Eur. Phys. J. C (2020) 80:418 https://doi.org/10.1140/epjc/s10052-020-7947-1 Regular Article - Theoretical Physics Unitary toy qubit transport model for black hole evaporation Bogusław Brodaa Department of Theoretical Physics, Faculty of Physics and Applied Informatics, University of Łód´z, Pomorska 149/153, 90-236 Lodz, Poland Received: 5 December 2019 / Accepted: 19 April 2020 / Published online: 13 May 2020 © The Author(s) 2020 Abstract In a recent paper Osuga and Page have presented In the present paper, we propose another toy qubit trans- an explicitly unitary toy qubit transport model for transfer- port model for black hole evaporation, which is also explic- ring information from a black hole to the outgoing radiation. itly unitary. Additional nice characteristic features include: Following their idea we propose a unitary toy model which (i) precisely (fermionic) Hawking states are involved; (ii) involves (fermionic) Hawking states. classical shrinking of a black hole, in quantum formalism, corresponds to transition to a vacuum state (in the black hole sector); (iii) primary entanglement (in the Hawking states) between modes inside and outside of the black hole vanishes. 1 Introduction For reader’s convenience, as well as for ours, we follow conventions and notation of [12] as closely as possible. In The black hole information (loss) paradox/problem/puzzle particular, we denote black hole qubits (once the black hole concerns difficulties in answering the question: “Is black hole forms) by a , where i = 1, 2,...,n, whereas qubits for the evaporation unitary?” Various answers and explanations have i infalling radiation and for the outgoing modes, by b and c , been proposed to date, “for” its unitarity as well as “against” i i respectively, where i = 1, 2,...,N. For technical simplic- it (for recent reviews see [1–4]). (The latter possibility, i.e., ity, from now on we confine ourselves to a Schwarzschild loss of unitarity, presumably requires some radical modifica- black hole, but this restriction is not crucial for our further tions of quantum mechanics, and presently this option seems considerations. to be less popular, however see [5]). To analyze the issue of unitarity in the context of black hole evaporation, the idea of toy qubit models has been proposed, developed and popular- ized, especially in [6–10], and nicely reviewed in [11]. Toy 2 The model qubit models give a possibility to mimic, in a simplified way, black hole evolution, and to trace, in detail, their (depending In the beginning, to each black hole mode ai , we associate a (fermionic) Hawking state |H (Ω ) , i.e., a pair of entan- on the model) breaking or maintaining unitarity. i bi ci In a recent paper [12], Osuga and Page have proposed gled radiation qubits, bi (infalling) and ci (outgoing), in the a strikingly simple toy qubit model demonstrating transfer- state ring information from a black hole to the outgoing radiation, |H (Ω ) = cos Ω |0 |0 + sin Ω |1 |1 , (1) which is explicitly unitary. Characteristic features of their i bi ci i bi ci i bi ci model, in particular, include: (i) Hilbert spaces (implicitly) involved in their analysis are actually fixed, though in their which is precisely the fermionic state created by a black hole, according to the Hawking effect. Alternatively, we could final state |Ψ1 some part is dropped; (ii) the model has a sim- ple (tensor) product structure (no mixing of different modes interpret (1) in terms of a two-term (qualitative) approx- “i”); and most noteworthy, (iii) unitarity of their model is imation of the bosonic Hawking state (see, e.g., the sec- Ω shown in the most direct way, i.e., a corresponding unitary ond sentence after Eq. (2.2) in [13]). The parameter i is operator is explicitly constructed. a function of Bogoljubov coefficients following from space- time geometry. From more physical point of view, Ωi is a i-mode dependent function of gravitational field embodied a e-mail: [email protected] (corresponding author) (in the Schwarzschild case) by the black hole mass Mbh. URL: http://merlin.phys.uni.lodz.pl/BBroda The above association is equivalent to the assumption that 