, in 0.362 = 12.96% of all cases g c, in 0.64− 0.36 = 23.04% of all cases c c − · − C. Classical perception of quantum musical or g g, respectively. − parallelism

In the following, we shall assume that quantum mu- III. QUANTUM MUSICAL ENTANGLEMENT sic is “reduced” to the continuous infinity of its classical forms. Then, if a classical auditorium listens to the quan- Quantum entanglement [8] is the property of multi- tum musical state ψ in Eq. 1, the individual classical partite quantum systems to code information “across listeners may perceive| i ψ very differently; that is, they quanta” in such a way that the state of any individual will hear only a single| onei of the different tones with quantum remains irreducibly indeterminate; that is, not 2 2 2 probabilities of αc , αd , ..., and αb , respectively. determined by the entangled multipartite state [8–11]. | | | | | | Indeed, suppose that classical recipients (aka “listen- Thus the entangled whole should not be thought of as ers”) modeled by classical measurement devices acting as composed of its proper parts. Formally, the composite information-theoretic receivers are assumed. Then any state cannot be expressed as a product of separate states perception (aka “listening” or reception) of a quantum of the individual quanta. musical state that is in a coherent superposition—with A typical example of an entangled state is the Bell some coefficients 0 < αi < 1—because of the supposedly state, Ψ− or, more generally, states in the Bell basis | | | i 4

2 2 3 4 5 6 7 8 G ˇˇ ˇ G ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ Piano (a) ăG ˇ ˇ 2 3 4 5 ˇ ˇ ˇ G ˇ ˇ ˇ ˇ ˇ (b) + FIG. 4. Quantum musical entangled states |Ψea− i and |Ψeai in + the first bar, and |Φea− i and |Φeai in the second bar (without relative phases). FIG. 6. Temporal succession of complementary tones (a) for 2 2 binary occupancy |φai = αa|0ai+βa|1ai, with |αa| +|βa| = 1 with increasing |αa| (decreasing occupancy), (b) in the bun- ˇ 2 ˇ dled octave model, separated by bars. G ˇ ˇ Piano are complementary. For the dichotomic field approach, ˇ ˇ Fig. 6 represents a configuration of mutually complemen- ă tary quantum tones for the note a in the C major scale G (a), and mutually complementary linear combinations as ˇ ˇ introduced in Section II (b). Complementarity can be extended to more advanced configurations of contexts. These quantum configurations and their associated quantum probability distribu- FIG. 5. (Color online) Quantum musical entangled states for + tions, if interpreted classically, either exhibit violations bundled octaves |Ψea− ′ i and |Ψea′ i in the first bar, and |Φea− ′ i + of classical probability theory, classical predictions, or and |Φ ′ i in the second bar (without relative phases). ea nonisomorphic embeddability of observables into classical propositional structures [12–21]. spanned by the quantized notes e and a; that is 1 V. BOSE AND FERMI MODEL OF TONES Ψ± = ( 0e 1a 1e 0a ) , | i √2 | i| i ± | i| i (5) 1 An alternative quantization to the one discussed earlier Φ± = ( 0e 0a 1e 1a ) , is in analogy to some fermionic or bosonic—such as the | i √2 | i| i ± | i| i electromagnetic—field. Just as the latter one in quantum A necessary and sufficient condition for entanglement optics [44, 45] and quantum field theory [46] is quan- among the quantized notes e and a is that the coeffi- tized by interpreting every single mode (determined, for cients α1, α2, α3, α4 of their general composite state the electromagnetic field for instance by a particular fre- Ψga = α1 0e 0a + α2 0e 1a + α3 1e 0a + α4 1e 1a quency and polarization) as a sort of “container”—that | i | i| i | i| i | i| i | i| i obey α1α4 = α2α3 [4, Sec. 1.5]. This is clearly satisfied is, by allowing the occupancy of that mode to be either by Eqs. (5).6 Fig. 4 depicts the entangled musical Bell empty or any positive integer (and a coherent superposi- states. tion thereof)—we obtain a vast realm of new musical ex- Entanglement between different octaves can be con- pressions which cannot be understood in classical terms. structed similarly. Fig. 5 depicts this configuration for Whereas in a “bosonic field model” occupancy of field an entanglement between e and a′. modes is easy to be correlated with the classical volume of the corresponding tone, in what follows we shall restrict ourselves to a sort of “fermionic field model” of music IV. QUANTUM MUSICAL which is characterized by a binary, dichotomic situation, COMPLEMENTARITY AND CONTEXTUALITY in which every tone has either null or one occupancy, represented by 0 = (0, 1) or 1 = (1, 0), respectively. Although complementarity [43] is mainly discussed in Thus every state| ofi such a tone| cani thus be formally rep- the context of observables, we can present it in the state resented by entities of a two-dimensional Hilbert space, formalism by observing that, as mentioned earlier, any C2, with the Cartesian standard basis B = 0 , 1 . {| i | i} pure state ψ corresponds to the projector Eψ = ψ ψ . Any note Ψi of the octave consisting of Ψc , Ψd , ..., | i | ih | | i | i | i In this way, any two nonvanishing nonorthogonal and Ψb , in the C major scale can be represented by the noncollinear states ψ and φ with 0 < φ ψ < 1 coherent| i superposition of its null and one occupancies; | i | i |h | i| 5