123 418 Page 2 of 3 Eur. Phys. J. C (2020) 80 :418 | = | ω | | + ω | | the black hole modes ai and the Hawking states (1) can E0 ai bi ci 0 ai cos i 0 bi 0 ci sin i 1 bi 1 ci = | | | | be paired, i.e., N n. It is true (or even obvious), at E1 a b c = 0 a 0 b 1 c ∼ i i i i i i least approximately, i.e. N n. Indeed, the number of |E = | | | 2 2 ai bi ci 0 ai 1 bi 0 ci black hole modes n ∼ SBH = 4π M , where SBH is the bh |E = |0 − sin ω |0 |0 + cos ω |1 |1 Bekenstein–Hawking entropy (we have used Planck units 3 ai bi ci ai i bi ci i bi ci = = = = | = | ω | | + ω | | in which h¯ c G kBoltzmann 1), whereas estimated E4 ai bi ci 1 ai cos i 0 bi 0 ci sin i 1 bi 1 ci number of outgoing radiation modes N ∼ M2 (see [14,15]). | = | | | bh E5 ai bi ci 1 ai 0 b 1 ci Consequently, the initial quantum state of black hole modes | | | | E6 a b c = 1 a 1 b 0 c and Hawking radiation is (cf. [12]) i i i i i i | = | − ω | | + ω | | . E7 ai bi ci 1 ai sin i 0 bi 0 ci cos i 1 bi 1 ci 1 n n (4) |Ψ = | | (Ω ) , 0 Aq q ···qn qi a H i b c 1 2 i i i The second (primed) orthonormal base parameterized by q1,q2,...,qn=0 i=1 i=1 7 θi is EΛ (θi ) a b c , and it is defined as: (2) i i i Λ=0 = | | | E0 a b c 0 ai 0 bi 0 ci where A ··· are the amplitudes for the black hole modes i i i q1q2 qn θ θ a . i | | | i | | | i E = cos 0 a 0 b 1 c − sin 1 a 0 b 0 c 1 ai bi ci 2 i i i 2 i i i Let us note that the state |Ψ0 is a tensor product of = | | | n + 1 (block-)states, which are, in general, entangled states. E2 0 ai 1 bi 0 ci ai bi ci Schematically, we can express it explicitly as |Ψ = |A ⊗ | | | 0 E = 0 a 1 b 1 c | ⊗···⊗| 3 ai bi ci i i i H 1 H n. θ θ The elementary, i.e., for fixed i, process we propose to E = sin i |0 |0 |1 + cos i |1 |0 |0 4 ai bi ci ai bi ci ai bi ci consider is (cf. Eq. (3.3) in [12], and possibly also Eq. (3.3) 2 2 | | | E = 1 a 0 b 1 c in [16]) 5 ai bi ci i i | | | E = 1 a 1 b 0 c 6 ai bi ci i i i |q |H (Ω ) −→ |0 |0 |q . (3) i ai i bi ci ai bi i ci E = |1 |1 |1 . 7 ai bi ci ai bi ci Obviously, the whole qubit transport process is a (“tensor”) (5) product of n processes of the type (3) for each mode i.Evi- The explicitly unitary transformation U (θ ) can now be dently, the process (3) transports information (which could be i i constructed from the two bases (4) and (5) in a standard way previously scrambled by a unitary evolution of the black hole) as encoded in the (base) qubits |qi (qi = 0, 1) from the black 7 hole modes ai to the outgoing radiation modes ci . Moreover, −1 Ui (θi ) = EΛ (θi ) EΛ π θi Ωi , (6) the mode levels inside the black hole (ai and bi ) become grad- ai bi ci a b c Λ= i i i ually emptied (classically, the black hole gradually shrinks), 0 and furthermore, the primarily entangled initial state on the where the auxiliary parameter (“time”) θi satisfies: 0 ≤ θi ≤ LHS of (3) becomes unentangled on the RHS. π (cf. [12]). Evidently, for θi = π, the unitary operator (6) performs the required transformation (3). Thus, finally, expressing the total unitary transformation (π) = n (π) 3 Unitarity as a tensor product U i=1 Ui , we can write U (π) |Ψ0 = |Ψ1, where the final state |Ψ1 assumes the To explicitly show unitarity of the qubit transformation (3), explicit form it is sufficient to construct a corresponding unitary operator n n 1 n performing the required transformation (3). To this end, let us |Ψ = | | ··· | . 1 0 ai 0 bi Aq1q2 qn qi ci first define two auxiliary (parameter-dependent) orthonormal = = , ,..., = = H = H × H × i 1 i 1 q1 q2 qn 0 i 1 bases in the tensor product Hilbert space i ai bi H (7) ci , where we have introduced the three Hilbert spaces (of complex dimension 2, each) for all types of the involved It follows from Eq. (7) that the whole black hole informa- modes: a ∈ H , b ∈ H , c ∈ H (dimC H = 8). The i ai i bi i ci i tion encoded in the amplitudes Aq1q2···qn has been transferred H = n H total Hilbert space is then i=1 i . from the black hole modes ai to the outgoing radiation modes The first (unprimed) orthonormal base parameterized by ci , whereas the black hole modes ai , bi are now in the vacuum 7 ω |EΛ (ω ) i is i ai bi ci Λ=0, and it consists of the following state.