Let us state up front that quantum visual art, and, in particular, quantum parallelism, is not about additive G (ˇ (ˇ ˇ color mixing, but it is about the simultaneous existence of ¯ (ˇ (ˇ (ˇ (ˇ diﬀerent, classically mutually exclusive “colors”, or visual |Ψci |Ψdi |Ψei |Ψf i |Ψgi |Ψai |- Ψbi impressions in general. Quantum visual arts use the same central assets or capacities (i)–(iv) mentioned earlier in Section I. It can be developed very much in parallel to FIG. 7. Temporal succession of tones |Ψci, |Ψdi, . . ., |Ψbi in quantum music but requires the creation of an entirely an octave in the C major scale with dicreasing mean occu- new nomenclature. The perception of quantum visual art pancy. is subject to the same assumptions about the cognitive capacities to comprehend these artifacts fully quantum mechanically or classically. This will be shortly discussed that is, in the following section.

Ψi = αi 0i + βi 1i , (6) | i | i | i with α 2 + β 2 = 1, α .β C. VII. CAN QUANTUM ART RENDER i i i i COGNITIONS AND PERCEPTIONS BEYOND | | | | ∈ α Every tone is characterized by the two coefficients CLASSICAL ART? and β, which in turn can be represented (like all quantized two-dimensional systems) by a Bloch sphere, with two angular parameters. If we restrict our attention Suppose for a moment that humans are capable to (somewhat superficially) to real Hilbert space R2, then sense, receive and perceive quantum signals not only clas- the unit circle, and thus a single angle ϕ, suffices for a sically but in a fully quantum mechanical way. Thereby, characterization of the coefficients α and β. Furthermore, they would, for instance, be capable of simultaneously we may very compactly notate the mean occupancy of the “holding” different classically distinct tones at once—not notes by gray levels. Now, in this “fermionic setting”, just by interference but by parallel co-existence. This with the mean occupation number of any tone between would result in a transgression of classical art forms, and 0 and 1 the gray level does not indicate the volume of in entirely novel forms of art. the corresponding tone but the mere chance of it being The existence of such possibilities depends on the present or not, see also Section II. Fig. 7 depicts a se- neurophysiology of the human, or, more generally, sen- quence of tones in an octave in the C major scale with tient, perception apparatus. Presently the question as to decreasing occupancy, indicated as gray levels. whether or not this is feasible is open; the answer to it is In this case, any nonmonotonous unitary quantum mu- unknown. sical evolution would have to involve the interaction of In the case that merely classical perceptions are feasi- different tones; that is, in a piano setting, across several ble, we would end up with a sort of Church-Turing thesis keys of the keyboard. for music. In particular, quantum music would not be able to “go beyond” classical music for a single observer, as only classical renditions could be perceived. Of course, VI. QUANTUM VISUAL ARTS as we mentioned earlier, quantum music might “sound differently for different observers”. To this end, we might Just as for the performing arts such as music one could conceptualize a kind of universal musical instrument that contemplate the quantum options and varieties for the is capable of rendering all possible classical notes. Pianos visual arts. Suffice it to say that the notion of “color” and organs might be “for all practical purposes good” ap- experience can be extended to the full quantum opti- proximations to such a universal device. cal varieties that result from the electromagnetic field Quantum music and quantum arts, just like quantum quantization, as already mentioned earlier. Incidentally, computing [52], or computations starting and ending in Schrödinger published a series of papers on classical color real numbers but using imaginary numbers as interme- perception [47, 48] until around 1925. Yet to our best diaries [53], might be a sort of bridge crossing elegantly knowledge he never considered the particular quantum a gap between two classical domains of perception. And aspects of human color and light perception. yet they could be so much more if only the quantum could Human rod cells respond to individual photons [49, 50]. be “heard” or “sensed”. Moreover, recent reports suggest that humans might be capable of “being aware” of the detection of a single- photon incident on the cornea with a probability signifi- VIII. SUMMARY cantly above chance [51]. It thus may be suspected that this area of perception presents the most promising path- We have contemplated the various extensions of mu- way into truly quantum perception. Speculations how sic, and arts in general, to the quantum domain. Thereby this issue may be transferred to the perception of sound we have located particular capacities which are genuine are compelling. properties. These involve parallelization through coher- 6 ent superposition (aka simultaneous linear combination), tion in general and human neurophysiology, in particular. entanglement, complementarity and contextuality. We have reviewed the nomenclature introduced previously [2] and considered particular instances of quantum music. Then we have briefly discussed quantum visual arts. ACKNOWLEDGMENTS The perception of quantum arts depends on the capacity of the audience to either perceive quantum physical This research was funded in whole, or in part, by the states as such, or reduce them to classical signals. In the Austrian Science Fund (FWF), Project No. I 4579-N. first case, this might give rise to entirely novel artistic For the purpose of open access, the author has applied a experiences. We believe that these are important issues CC BY public copyright licence to any Author Accepted that deserve further attention, also for sentient percep- Manuscript version arising from this submission.

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